rate-adaptive multiple access for uplink grant-free...

Research ArticleRate-Adaptive Multiple Access for Uplink Grant-FreeTransmission

Neng Ye 1 Aihua Wang1 Xiangming Li 1 Wenjia Liu2

Xiaolin Hou2 and Hanxiao Yu 1

1School of Information and Electronics Beijing Institute of Technology Beijing China2DOCOMO Beijing Communications Laboratories Co Ltd Beijing China

Correspondence should be addressed to Xiangming Li xmlibiteducn

Received 30 November 2017 Accepted 11 January 2018 Published 3 July 2018

Academic Editor Michel Kadoch

Copyright copy 2018 Neng Ye et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Grant-free transmission which simplifies the signaling procedure via uplink instant transmission has been recognized as apromising multiple access protocol to address the massive connectivity and low latency requirements for future machine typecommunications The major drawback of grant-free transmission is that the contaminations among uncoordinated transmissionscan reduce the data throughput and deteriorate the outage performance In this paper we propose a rate-adaptive multiple access(RAMA) scheme to tackle the collision problems caused by the grant-free transmission Different from the conventional grant-free (conv-GF) scheme which transmits a single signal layer RAMA transmits the signals with a multilayered structure wheredifferent layers exhibit unequal protection property At the receiver the intra- and interuser successive interference cancellation(SIC) receiving algorithm is employed to detect multiple data streams In RAMA the users can achieve rate adaptation withoutthe prior knowledge of the channel conditions since the layers with high protection property can be successfully recovered whenthe interference is severe while other layers can take advantage of the channel when the interference is less significant BesidesRAMA also facilitates the SIC receiving since the multiple layers in the transmission signals can provide more opportunities forinterference cancellation To evaluate the system performance we analyze the exact expressions of the throughout and the outageprobability of both conv-GF and RAMA Finally theoretical analysis and simulation results validate that the proposed RAMAscheme can simultaneously achieve higher average throughput and lower outage performance than conv-GF Meanwhile RAMAshows its robustness with large user activation probability where the collisions among users are severe

1 Introduction

Thenext generation wireless communication network (5G) isexpected to support various diversified usage scenarios withdifferent performance requirements Specifically the mostimportant usage scenarios for radio access are categorizedinto three families by Third-Generation Partnership Project(3GPP) enhanced mobile broadband (eMBB) ultrareliableand low latency communication (uRLLC) and massivemachine type communication (mMTC) [1] While eMBBaims at offering higher peak data rates and higher systemthroughput in mobile hotspots the remaining two scenariosfocus on the machine type communications where mMTCis about serving massive devices with small and sporadicpackets and URLLC addresses the applications with very

rigorous requirements on latency and reliability Accordinglythe machine type communications are the extended use casesfor 5G compared with the current 4G system

Currently a scheduling request and grant-based accessmechanism is employed in uplink data transmission How-ever as shown in [2] the grant-based access mechanismexpends tens of milliseconds on the signaling intersectionand the signaling overhead ratio approaches nearly a half withsmall packets (eg packets which contain dozens of bytes)Therefore the latency and overhead requirements cannot besatisfied with grant-based access mechanism

Recently the grant-free access mechanism has attractedmuch attention from both industry and academia [3 4]where the signaling procedure is greatly simplified In uplinkgrant-free access once the users have data in buffer they

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 8978207 21 pageshttpsdoiorg10115520188978207

2 Wireless Communications and Mobile Computing

M1

M2

S1 = 0 1

S2 = 0 1

ENC 1

ENC 2

noise DEC

(a) Grant-free model

M1

M2

Sourcepacket 1

Sourcepacket 2

Recovery ifno collisionor erasure

S1 = 0 1

S2 = 0 1

(b) Network-oriented model (erasure channel model)

M1

M2

ENC 1

ENC 2

DECnoise

(c) MAC model

M1

M2

ENC 1 DEC 1

DEC 2

DEC 12

ENC 2

noise

noise

noise(d) RAC model

Figure 1 Illustrations of MAC grant-free transmission and RACmodels1198721 and1198722 are the messages to be transmitted

instantly transmit their signals on the preconfigured phys-ical resources instead of waiting for grant signaling Thusthe grant-free access mechanism is especially suitable foruRLLC and mMTC since it reduces the latency and signalingoverhead by avoiding complicated signaling intersectionsbetween users and the base station (BS) Nevertheless dueto the decentralized access unexpected collisions may occurin grant-free access and may deteriorate the system perfor-mance As a result the prospects and challenges of grant-free access have motivated the researchers to make furtherinvestigations on the grant-free transceivers to fulfill therequirements of machine type communications in 5G

11 RelatedWork andMotivation of Our Research During thepast several decades there have already been two researchdirections towards the problem of multiuser access in wire-less networks ie network-oriented research and Shannontheory oriented research

In the former research direction packet transition chan-nel model is assumed and the researchers mainly focus onthe protocol design of media access control layer One typicalexample is the slotted ALOHA (SA) protocol proposed fornetwork communications [5] After that the listening andbackoff based SA is adopted in WIFI system [6] In morerecent years a class of graphical based SA schemes havebeen proposed [7] where the random packet transmission isdescribed by a Tanner graph

In the latter research direction multiuser informationtheory is exploited to design capacity achieving multipleaccess technologies In the 1970s the capacity region ofuplink multiple access channel (MAC) is derived known asthe Cover-Wyner region [8] To approach the corner pointsor the hypotenuse of the Cover-Wyner region successiveinterference cancellation (SIC) detection or maximum like-lihood (ML) detection should be deployed at the receiver

However these advanced receivers are not widely acceptedby the industry and the academia due to the concernof high computational complexity until half a century ofMoorersquos law has made the complexity less noticeable In themost recent decade nonorthogonal multiple access (NOMA)technologies which are promising to achieve the entireCover-Wyner region have attracted much attention By mul-tiplexing different usersrsquo signals on the samephysical resourceand employing advanced detector at the receiver NOMApossesses higher spectral efficiency and higher overloadingcompared to the conventional orthogonal multiple access(OMA)

However none of the above researches can fully describethe grant-free transmission as pointed out in [9 10] Wecompare the grant-free access model with the access modelsdealt with in the above two research directions in Figure 1First of all a two-user grant-free access model is presentedin Figure 1(a) where each user is activated according to arandom variable 119878119894 119894 = 1 2 and the received signal is pol-luted by the noise In network-oriented research the grant-free access channel ismodeledwith erasure channelmodel asshown in Figure 1(b) where the discontinuous packet arrivalcan be described but the underlaid physical layer proceduresare ignored ie the noise and the potential near-far effectbetween two users cannot be modeled Similarly the grant-free access cannot be modeled by the MAC model either asshown in Figure 1(c) since the MAC model assumes full-buffer traffic and neglects the random packet arrivals

Efforts have been made to provide a universal descriptionof grant-free access by bridging the gap between thesetwo research directions A two-user random access channel(RAC) model is proposed in [10] as presented in Figure 1(d)where three auxiliary receivers are introduced to representthree different user activation conditions ie user-1 isactivated user-2 is activated and both users are activated

Wireless Communications and Mobile Computing 3

The RAC model roots in the Shannon information theorywhile it also reflects the burst transmissions of grant-freeaccess Since the RAC model involves multiple transmittersandmultiple receivers it is similar to the interference channel(IC) model proposed in multiuser information theory [11]The capacity region of RAC is defined by taking the closureof the unions of achievable rate tuples at user-1 and user-2 with respect to the above receivers which is a classicalinformation theoretic approach Obviously different fromin the MAC model where the NOMA technologies canapproach the entire capacity region of MAC they are notcapacity achieving inRACAs illustrated in [10] rate-splittingtechnique is required to approach the Shannon limit at leastin two-user RAC

However the existing literatures mainly focus on verytheoretical cases and do not provide practical designs toincorporate the rate splitting in grant-free transmission Alsothey usually assume a grant-free access system with up totwo users which is far from the requirements of 5G Andthe advantages are not clarified when the number of users ismore than two Nevertheless they do provide the insightfulhint that is to use rate splitting for grant-free users Inthis paper we aim to design a practical multiple accessscheme to address the unpredictable interference in grant-free access and to fully utilize the underlaid physical channelIn the meantime the performance gain of the proposedscheme over conventional grant-free access (conv-GF) is alsoclarified

12 Contributions

121 Novel Grant-Free Access Scheme To combat the unpre-dictable interference in grant-free access we propose a novelmultiple access scheme namely rate-adaptive multiple access(RAMA) The main idea of RAMA is to incorporate ratesplitting at the transmitter and employ SIC receiving at thereceiver With rate splitting the total transmission power isunequally split into several layers and each layer is assignedwith an independent codeword where the multiple layers ofa single user hold unequal protection property (UEP) Thelayers with UEP can be successfully recovered under differentlevels of interference ie when the collision is high the layerswith high priority can be recovered while when collision islow the layers with low priority can take the advantage ofthe channel Besides SIC receiver is employed to mitigatethe interference among the layers and the users In this wayRAMA can enhance the system throughput while reducingthe outage probability simultaneously Although the grant-free users cannot know the channel conditions in advance theactual achievable transmission rates can still adapt to the real-time conditions of channel Thus RAMA actually achievesthe rate adaptation which is similar to the link adaptationin the grant-based transmission Also introducing morelayers at the transmitter can provide more opportunities forinterference cancellation which makes RAMA more robustthan conv-GF with high user activation probability

122 Clarification on Performance Gain Another contribu-tion of ourwork is thatwe analytically clarify the performance

gain of RAMA over conv-GF The existing literatures haveshown that incorporating rate splitting in grant-free accesswith two users in an information theoretical approach canachieve performance gain However it is hard to illustratethe gain with multiple users since the information theo-retical model of being grant-free is rather complicated withmore than two transmitters In this paper the throughputperformance and the outage probability are analyzed withstatistical methods where users are randomly deployed in thecell We then formulate the exact expressions of the outageprobability and throughput of RAMA and conv-GF Analysisand simulation results reveal that RAMA can achieve highersum throughput as well as lower outage probability whichillustrates that the fairness among users is also improved

123 Constellation Design and Parameter Optimization Tofacilitate the proposed RAMA scheme we propose twoRAMA amenable constellation design methods namelyoverlapping method and bundling method respectively Inparticular the constellations designed by the overlappingmethod are composed of several base constellations wherethe parameter optimization methods are discussed includingthe optimizations of power coefficients and relative rotationangles among the base constellations

The rest of this paper is organized as follows Section 2describes the system and channel model of grant-free trans-mission followed by the description of the proposed RAMAscheme in Section 3 Some implementation issues are alsodiscussed in Section 3 as well Section 4 conducts the theo-retical analysis on conv-GF and RAMA Section 5 providestwo design methods on RAMA amenable constellations Thesimulation results are presented in Section 6 Finally theconclusions and the future works are illustrated in Section 7

2 System Model

Consider a grant-free access network as shown in Figure 2Suppose that a total of 119872 users are randomly distributed inthe cell with the maximum distance 1198771 and the minimumdistance1198772 and theBS is deployed in the center of the cellWemodel packet arrivals at the user side as Poisson distributionand define Poisson arrival rate as 120574 In the network theusers are always in inactive mode to save energy if theirbuffer is empty Once there are packets in the buffer theusers transfer to active mode and instantly transmit the datapackets without grant signaling from the BS Without lossof generality we suppose that the BS has full knowledge ofuser activation information eg via user-specific preambleor simplified random access procedure [10 12] while theusers have no knowledge about other activated users Notethat in this paper we focus on the one-shot grant-freetransmission [13] and do not consider the retransmission orhybrid automatic repeat request (HARQ)

At each time index-119905 one physical resource block isdefined for grant-free access Define b119905

119898 isin 0 1119896 as theinformation bit sequence of an active user-119898 (1 le 119898 le119872) with length-119896 at time index-119905 In conv-GF b119905

119898 is firstencoded into a single coded bit sequence c119905119898 isin 0 1119899


Grant-free

Grant-free transmission

transmission

Active UE

Inactive UE

Inactive UE

Active UENo data arrival

No data arrival

Inactive UE

No data arrival

Inactive UE

No data arrival

Figure 2 Grant-free access system model

with coding rate 119903conv119898 and then modulated into a complexsymbol sequence x119905

119898 isin X119899log2(|X|) where X is theconstellation with cardinality |X| Each symbol sequenceoccupies the entire physical resource block Furthermorewe assume Rayleigh block fading channel model where theRayleigh distributed small-scale fading coefficient remains aconstantwithin each block and the fading is independent andidentically distributed (iid) among different blocks or usersTherefore the channel coefficient between the BS and user-119898at time index-119905 is given by ℎ119905

119898 = 119892119905119898radic119889120572

119898 [14] where 119892119905119898 is

Rayleigh fading coefficient 119889119898 is the distance between the BSand the user-119898 and 120572 is the decay exponent Without loss ofgenerality we assume 120572 = 2 [15] and assume the transmissionpower of each user is 119875 Therefore the received signal at theBS at time index-119905 is formulated as follows

y119905 = 119872sum119898=1

119868119905119898ℎ119905119898radic119875x119905

119898 + 119899119905 119868119905119898 = 1 active

0 inactive (1)

where 119868119905119898 indicates the user activation and 119899119905 is the additivewhite Gaussian noise with zero mean and variance 1205902

At the receiver advanced multiuser detector (MUD) isusually employed to mitigate the mutual interference amongthe users and to distinguish different usersrsquo data streams Forsimplification we define 119879119905 as the normalized throughput ofthe network which equals the number of successfully trans-mitted packets at time index-119905 and the average normalizedthroughput is given by 119879 = 119864119905119879119905 The outage is defined asthe event that a packet is not successfully decoded in giventime index

In this paper we assume that the BS deploys the SICreceiver which is a typicalMUD inNOMA [4] to accomplisha good tradeoff between the detection accuracy and thecomputational complexity The main idea of SIC is to firstlyrecover and cancel the data streams with high priority whileregarding the other signals as noise and then take advantagesof the residual signals In the conventional grant-basedNOMA the users are scheduled by the BS to deliberately

and cautiously multiplex together to ensure low detectionerror probability However as one may expect the randomsuperposition of signals in grant-free data transmission maynot facilitate SIC receiving Thereforemore elaborate designsshould be made at the transmitters to enhance the totalthroughput and reduce the outage probability of grant-freeaccess system

3 The Proposed Rate-AdaptiveMultiple Access

In grant-based access each user transmits a codeword in aslot with a certain coding rate derived by channel estimationHowever in grant-free access the users cannot anticipatethe real-time traffic load and may experience unexpectedinterference from other active users We illustrate the achiev-able data rates versus the interference with different grant-free access schemes in Figure 3 Conv-GF adopts the sametransmission strategy as in grant-based access which maylead to performance loss compared to the theoretical limit asshownby the red arrows in Figure 3(a)When the interferenceis lower than the threshold the user cannot fully utilize thepotential of channel and when the interference is higher thanthe threshold the data cannot even be successfully recoveredOne may expect an ideal grant-free access scheme wherethe achievable data rate at receiver can automatically adaptto the interference and thus follow the theoretical limits asshown in Figure 3(b) However this is not realistic In thissection we propose a rate-adaptive multiple access (RAMA)scheme for grant-free data transmission which is based onthe rate-splitting technique and can be regarded as the anapproximation to the ideal grant-free access as illustratedin Figure 3(c) With this aim we firstly introduce the rate-splitting technique before presenting the proposed RAMAscheme

31 Rate Splitting Rate splitting (RS) was originally intro-duced in the multiuser information theory as a technique


Dat

a rat

eTheoretical limit

Achievabledata rate

ThresholdInterference

(a) Conv-GF

Dat

a rat

e

Theoretical limitAchievabledata rate

Interference(b) Ideal grant-free access

Dat

a rat

e


Interference(c) RAMA

Figure 3 Achievable data rate versus interference with different grant-free access schemes

Informationbits

Step 1 Step 2Step 3

Data layer 1

Data level l

1

l

Option 1

Option 2

－＄1

－＄2

Level Mappingmultiple coded bit streams

to a single stream

Level Mappingmultiple modulation

streams to a single stream

High orderModulation

LmData level

middot middot middot


middot middot middottm t

ml

L

tml

t

t

－＄L

(a) Transmitter

Inter-user SIC

Inter-userSIC

Intra-user SIC

priority

Signal ofweak user

Signal ofstrong user

MUD

Demapping

Signalcancel

Signalcancel

Signalrebuild

Signalrebuild

decodedbits

decodedbits

decodedbits high

low

＄1

＄l


t

tmlt

ml

＄L

(b) Receiver

Figure 4 RAMA transmitter and receiver structure

to prove the capacity bounds of broadcasting channel (BC)multiple access channel (MAC) and interference channel(IC) [16] The core idea of RS is to split the original messageinto two or several independent layers and transmit themsimultaneously During these years RS has attracted theattention of researchers for its potentials to reach every pointin MAC [17] to enhance the fairness among the users in the

network [18] and to promote the security in MIMO network[19] etc

32 RAMA for Grant-Free Transmission The proposedRAMA scheme is demonstrated in Figure 4 where RStechnique and SIC are adopted at the transmitters and thereceiver respectively When each user has data in buffer it


Require the received signal y119905Ensure the estimated information bits(1) Transverse all possible SIC orders and conduct SIC receiving for each SIC order(2) Find the optimal SIC order which achieves the largest throughput and output the estimated information bits

Algorithm 1 Optimal SIC-Based Detection Algorithm

instantly transmits signals according to the RAMA schemeas shown in Figure 4(a) with three steps

Step 1 (data reorganization) At the active user-119898 informa-tion bit sequence b119905

119898 isin 0 1119899 is partitioned and reorganizedby a bijection B

B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)

where 119871119898 is the number of layers and 1205731199051198982 (1 le 119897 le 119871119898) isthe information bit sequence for 119897th layer We assign differentpriority levels to data layers where layers with higher prioritywill experience greater protection

Step 2 (single-layer channel coding) Each subsequence 120573119905119898119897

is encoded with a channel encoder ENC119897 with rate 119903RAMA119898119897

ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)

where 119888119905119898119897 is the coded bit sequence Generally high prioritylayers are encoded with low coding rate

Step 3 (layer-aggregation) Two options can be used toaggregate different layers into one symbol sequence

In option 1 each coding layer 119888119905119898119897 is independentlymodulated with modulator MOD119897

MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)

where x119905119898119897 is the modulation symbol sequence with each ele-

ment drawn from the constellation X119898119897 All layers x119905119898119897 (1 le119897 le 119871119898) are then superimposed together to get the composite

constellation symbol sequence x119905119898 with certain power coeffi-

cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898

] and phase rotation angle120599119905119898 = [120599119905

1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)

andX(120582119905119898 120599119905119898)is the composite constellation defined by120582119905119898 and

120599119905119898 The layers with higher priority are assigned with largerpower coefficients

In option 2 multiple coding layers are firstly mappedto a single bit stream and then modulated with high-order constellation similar to the coded modulation as(c1199051198981 c1199051198982 c119905119898119871119898

) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The

mapping ensures that the coded bits of higher priority layershold larger minimum Euclidean distances

At the receiver multiuser detection algorithm should beemployed to distinguish different usersrsquo signals since multi-ple users may collide due to the uncoordinated transmissionWe show an optimal SIC-based detection algorithm in Algo-rithm 1 which requires traversing all possible SIC orders Toreduce the computational complexity we propose a simplifieddetection algorithm as demonstrated in Algorithm 2 Notethat by employing Algorithm 2 we can simplify the analysisof outage performance in Section 5

As shown in Figure 3(c) the advantage of RAMA can beintuitively illustrated as follows In RAMA the data of eachuser is transmitted with multiple signal layers where all thelayers have different reliability As for a certain user when theexternal interference is significant only the signal layer withhigher reliability can be solved Otherwise when the externalinterference is not significant the signal layers with lowerreliability can also be successfully recovered Besides thelayered structure can also mitigate the mutual interferencesince the recovered signal layers can be reconstructed andcancelled via SIC receiving As a result even if the grant-freeuser cannot foresee the channel occupancy each userrsquos actualtransmission rate can still adapt to the real-time conditions ofchannel

33 Implementation Issues

331 Priority Setting As mentioned above the multiplelayers in the transmission signals of RAMA exhibit UEPand the data with disparate priority shall be mapped tocorresponding layers Therefore the order of the importanceof data sets shall be decided in practice The data setscan be randomly assigned with different priority Moreoverin some usage scenarios different data sets naturally havedifferent levels of importance For example in mMTC datasets may contain user identity and application data Thedata set containing the user identity is regarded as havinghigh priority Once the BS knows the user identities theBS may schedule these users with grant-based transmissionto mitigate the collision [20] Another example happenswhen grant-free uplink transmission collides with the uplinkcontrol information (UCI) on the same resources [21] In thiscase the user may piggyback the UCI report into grant-freedata transmission and the UCI report and grant-free datahave different priority eg if the data is for URLLC and theUCI is for eMBB the former has higher priority

332 Frame Structure The proposed RAMA scheme canbe incorporated into existing frame structure designed forgrant-free transmission [22] However the transmission


Require the received signal y119905 the number of active user119872ac maximum SIC iteration number 119878max maximum layer number119871max and flag 119891Ensure the estimated information bits

Set 119891 = 1(2) while 119891 = 1 amp 1 le 119904 le 119878max do

Set 119891 = 0(4) for 1 le 119897 le 119871max do

for 1 le 119898 le 119872ac do(6) Step 1 Detect the 119897th signal layer of the user with 119898th strongest channel gain while regarding interference as noise

if this signal layer is successfully recovered then(8) Step 2 Output the estimated information bits of this signal layer

Step 3 Reconstruct and cancel this signal layer from y119905(10) Step 4 Set flag 119891 = 1

end if(12) end for

end for(14)end while

Algorithm 2 The Proposed SIC-Based Detection Algorithm

block sizes (TBSs) defined for LTE may not satisfy the needof RAMA since RAMA contains more than one data block ineach transmission and some data blocks may only have muchless amount of bits Thus more TBSs should be defined forRAMA

333 Retransmission Due to the UEP of RAMA somesignal layers may not have enough signal to interference andnoise ratio (SINR) to be recovered and retransmission canbe employed to make use of the received signals For theretransmission either grant-based or grant-free transmis-sion is available depending on specific reliability or latencyrequirements

4 Performance Analysis of Conv-GFand RAMA

In this section we analyze and compare the outage andthroughput performance of both conv-GF and RAMA andshow the advantages of RAMA

41 Outage Performance Analysis of Grant-Free Access Theperformance of conv-GF and RAMA is analyzed in incre-mental steps First of all we study the channel statistics inLemma 1

Lemma 1 For each active user the probability density function(PDF) of channel gains at time index-119905 ie |ℎ119905|2 is given by

119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)

Empirical distributionAnalytical distribution

0

500

1000

1500

2000

2500

3000

PDF

10minus4 10minus3 10minus2 10minus1 10010minus5

|ℎ2|

Figure 5 Comparison between empirical and analytical distribu-tions of |ℎ119905|2 with 1198771 = 100 1198772 = 10 and 108 samples

and the cumulative density function (CDF) of |ℎ119905119898|2 is

119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|

2 (119909) 119889119909= 1 + 2119890minus(11987721119911)2 minus 2119890minus(11987722119911)2119911 (1198772

1 minus 11987722) (7)

Proof Please refer to Appendix A

The empirical and analytical distributions of |ℎ119905|2 arecompared in Figure 5 which shows a perfect match In thefollowing we omit the index 119905 for simplicity


Denote the event that 119872ac users become active at timeindex-119905 as119870119872ac

and its probability can be given by

119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)

Further define 119866|ℎ1|2|ℎ119872ac |

2 as the event where thechannel gains of the 119872ac active users form the set 119867 =|ℎ1|2 |ℎ119872ac

|2 at time index-119905 Note that 119866|ℎ119894|2|ℎ119895|

2

and 119866|ℎ119895|2|ℎ119894|

2 are exactly the same event

Proposition 2 The PDF of 119866|ℎ1|2|ℎ119872119886119888 |

2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)

Proof For an active user the PDF of |ℎ119898|2 is 119891|ℎ|2(|ℎ119898|2)Since the channel coefficients of different users are inde-pendent the joint probability density of channel gain vec-tor (|ℎ1|2 |ℎ2|2 |ℎ119898|2) is prod119872ac

119898=1119891|ℎ|2(|ℎ119898|2) We note thatsweeping the elements in 119867 does not make it a distinctevent and there are a total number of 119872ac permutations of(|ℎ1|2 |ℎ2|2 |ℎ119898|2) which corresponds to the same event119866|ℎ1|

2 |ℎ119872ac |2 Thus the PDF of 119866|ℎ1|

2|ℎ119872ac |2 is given by

multiplexing 119872ac with prod119872ac119898=1119891|ℎ|2(|ℎ119898|2)

Up to now the channel gains of119872ac users are unorderedand thus analysis with SIC receiver is hard However sincepermuting the elements in 119867 does not change the set itselfwe can always assume that the set |ℎ1|2 |ℎ119872ac

|2 is sortedwith |ℎ119894| lt |ℎ119895| if 119894 gt 119895 The normalization condition of 119875119866

holds as follows

int1199111gesdotsdotsdotge119911119872acge0

119875 (1198661199111119911119872ac ) 1198891199111 sdot sdot sdot 119911119872ac

= 1 (10)

Few literatures have considered the outage probability ofuplink NOMA The work [14] studied the outage probabilityof NOMA where the outage events of the users in the uplinkNOMA system are considered to be mutually independentand the user which is successfully recovered is consideredas having no correlation with the remaining users Howeverthis argument is incomplete since the outage event of theprevious users may indicate the channel conditions of theremaining users In the following example we show theargument in [14] is incomplete

Example 1 Consider a special two-user uplink NOMA sys-tem with unit transmission power and unit-variance Gaus-sian noise The ordered channel coefficients namely ℎ1 andℎ2 (ℎ1 gt ℎ2) may choose values from the set 2 3 withequal probability Define the outage event of user-119894 as 119864119894119894 = 1 2 and 119864119888

119894 as the complementary set of 119864119894 Assume thatthe target data rates of both users are 1199031 = log2(1 + 1) = 1and 1199032 = log2(1 + 3) = 2 respectively Then we find thatevent 119864119888

1 happens only when ℎ1 = 3 and ℎ2 = 2 and in thiscase 1198642 must happen Otherwise when 1198641 happens we haveℎ1 = 3 and ℎ2 = 3or ℎ1 = 2 and ℎ2 = 2 where 1198642 happens

with half probability As a result 1198641 is correlated to 1198642 Inthis example we see that the outage events of previous usersactually constrain the probability spaces of the channel gainsof the rest of the users and thus influence the outage events

As illustrated in Example 1 to get the outage probabilityof 119898th user it is not appropriate to decompose the outageevent into several independent events Instead we directlydeal with the outage event of the active users by applying highdimensional integration in the following derivations Withthis aim we define the following outage events to analyze theoutage probability of conv-GF

Without loss of generality we assume that all users adoptthe same transmission rate ie 119903conv119898 = 119903conv forall119898 [22]We use119864conv

119898119872acto represent the outage event where the signals of 1st to(119898 minus 1)th users are successfully recovered and the signals of119898th to119872acth users cannot be recovered In the following we

assume capacity achieving channel coding and modulationif not specified Thus 119864conv

119898119872acis given by

119864conv119898119872ac

≜ 119903conv119895 ge 119903conv119895 1 le 119895 lt 119898 119903conv119898 lt 119903conv119898 (11)

where 119903conv119895 = log2(1 + SINRconv119895 ) and SINRconv

119895 is thereceived SINR of user-119898 According to (11) we readily havethe following proposition

Proposition 3 The conditional probability of event 119864119888119900119899V119898119872119886119888

given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)

where 120601 = 2119903119888119900119899V minus 1 1 ge 119894 ge 119872119886119888 and it is given by

C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)

Averaging (12) over the entire probability space of1198661199051199111119911119872119886119888

we have

119875 (119864119888119900119899V119898 ) = int

1199111gesdotsdotsdotge119911119872119886119888ge0119875 (119864119888119900119899V

119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)

and the exact expression of 119875(119864119888119900119899V119898119872119886119888

) is given by (15)

Due to the noncontinuous max operations in the integralregions it is generally difficult to integrate (15) with eithernumerical intergeneration or approximation Therefore theexact expressions of (15) which do not contain the max


operations should be derived When 120601 lt 1 the exactexpressions of 119875(119864119888119900119899V

119898119872ac) can be recursively derived as shown

in Appendix B which does not involve the max operationsWhen 120601 ge 1 the exact expression of 119875(119864119888119900119899V

119898119872ac) is given by

(16) Without loss of generality we assume 120601 ge 1 in thefollowing analysis

119875 (119864119888119900119899V119898119872ac

) = 119872ac int+infin

0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac

119891|ℎ|2 (119911119872acminus1) sdot sdot sdot int+infin

119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))

119891|ℎ|2 (119911119898minus1)sdot sdot sdot int+infin

1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))

119891|ℎ|2 (1199111) 1198891199111 sdot sdot sdot 119911119872ac

(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ|2 (1199111) 1198891199111 sdot sdot sdot 119889119911119898 sdot sdot sdot 119911119872ac

(16)

However it is still nontrivial to derive a general andclosed-form expression of (16) However according to therequirements in [5G traffic model] the average number ofnew packets in each time index is at the level of 100 where2 is a typical value Therefore in Appendix C we derivethe compact outage expressions of the outage probabilitiesfor some special cases where active user number is smallerthan or equal to 3 ie 119872ac le 3 which may constitutethe mainstreams of grant-free transmission in the practiceDefine the outage event of an active user with conv-GF as119864RAMA Nowwe are ready to give the expressions of the outageand throughput performance of conv-GF

Theorem 4 The average outage probability and the through-put of conv-GF are given by

119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively

Proof When119872ac users are active in a time index the averageamount of the outage users is sum119872ac

119898=1 119875(119864conv119898119872ac

)(119872ac minus 119898 + 1)Averaging the above value over (8) we get (17) Similarly119879RAMA is given bymultiplexing the transmission rate of userswith the successfully recovered users as derived in (18)

With the similar approach the outage probability ofRAMA can also be derived as follows

CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)

42 Outage Performance Analysis of RAMA In this subsec-tion we analyze the outage performance of RAMA Withoutloss of generality we assume that each user splits its signalinto two layers with RAMA since introducing more layersmay lead to severe error propagation [23] Besides weassume all users adopt the same transmission procedure withthe same coefficients and denote 119903RAMA

1 and 119903RAMA2 as the

transmission rate of the layer-1 and layer-2 at each userrespectively The power splitting ratio is defined as 120572 for eachuser ie the transmission power of layer-1 and layer-2 is 120572119875and (1 minus 120572)119875 respectively

Similar to conv-GF we denote 119864RAMA11989811198982119872ac

as the outageevent where the 1st to (1198981 minus1)th userrsquos first layers and the 1stto (1198982 minus 1)th userrsquos second layers are successfully recoveredrespectively while the rest of the layers cannot be recoveredFurthermore we assume that layer-1 exhibits higher pro-tection than layer-2 ie layer-1 can always be successfullydetected once layer-2 can be successfully detectedTherefore119864RAMA

11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)

where 119903RAMA119895119897 = log2(1 + SINRRAMA

119895119897 ) and SINRRAMA119895119897 is the

received SINR of the 119897th layer of the 119895th user Due to the factthat the 1st layer exhibits higher protection than the 2nd layerwe always have 1198981 ge 1198982 Besides we define the outage eventof active users namely 119864RAMA

119898 which happens when at least1 layer of the 1st to (119898 minus 1)th users is recovered and none ofthe layers of119898th to119872acth users is successfully decoded and119864RAMA

119898 can be readily defined as

119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)

As mentioned before Algorithm 1 is the optimal SIC-based multiuser detection algorithm for RAMA Howeversince Algorithm 1 involves a traversing operation which doesnot have an exact mathematical expression we study theperformance of RAMAby assumingAlgorithm2 Before thatwe first show the optimality of the proposed Algorithm 2 byLemma 5

Lemma 5 Assume that all users split and encode their signalswith the same power coefficients and the same transmission

rates respectively the proposed Algorithm 1 is the optimal SIC-basedmultiuser detection algorithm of RAMA ie Algorithm 1achieves the same outage and throughput performance asAlgorithm 2

Proof First of all we note that Algorithm 1 traverses allpossible successive cancellation orders and therefore it is theoptimal multiuser detection algorithm based on SIC Nextwe show the optimality of Algorithm 2 by contradictionAssume that the 119897th layer of the 119898th user happens to bedecoded by Algorithm 1 but not by Algorithm 2 According tothe assumptions of this paper the 1st to 119897th layers of the 1st to(119898minus1)th users and the 1st to (119897minus1)th layers of the119898th user canbe successfully recovered by both Algorithms 1 and 2 Aftercancelling the aforementioned layers the 119897th layer of 119898thusers is the most reliable layer among the remaining signallayers Therefore Algorithm 1 will recover this layer whileregarding other layers as noise according to the assumptionHowever Algorithm 2 can also decode this layer just asAlgorithm 1 which contradicts the assumption After allAlgorithm 2 is optimal

Tomodel the effect of SIC receiving we define119864RAMA(119904)11989811198982119872ac

as the outage event that after 119904th iteration in Algorithm 21198981th userrsquos first layer and1198982th userrsquos second layer cannot besuccessfully decoded with the 1st to (1198981 minus 1)th userrsquos firstlayers and the 1st to (1198982 minus 1)th userrsquos second layers when119872ac users are active In this case1198981 ge 1198982

Proposition 6 The conditional probability of event119864119877119860119872119860111989811198982119872119886119888

given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888

is a region given by (19) where 120593119894 = 2119903119877119860119872119860119894 minus1 1 le 119894 le 2Similarly 119875(119864119877119860119872119860(2)

11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888

is a region given by (20)


Using mathematical induction the general expression of119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888

is a region given by

C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888

) is given by averaging (25) over (9) asfollows

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0

int1199111gtsdotsdotsdotgt119911119897gt0

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )

sdot 119875 (1198661199111119911119872119886119888 ) 1198891199111 sdot sdot sdot 119911119872119886119888

(27)

We define the outage event of an active user ie 119864119877119860119872119860to be event where none of its signal layers are successfullydecoded by the BS The outage performance of RAMA isshown in Theorem 7

Theorem 7 The exact expressions of the average outageprobability and the average throughput of RAMA after 119904th SICare given by (28) and (29) respectively

119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)

Proof The proof is similar to the proof of Theorem 4

As mentioned before RAMA introduces multiple layersat the transmitters therefore the power and transmissionrates allocation among different layers can act as an additionaldegree of freedom to match the statistical characteristics ofinterference and to further enhance the network throughputof RAMA The optimal power and transmission rates ofRAMA are given by

max120572119903RAMA1 119903RAMA

2

119879RAMAst 119903RAMA

1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)

Note that conv-GF only considers a single signal layerand thus it is not as adjustable as RAMA

43 Comparisons In this subsection we compare the outageperformance achieved by conv-GF and RAMA First of allwe note that by setting 120572 = 1 1205931 = 120601 and 1205932 = 0 in (30)the outage and throughput performance of RAMAare exactlythe same as conv-GF Therefore it is straightforward thatRAMA outperforms conv-GF with sophisticatedly designedparameters

To visually illustrate the advantage of RAMA withrespect to the outage performance we compare the com-plementary outage regions of conv-GF and RAMA ie(⋃1le119898le119872ac

Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with


119872ac = 2 and 3 as shown in Figures 6 and 7 respectivelyIn the simulation conv-GF and RAMA are set with thesame transmission rates which takes value in 08 1 12Thetransmission rates and the power coefficients of RAMA areoptimized according to (30) In Figure 6 the black cyclesrepresent all possible realizations of 119867 with 119872ac = 2and the red crosses represent the realizations of 119867 suchthat the signals of the two users are successfully recoveredWe found that with the increase of total transmissionrate ie 120601 the outage performance of conv-GF becomesworse while RAMA can achieve successful transmissionswith almost all channel realizations In Figure 7 the blackdots represent all possible realizations of 119867 with 119872ac =3 and the red cycled points represent the realizations of119867 such that the three users can be successfully recoveredThe advantage of RAMA over conv-GF with respect to theoutage performance with 119872ac = 3 is more significant than119872ac = 2 which validate the robustness of RAMA We alsoobserve that when the interference among users is severeie the areas selected by cycles in Figures 6(b) 7(b) 6(c)and 7(c) conv-GF cannot ensure successful transmissionswhile RAMA still achieves high throughput To sum upRAMA achieves high data rate in low interference regionwhile the robustness can also be assured in high interferenceregion

5 RAMA Amenable Constellations

In the above analysis we have considered the ideal situationwhere Gaussian-distributed continuous alphabet is assumedat the transmitter However only finite alphabets can bedeployed in practice Therefore we focus on the designand optimization of RAMA amenable constellations in thissection

The RAMA amenable constellations are composed ofseveral subconstellations where each subconstellation corre-sponds to a signal layer at transmitter We call the equivalentchannel experienced by each signal layer as a subchannelas mentioned in Section 32 To facilitate RAMA our aimis to construct the subchannels such that they have UEPCorresponding to the two options in Step 3 of the RAMAscheme in Section 32 we propose two methods to design

RAMA amenable constellations as well as the subchannels inthe following

51 Overlapping Method Corresponding to option 1 in Step3 of RAMA we propose the overlapping method wherethe composite constellation is constructed by overlappingseveral base constellations ie X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |) as

shown in (6) In Figure 8 we show two examples of theRAMA amenable constellations where BPSK and QPSK areemployed as the basic building blocks When the powercoefficients ie 1205821 and 1205822 of different base constellationsare different the bits in the composite constellation normallyhave different constellation constrained capacity Thereforewe regard each bit (or several bits) as a subchannel whereone signal layer can be transmitted as shown in Figures8(a) and 8(b) respectively We note that with the proposedconstellations even all signal layers are encoded with thesame coding rate (to save the hardware resources) and theystill exhibit UEP property which facilitates the SIC receivingof RAMA

Furthermore the power coefficients ie120582119905119898 and rotationangles ie 120599119905119898 shall be optimized to adapt to RAMA Inthis paper we sequentially optimize these coefficients Theoptimization of 120582119905119898 includes the following three steps

Step 1 Define different reliability levels for different sub-channels eg different BLER targets for the codewordstransmitted in different subchannels

Step 2 Map the reliability levels to the capacity of differentsubchannels

Step 3 Adjust 120582119905119898 to meet the capacity requirements ofdifferent subchannels

With the fixed 120582119905119898 120599119905119898 should be optimized to achieve

optimal constellation constrained capacity When the over-lapped layers at transmitter are set to 2 optimal rotationangles can be derived by

max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)

log( sum1199092isinX

119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)

52 Bundling Method When high-order constellations areapplied in RAMA as illustrated in option 2 of Step 3 inSection 32 we propose the bundling method to constructsubchannels where different numbers of bits are bundledfor different subchannels We show an example in Figure 9where a 16-QAM constellation is employed as the compositeconstellation of RAMA We use the first bit as subchannel-1

and the remaining three bits as subchannel-2 Suppose thathigh priority data stream and low priority data stream aretransmitted in subchannel-1 and subchannel-2 respectivelyTo ensure that high priority data stream is better protectedlow-rate channel coding should be adopted After cancellingthe signal transmitted in subchannel-1 the residual constel-lation is shown at the right-hand side of Figure 9


HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074

Figure 6 Comparison on the complementary outage regions of conv-GF and RAMA with119872ac = 2 The SNR of each user is 10 dB


1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



Subchannel-1Subchannel-2

00 2

1

11

1001

= (0∘ 90∘)

(a) 2 bits per symbol

Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)

(b) 3 bits per symbol

Figure 8 Illustrations of RAMA amenable constellations which are constructed by the overlapping method 1205821 and 1205822 are the powercoefficients for each of the base constellations and 120599 is the rotation angle vector

10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001

000000Subchannel-1 Subchannel-2 Subchannel-2

Cancel thesignal insubchannel-1

Figure 9 An illustration of RAMA amenable constellation with 4 bits per symbol which are constructed by the bundling method

We note that the distances between the constellationpoints in the composite constellation cannot be arbitrarilyadjusted Therefore the UEP property is mainly offered byusing different coding rates at different signal layers whichmay lead to heavy burden on hardware resources This factmakes the bundling method less flexible compared with theoverlapping method

6 Simulation Results

In this section we evaluate the performance of the proposedRAMA scheme First of all we assume that Gaussian-distributed continuous alphabet and ideal SIC receiving areapplied at the transmitter and the receiver respectively Basedon these settings and the theoretical analysis in Section 4we compare the outage and throughput performance ofRAMA with that of the conv-GF [22] Next we validatethe advantages of RAMA in practical situations where theRAMA amenable constellations designed in Section 5 andrealistic MMSE-SIC receiver are assumed

61 Ideal Settings As mentioned before we assume Gaus-sian-distributed continuous alphabet and ideal SIC receivingFigures 10 and 11 compare the analytical and simulationresults of conv-GF and RAMA with 119872 = 100 1198771 = 100and 1198772 = 10 The average SNR of each user is assumed as

10 dB The analytical results of conv-GF and RAMA whichare respectively shown by ldquo◻rdquo and ldquo ⃝rdquo is derived by (17)and (28) respectively via Monte-Carlo sampling methodThe upper bounds of the outage performance of conv-GFwhich are shown by dashed lines are derived by integratingthe results in Appendix C into (17) and setting 119875(119864conv

1119872ac) =1 119872ac gt 3 The upper bound is more tight when 120601

becomes larger since the outage probability raises sharplywhen 119872ac gt 3 The simulation results show that RAMA cansimultaneously achieve higher throughput and lower outageprobability than conv-GF

In the simulation we consider a special case namelyldquosingle-layerrdquo where only layer-1 of each user is transmittedin RAMA and layer-2 is assumed as noise Figures 10(b) and11(b) show that the outage performance can be enhancedcomparedwith conv-GF however the gains aremuch smallerthan RAMA These results validate that the outage perfor-mance gain of RAMA comes from two aspects first of alllayer-1 of each user has high protection property due to itslow coding rate and high power ratio secondly the layeredstructure facilitates the interference cancellation and furtherenhances the outage performance since the users with smallchannel gains can benefit from the cancellation of highlyprotected layers of the users with large channel gains andthis is different from conv-GF where the signals of the usersshould be entirely recovered before cancellation


Conv-GF simulationRAMA simulation

03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)

1 15 2 25 3 35 405Average number of active users

(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y


RAMA analyticalConv-GF upper boundSingle layer

Conv-GF simulationRAMA simulationConv-GF analytical

(b) Outage performance

Figure 10 Comparisons of the throughput and outage performance between conv-GF and RAMA with 120601 = 12 119903RAMA1 = 06 119903RAMA

2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 1 and120572 = 07562 Realistic Settings In this subsection we conduct alink level simulation of conv-GF and RAMA with realisticsettings We assume a single-cell OFDM-based uplink systemwith single antenna at both the BS and the user The numberof users is 100 and the activation probability varies from0002 to 006 The average received SNR of each user isassumed to take value from [4 20] dB uniformly [25] Wealso assume that the small-scale channel coefficients followsRayleigh fading and the correlation coefficients among the

small-scale channels of the symbols within a transmissionblock are set as 02 05 and 08 where the larger thecorrelation coefficient the flatter the wireless channel

For conv-GF we apply 13 rate Turbo coding and QPSKmodulation while for RAMA we assume 119871119898 = 2 with 13rate Turbo coding for both layers and apply the constellationprovided in Figure 8(a) with 12058211205822 = 2 4 6 The length ofinformation bits is assumed as 1024 At the receiver we applyAlgorithm 2 to separate different users signals Specifically


Table 1 Simulation parameters

Parameters Values or AssumptionsUser number 100Activation probability 0002 0002 006Waveform OFDMAverage received SNR Uniformly distributed in [4 20] dBSmall scale channel model Rayleigh fading with correlation coefficients equals 02 05 and 08Transmission mode TM 1 (SISO)Length of information bits 1024Channel coding 13 rate TurboModulation of conv-GF QPSKModulation of RAMA Use Figure 8(a) with 12058211205822 = 2 4 6Channel estimation IdealReceiver MMSE-SIC

minimummean square error (MMSE) detection is employedin Step 1 of Algorithm 2 which is given by [26]

x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)

where x119898119897 is the estimated signal of 119898th user Note thatanother advanced demodulation technique eg messagepassing algorithm (MPA) is not precluded in RAMA Thedetailed simulation settings can be found in Table 1

Figures 12 and 13 compare the throughput and the outageperformance of conv-GF and RAMA Generally the systemthroughput enhances with the increase of the channel cor-relation coefficients ie 120574 With elaborately selected powercoefficients of constellation RAMA achieves larger through-put as well as lower outage probability than conv-GF Thisresult also reflects that the fairness among grant-free users isimproved with RAMA Moreover the performance gains ofRAMA are more significant when the underlaying physicalchannel is not flat Besides we also observe that RAMAachieves high robustness when the activation probability goeslarger (eg 120574 = 006) while conv-GF experiences significantdrop in total throughput

7 Conclusion and Future Work

In this paper we have proposed the RAMA scheme for uplinkgrant-free data transmission By employing layered signalstructure at the transmitter and intra- and interuser SIC atthe receiver the proposed RAMA scheme achieves significantthroughput and outage performance gain over conv-GFwhich have been validated by analysis and simulations Theactual transmission data rate can adapt to the actual channelconditions of active users which cannot be foreseen RAMAalso achieves high robustness when the activation probabilityof the users is large Despite all this improvement we discusssome open issues of RAMA that are worth further study inthe following

Joint Design with Spreading-Based NOMA In this paperwe have mainly focused on the grant-free transmissionbased on the power domain NOMA where symbol levelspreading is not included RAMA can also codeploy withother state-of-the-art NOMA schemes where the main ideais to incorporate multiple independent signal layers at thetransmitter

Location-Based Access Although the grant-free users cannotacquire the accurate channel information they may estimatetheir large scale channel coefficients eg via reference signalreceiver power (RSRP) at the downlinkThis side informationmay serve as an important factor based on which the grant-free users choose suitable power and transmission data ratesfor different layers in RAMA

Error Propagation Since multiple layers are introduced inRAMA a natural problem is the error propagation amongdifferent layers during SIC receiving This issue may beaddressed by deploying the joint-detection based receiverwhich also requires further study

Appendix

A Proof of Lemma 1

As assumed the users are uniformly distributed in the cellTherefore the PDF of the distance between the users and theBS is given by

119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)

The PDF of the square of the distance ie 119909 = V2 is

1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)

002 003 004 005 006001Activation probability

Offered trafficconv-GFRAMA 12 = 2

RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6

(a) Correlation coefficient = 02

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

(b) Correlation coefficient = 05



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)

(c) Correlation coefficient = 08

Figure 12 Comparison on the throughput performance between conv-GF and RAMA

The square of the magnitude of the small-scale fadingcoefficient follows the 1205942(V) distribution with two degrees offreedom ie V = 2 and is given by

119891|119892|2 (119909) = 12119890minus1199092 (A3)

Then by integrating (A2) and (A3) we have

119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)

where the last equation is due to int 119890minus119905119905 119889119905 = minus119905119890minus119905 minus 119890minus119905 + 119862


0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y


Figure 13 Comparison on the outage probability between conv-GF and RAMA

The CDF of 119891|ℎ|2(119911) is calculated as follows We note thatthe indefinite integration of 119891|ℎ|2(119911) is given by ℎ (119911) = int119891|ℎ|2 (119911) 119889119911 = 2119890minus(11987721119911)2 minus 2119890minus(11987722119911)2119911 (1198772

1 minus 11987722) (A5)

where ℎ(119911) 997888rarr minus1 119911 997888rarr 0+ The CDF of |ℎ|2 is given as

119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac

) = 119872ac times int+infin

0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac

119891|ℎ119905|2 (119911119872acminus1) times sdot sdot sdot times int+infin

119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))

119891|ℎ119905|2 (119911119898minus1) times sdot sdot sdot


times [[[[[int+infin

1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)

sdot int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)

B Recursive Expression of (15)

We assume that 120601 ge 12 in the following derivation wherethe derivation with 120601 lt 12 follows the same approach

To eliminate the max operations and derive the exactexpressions of (15) we consider two cases ie 1199112 ge120601(sum119872ac

119895=2 +1205902119875) and 1199112 lt 120601(sum119872ac119895=2 +1205902119875) where 1199112 ge(120601(1 minus 120601))(sum119872ac

119895=3 +1205902119875) and 1199112 ge (120601(1 minus 120601))(sum119872ac119895=3 +1205902119875)

respectively Therefore (15) is given by (A7) where 1198681 and 1198682can be further simplified in the following

Observe that 1 gt 120601 ge 12 1198681 can be simplified asderived in (A8) Besides to eliminate the max operationin 1198682 we consider two cases ie 1199113 ge 120601(sum119872ac

119895=3 +1205902119875) and1199112 lt 120601(sum119872ac119895=3 +1205902119875) where 1199113 ge (120601(1 minus 120601))(sum119872ac

119895=4 +1205902119875)and 1199113 ge (120601(1 minus 120601))(sum119872ac

119895=4 +1205902119875) respectively And 1198682 isgiven by (A9) Till now the max operations related to 1199111 and1199112 are completely eliminated By recursively conducting theprocedures between (A7) and (A9) the exact expression of(15) can be derived

C Proof of Proposition 6

Assuming119872ac = 1 119875(119864conv11 ) is given by

119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)

Furthermore if119872ac = 2 119875(119864conv12 ) and 119875(119864conv

22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112

= 2 (F (+infin) minusF (0)) minus 1

(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (+infin) minus 119865|ℎ119905|2 (120601(1199112 + 1205902

119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)

Assume high SNR and 120601 = 1 and (C4) can be calculated asF(119911) = (12)(119865(119911))2

When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be

derived with the similar approaches which is omitted due tothe space limitation

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (NSFC) under Grant 61771051

References

[1] ldquoTR 38913 Study on scenarios and requirements for nextgeneration access technologies 3GPP May 2015 (Release 14)rdquo

[2] K Au L Zhang H Nikopour et al ldquoUplink contention basedSCMA for 5G radio accessrdquo in Proceedings of the 2014 IEEEGlobecom Workshops GC Wkshps 2014 pp 900ndash905 USADecember 2014

[3] C Bockelmann N Pratas H Nikopour et al ldquoMassivemachine-type communications in 5g Physical and MAC-layersolutionsrdquo IEEE Communications Magazine vol 54 no 9 pp59ndash65 2016

[4] Y Yuan Z Yuan G Yu et al ldquoNon-orthogonal transmissiontechnology in LTE evolutionrdquo IEEECommunicationsMagazinevol 54 no 7 pp 68ndash74 2016

[5] Y-J Choi and K G Shin ldquoJoint collision resolution andtransmit-power adjustment for Aloha-type random accessrdquoWireless Communications and Mobile Computing vol 13 no 2pp 184ndash197 2013

[6] C Zhu L Shu T Hara L Wang S Nishio and L T YangldquoA survey on communication and data management issues inmobile sensor networksrdquoWireless Communications and MobileComputing vol 14 no 1 pp 19ndash36 2014

[7] E Paolini G Liva and M Chiani ldquoCoded slotted ALOHAa graph-based method for uncoordinated multiple accessrdquoInstitute of Electrical and Electronics Engineers Transactions onInformation Theory vol 61 no 12 pp 6815ndash6832 2015

[8] A D Wyner ldquoRecent Results in the Shannon Theoryrdquo IEEETransactions on InformationTheory vol 20 no 1 pp 2ndash10 1974

[9] M Medard J Huang A J Goldsmith S P Meyn and TP Coleman ldquoCapacity of Time-Slotted ALOHA PacketizedMultiple-Access Systems Over the AWGN Channelrdquo IEEETransactions onWireless Communications vol 3 no 2 pp 486ndash499 2004

[10] P Minero M Franceschetti and D N C Tse ldquoRandom accessAn information-theoretic perspectiverdquo IEEE Transactions onInformation Theory vol 58 no 2 pp 909ndash930 2012

[11] R H Etkin D N C Tse and H Wang ldquoGaussian interferencechannel capacity to within one bitrdquo IEEE Transactions onInformation Theory vol 54 no 12 pp 5534ndash5562 2008

[12] ldquoZTE R1-1701608 Performance evaluation of nonorthogonalmultiple access in 2-step random access procedure 3GPP 2017rdquo

[13] Y Wu and J Fang Large-Scale Antenna-Assisted Grant-freeNon-Orthogonal Multiple Access via Compressed Sensing 2016httpsarxivorgabs160900452

[14] N Zhang JWangG Kang andY Liu ldquoUplinkNonorthogonalMultiple Access in 5G Systemsrdquo IEEE Communications Lettersvol 20 no 3 pp 458ndash461 2016

[15] H Zheng X Li and N Ye ldquoRandom Subchannel Selection ofStore-Carry and Forward Transmissions in Traffic HotspotsrdquoIEEECommunications Letters vol 21 no 9 pp 2073ndash2076 2017

[16] A El Gamal and Y-H Kim Network information theoryCambridge University Press Cambridge UK 2011

[17] B Rimoldi R Urbanke and B H Rimoldi ldquoA Rate-SplittingApproach to the Gaussian Multiple-Access Channelrdquo IEEETransactions on Information Theory vol 42 no 2 pp 364ndash3751996

[18] H Joudeh and B Clerckx ldquoRate-Splitting for Max-Min FairMultigroup Multicast Beamforming in Overloaded SystemsrdquoIEEE Transactions on Wireless Communications 2017

[19] B Clerckx H Joudeh C Hao M Dai and B Rassouli ldquoRatesplitting for MIMO wireless networks A promising PHY-layerstrategy for LTE evolutionrdquo IEEE Communications Magazinevol 54 no 5 pp 98ndash105 2016

[20] NTT DOCOMO INC R1-1713952 UL data transmission with-out UL grant 3GPP 2017

[21] ldquoIntel R1-1712592 UL data transmission without grant 3GPP2017rdquo

[22] NTT DOCOMO INC R1-167392 Discussion on multiple accessfor UL mMTC 3GPP 2016

[23] A Benjebbour Y Saito Y Kishiyama A Li A Harada andT Nakamura ldquoConcept and practical considerations of non-orthogonal multiple access (NOMA) for future radio accessrdquo inProceedings of the 2013 21st International Symposium on Intel-ligent Signal Processing and Communication Systems ISPACS2013 pp 770ndash774 Japan November 2013

[24] N Ye A Wang X Li W Liu X Hou and H Yu ldquoOnConstellation Rotation of NOMA with SIC Receiverrdquo IEEECommunications Letters pp 1-1

[25] Z Yuan G Yu W Li Y Yuan X Wang and J Xu ldquoMulti-usershared access for internet of thingsrdquo in Proceedings of the 83rdIEEEVehicular Technology Conference VTC Spring 2016 ChinaMay 2016

[26] K Higuchi and A Benjebbour ldquoNon-Orthogonal MultipleAccess (NOMA) with successive interference cancellation forfuture radio accessrdquo IEICE Transactions on Communicationsvol E98B no 3 pp 403ndash414 2015

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M1

M2

S1 = 0 1

S2 = 0 1

ENC 1

ENC 2

noise DEC

(a) Grant-free model

M1

M2

Sourcepacket 1

Sourcepacket 2

Recovery ifno collisionor erasure

S1 = 0 1

S2 = 0 1

(b) Network-oriented model (erasure channel model)

M1

M2

ENC 1

ENC 2

DECnoise

(c) MAC model

M1

M2

ENC 1 DEC 1

DEC 2

DEC 12

ENC 2

noise

noise

noise(d) RAC model

Figure 1 Illustrations of MAC grant-free transmission and RACmodels1198721 and1198722 are the messages to be transmitted

instantly transmit their signals on the preconfigured phys-ical resources instead of waiting for grant signaling Thusthe grant-free access mechanism is especially suitable foruRLLC and mMTC since it reduces the latency and signalingoverhead by avoiding complicated signaling intersectionsbetween users and the base station (BS) Nevertheless dueto the decentralized access unexpected collisions may occurin grant-free access and may deteriorate the system perfor-mance As a result the prospects and challenges of grant-free access have motivated the researchers to make furtherinvestigations on the grant-free transceivers to fulfill therequirements of machine type communications in 5G

11 RelatedWork andMotivation of Our Research During thepast several decades there have already been two researchdirections towards the problem of multiuser access in wire-less networks ie network-oriented research and Shannontheory oriented research

In the former research direction packet transition chan-nel model is assumed and the researchers mainly focus onthe protocol design of media access control layer One typicalexample is the slotted ALOHA (SA) protocol proposed fornetwork communications [5] After that the listening andbackoff based SA is adopted in WIFI system [6] In morerecent years a class of graphical based SA schemes havebeen proposed [7] where the random packet transmission isdescribed by a Tanner graph

In the latter research direction multiuser informationtheory is exploited to design capacity achieving multipleaccess technologies In the 1970s the capacity region ofuplink multiple access channel (MAC) is derived known asthe Cover-Wyner region [8] To approach the corner pointsor the hypotenuse of the Cover-Wyner region successiveinterference cancellation (SIC) detection or maximum like-lihood (ML) detection should be deployed at the receiver

However these advanced receivers are not widely acceptedby the industry and the academia due to the concernof high computational complexity until half a century ofMoorersquos law has made the complexity less noticeable In themost recent decade nonorthogonal multiple access (NOMA)technologies which are promising to achieve the entireCover-Wyner region have attracted much attention By mul-tiplexing different usersrsquo signals on the samephysical resourceand employing advanced detector at the receiver NOMApossesses higher spectral efficiency and higher overloadingcompared to the conventional orthogonal multiple access(OMA)

However none of the above researches can fully describethe grant-free transmission as pointed out in [9 10] Wecompare the grant-free access model with the access modelsdealt with in the above two research directions in Figure 1First of all a two-user grant-free access model is presentedin Figure 1(a) where each user is activated according to arandom variable 119878119894 119894 = 1 2 and the received signal is pol-luted by the noise In network-oriented research the grant-free access channel ismodeledwith erasure channelmodel asshown in Figure 1(b) where the discontinuous packet arrivalcan be described but the underlaid physical layer proceduresare ignored ie the noise and the potential near-far effectbetween two users cannot be modeled Similarly the grant-free access cannot be modeled by the MAC model either asshown in Figure 1(c) since the MAC model assumes full-buffer traffic and neglects the random packet arrivals

Efforts have been made to provide a universal descriptionof grant-free access by bridging the gap between thesetwo research directions A two-user random access channel(RAC) model is proposed in [10] as presented in Figure 1(d)where three auxiliary receivers are introduced to representthree different user activation conditions ie user-1 isactivated user-2 is activated and both users are activated




12 Contributions






2 System Model






Grant-free


transmission

Active UE

Inactive UE

Inactive UE


No data arrival

Inactive UE

No data arrival

Inactive UE

No data arrival




119898 = 119892119905119898radic119889120572

119898 [14] where 119892119905119898 is


y119905 = 119872sum119898=1

119868119905119898ℎ119905119898radic119875x119905

119898 + 119899119905 119868119905119898 = 1 active

0 inactive (1)









Dat

a rat

eTheoretical limit

Achievabledata rate


(a) Conv-GF

Dat

a rat

e



Dat

a rat

e




Informationbits

Step 1 Step 2Step 3

Data layer 1

Data level l

1

l

Option 1

Option 2

－＄1

－＄2


to a single stream




LmData level




ml

L

tml

t

t

－＄L

(a) Transmitter

Inter-user SIC

Inter-userSIC

Intra-user SIC

priority

Signal ofweak user


MUD

Demapping

Signalcancel

Signalcancel

Signalrebuild

Signalrebuild

decodedbits

decodedbits

decodedbits high

low

＄1

＄l


t

tmlt

ml

＄L

(b) Receiver











B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)




ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)




MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)




cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898


1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)




) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The














end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





12 Contributions






2 System Model






Grant-free


transmission

Active UE

Inactive UE

Inactive UE


No data arrival

Inactive UE

No data arrival

Inactive UE

No data arrival




119898 = 119892119905119898radic119889120572

119898 [14] where 119892119905119898 is


y119905 = 119872sum119898=1

119868119905119898ℎ119905119898radic119875x119905

119898 + 119899119905 119868119905119898 = 1 active

0 inactive (1)









Dat

a rat

eTheoretical limit

Achievabledata rate


(a) Conv-GF

Dat

a rat

e



Dat

a rat

e




Informationbits

Step 1 Step 2Step 3

Data layer 1

Data level l

1

l

Option 1

Option 2

－＄1

－＄2


to a single stream




LmData level




ml

L

tml

t

t

－＄L

(a) Transmitter

Inter-user SIC

Inter-userSIC

Intra-user SIC

priority

Signal ofweak user


MUD

Demapping

Signalcancel

Signalcancel

Signalrebuild

Signalrebuild

decodedbits

decodedbits

decodedbits high

low

＄1

＄l


t

tmlt

ml

＄L

(b) Receiver











B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)




ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)




MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)




cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898


1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)




) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The














end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Grant-free


transmission

Active UE

Inactive UE

Inactive UE


No data arrival

Inactive UE

No data arrival

Inactive UE

No data arrival




119898 = 119892119905119898radic119889120572

119898 [14] where 119892119905119898 is


y119905 = 119872sum119898=1

119868119905119898ℎ119905119898radic119875x119905

119898 + 119899119905 119868119905119898 = 1 active

0 inactive (1)









Dat

a rat

eTheoretical limit

Achievabledata rate


(a) Conv-GF

Dat

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e



Dat

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Informationbits

Step 1 Step 2Step 3

Data layer 1

Data level l

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Option 1

Option 2

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to a single stream




LmData level




ml

L

tml

t

t

－＄L

(a) Transmitter

Inter-user SIC

Inter-userSIC

Intra-user SIC

priority

Signal ofweak user


MUD

Demapping

Signalcancel

Signalcancel

Signalrebuild

Signalrebuild

decodedbits

decodedbits

decodedbits high

low

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(b) Receiver











B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)




ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)




MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)




cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898


1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)




) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The














end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Dat

a rat

eTheoretical limit

Achievabledata rate


(a) Conv-GF

Dat

a rat

e



Dat

a rat

e




Informationbits

Step 1 Step 2Step 3

Data layer 1

Data level l

1

l

Option 1

Option 2

－＄1

－＄2


to a single stream




LmData level




ml

L

tml

t

t

－＄L

(a) Transmitter

Inter-user SIC

Inter-userSIC

Intra-user SIC

priority

Signal ofweak user


MUD

Demapping

Signalcancel

Signalcancel

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Signalrebuild

decodedbits

decodedbits

decodedbits high

low

＄1

＄l


t

tmlt

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＄L

(b) Receiver











B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)




ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)




MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)




cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898


1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)




) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The














end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








B b119905119898 997891997888rarr (12057311990511989811205731199051198982 120573119905119898119871119898

) (2)




ENC119897 120573119905119898119897 997891997888rarr c119905119898119897 (3)




MOD119897 c119905119898119897 997891997888rarr x119905119898119897 (4)




cient 120582119905119898 = [1205821199051198981 120582119905

1198982 120582119905119898119871119898


1198981 1205991199051198982 120599119905

119898119871119898] where x119905

119898 is given by

x119905119898 = 119871119898sum

119897=1

120582119905119898119897x

119905119898119897119890119895120599119905119898119897

x119905119898 isin X(120582119905119898120599

119905119898)119899log2(|X(120582119905119898120599119905119898) |)

(5)




) 997891997888rarr x119905119898 where x

119905119898 isin X119899log2(|X|) The














end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









end if(12) end for









119891|ℎ119905|2 (119911) = minus 1(11987721 minus 1198772

2) 1199112((119890minus119877211199112 (1198772

1119911 + 2))minus (119890minus119877221199112 (1198772

2119911 + 2))) (6)


0

500

1000

1500

2000

2500

3000

PDF


|ℎ2|



119865|ℎ119905|2 (119911) = int119911

119909=0119891|ℎ119905119898|


1 minus 11987722) (7)






119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119875 (119870119872ac) = (120582119872)119872ac 119890minus120582119872

119872ac (8)




2

and 119866|ℎ119895|2|ℎ119894|



2 is given by

119875 (119866|ℎ1|2|ℎ119872119886119888 |

2) = 119872119886119888119872119886119888prod119898=1

119891|ℎ|2 (1003816100381610038161003816ℎ11989810038161003816100381610038162) (9)








holds as follows



= 1 (10)











119864conv119898119872ac





given 119866|ℎ1|2|ℎ119872119886119888 |

2 is derived as

119875 (119864119888119900119899V119898119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2) =

1 C119888119900119899V1198981198721198861198880 119900119905ℎ119890119903119908119894119904119890

1 le 119898 le 119872119886119888(12)


C119888119900119899V119898119872119886119888

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) | 10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 119875sum119872119886119888119894=119895+1

1003816100381610038161003816ℎ11989410038161003816100381610038162 119875 + 1205902

ge 120601 1 le 119895 lt 119898 1003816100381610038161003816ℎ11989810038161003816100381610038162 119875sum119872119886119888

119894=119898+11003816100381610038161003816ℎ119894

10038161003816100381610038162 119875 + 1205902lt 120601

(13)


we have

119875 (119864119888119900119899V119898 ) = int


119898 | 1198661199111119911119872119886119888 )

times 119875 (1198661199051199111119911119872119886119888

) 1198891199111 sdot sdot sdot 119911119872119886119888 (14)


) is given by (15)








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




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Propagation






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Advances in

Multimedia








119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)sdot int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))


1199111=max(1199112120601(sum119872ac119895=2 119911119895+1205902119875))


(15)

119875 (119864119888119900119899V119898119872ac


0119891|ℎ|2 (119911119872ac

)sdot int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ|2 (119911119898)sdot int+infin

119911119898minus1=120601(sum119872ac119895=119898 119911119895+1205902119875)


1199111=120601(sum119872ac119895=2 119911119895+1205902119875)


(16)



119875 (119864119888119900119899V)= suminfin

119872119886119888=1 119875 (119870119872119886119888) (sum119872119886119888

119898=1 119875 (119864119888119900119899V119898119872119886119888

) (119872119886119888 minus 119898 + 1))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(17)

119879119877119860119872119860

= infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

119898=1

(119875 (119864119888119900119899V119898119872119886119888

) (119898119903119888119900119899V))) (18)

respectively


119898=1 119875(119864conv119898119872ac



CRAMA(1)11989811198982119872ac

= (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902lt 1205931

1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205932

10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1 le 119894 lt 1198981 1 le 119896 lt 1198982

(19)

CRAMA(2)11989811198982119872ac

= ⋃1le11989811lt11989811le11989821le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872ac

100381610038161003816100381610038162) |C

RAMA(1)1198981111989821119872ac

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872ac119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872ac119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=11989811

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 11989821) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 11989821) 1205932


10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



10038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872ac

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872ac119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 11989811 le 119894 lt 1198981 11989821 le 119896 le 1198982

(20)






11989811198982119872acis given by

119864RAMA11989811198982119872ac

≜ 119903RAMA1198951 ge 119903RAMA

1 1 le 119895 lt 1198981 119903RAMA1198962

ge 119903RAMA2 1 le 119896 lt 1198982 119903RAMA

11989811lt 119903RAMA

1 119903RAMA11989822

lt 119903RAMA2

(21)






119864RAMA119898 ≜ ⋃119864RAMA

1198981198982119872ac 1 le 1198982 le 119898 (22)








given 119866|ℎ1|2 |ℎ119872119886119888 |

2 is derived as

119875 (119864119877119860119872119860(1)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119872119886119888 |

2)=

1 (1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(23)

and C119877119860119872119860(1)1198981 1198982119872119886119888


11989811198982119872119886119888| 119866|ℎ1|

2 |ℎ119898|2|ℎ119872119886119888 |

2) is given by

119875 (119864119877119860119872119860(2)11989811198982119872119886119888

| 119866|ℎ1|2|ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(2)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(24)

where C119877119860119872119860(2)11989811198982119872119886119888




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

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hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

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bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

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bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

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hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

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babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

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06

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08

09

1

Out

age p

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

09

1

Out

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y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2) is given by

119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 119866|ℎ1|2 |ℎ119898|

2|ℎ119872119886119888 |2)

= 1 (1003816100381610038161003816ℎ1

10038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) isin C119877119860119872119860(119904+1)11989811198982119872119886119888

0 119900119905ℎ119890119903119908119894119904119890

1 le 1198981 le 119872119886119888 1 le 1198982 le 119872119886119888(25)

andC119877119860119872119860(119904+1)11989811198982119872119886119888


C119877119860119872119860(119904+1)11989811198982119872119886119888

= ⋃1le1198981119904lt11989811le1198982119904le1198982

(1003816100381610038161003816ℎ110038161003816100381610038162 10038161003816100381610038161003816ℎ119872119886119888

100381610038161003816100381610038162) |C

119877119860119872119860(119904)11989811199041198982119904119872119886119888

1003816100381610038161003816ℎ119894

10038161003816100381610038162 120572119875sum119872119886119888119895=119894+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902ge 1205931

10038161003816100381610038161003816ℎ1198981

100381610038161003816100381610038162 120572119875sum119872119886119888119895=1198981+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981119904

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205931 (1198982 minus 1198982119904) 1003816100381610038161003816ℎ11989610038161003816100381610038162 (1 minus 120572) 119875

sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=119896+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

ge (1198982 minus 1198982119904) 120593210038161003816100381610038161003816ℎ1198982

100381610038161003816100381610038162 (1 minus 120572) 119875sum119872119886119888

119895=1198982+1

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 120572119875 + sum119872119886119888119895=1198981

10038161003816100381610038161003816ℎ119895

100381610038161003816100381610038162 (1 minus 120572) 119875 + 1205902

lt 1205932 1198981119904 le 119894 lt 1198981 1198982119904 le 119896 le 1198982

(26)

119875(119864119877119860119872119860(119904+1)11989811198982119872119886119888


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

)= 119872119886119888 infinsum

119872119886119888=0


119875 (119864119877119860119872119860(119904+1)11989811198982119872119886119888

| 1198661199111119911119872119886119888 )


(27)



119875 (119864119877119860119872119860) = suminfin119872119886119888=1 (119875 (119870119872119886119888

)sum1198721198861198881198981=1 ((sum119872119886119888

1198982=1 119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

)) (119872119886119888 minus 1198981 + 1)))suminfin

119872119886119888=1 119875 (119870119872119886119888)119872119886119888

(28)

119879119877119860119872119860 = infinsum119872119886119888=1

(119875(119870119872119886119888) 119872119886119888sum

1198981=1

( 119872119886119888sum1198982=1

(119875 (119864119877119860119872119860(119904)11989811198982119872119886119888

) (11989811199031198771198601198721198601 + 1198982119903119877119860119872119860

2 )))) (29)



max120572119903RAMA1 119903RAMA

2


1 + 119903RAMA2 = 119903conv

0 le 120572 le 1(30)




Cconv119898119872ac

)119888 and (⋃1le11989811198982le119872acC

RAMA(119904)11989811198982119872ac

)119888 with








119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









119905119898)119899log2(|X(120582119905119898120599119905119898) |) as








max120599119898

119898(120599119905119898) (31)

where119898(120599119898) is given by [24]

119898(120599119898) = sum1199091isinX

119905119898(120582119898120599

119905119898)


119905119898(120582119898120599

119905119898)

110038161003816100381610038161003816X119905119898 (120582119905119898 120599119905119898)100381610038161003816100381610038162 exp (minus 141205902

10038171003817100381710038171199091 minus 119909210038171003817100381710038172)) (32)




HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














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11

12Su

m th

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hput

(bits

cha

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use

)


(a) Throughput

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2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

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x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

25

3

35

4

45

Achi

evab

le th

roug

hput

(pac

kets

slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



HConv-GF

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(a) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(b) Conv-GF

HConv-GF

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(c) Conv-GF

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1 ℎ

2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

HRAMA s = 2

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

H

01

02

03

04

05

06

07

08

09

1

ℎ2 2

02 03 04 05 06 07 08 09 101ℎ1

2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

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05

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09

1

11

12Su

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(bits

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)


(a) Throughput

02

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1

Out

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2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

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el u

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(a) Throughput

02

03

04

05

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1

Out

age p

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x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

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acke

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RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

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15

2

25

3

35

4

45

Achi

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roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

1

15

2

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35

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45

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(pac

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119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

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babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

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1

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

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05

06

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1

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(a) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(b) Conv-GF

1

08

06

041 08 06 02

0402

HConv-GF

02040608

1

ℎ3 2

ℎ22

ℎ1 2

(c) Conv-GF

1

08

06

04

0809 07 06 0502

04 0203 01

H

ℎ22

ℎ1 2

02040608

ℎ3 2

RAMA s = 2

(d) RAMA 120601 = 08 119903RAMA1 = 025 and 120572 = 05

1

08

06

041

0806 02

0402

H

ℎ22

ℎ1 202

040608

1

ℎ3 2

RAMA s = 2

(e) RAMA 120601 = 1 119903RAMA1 = 05 and 120572 = 07

1

08

06

041 08 06

0204 02

H

ℎ22

ℎ1 2

02

04

06

08

1

ℎ3 2

RAMA s = 2

(f) RAMA 120601 = 12 119903RAMA1 = 06 and 120572 = 074




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

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y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

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bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

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thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

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2

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3

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45

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(pac

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slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

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(pac

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slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

08

09

1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

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06

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




00 2

1

11

1001

= (0∘ 90∘)


Subchannel-1

Subchannel-2

0002

1001

010

011

110

111

101

100

= (0∘ minus45∘)



10000

0001

0101

0100 0110

0111

0011

0010 1010

1011

1111

1110 1100

1101

1001

010

011

111

110 100

101

001














03

04

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1

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)


(a) Throughput

02

03

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1

Out

age p

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y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

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1

Out

age p

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x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

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2

25

3

35

4

45

Achi

evab

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roug

hput

(pac

kets

slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

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2

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Achi

evab

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(pac

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slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

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09

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babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

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age p

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

01

02

03

04

05

06

07

08

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age p

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y




1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




03

04

05

06

07

08

09

1

11

12Su

m th

roug

hput

(bits

cha

nnel

use

)


(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y






2 = 06and 120572 = 074

035

04

045

05

055

06

065

07

075

08

Sum

thro

ughp

ut (b

itsc

hann

el u

se)



(a) Throughput

02

03

04

05

06

07

08

09

1

Out

age p

roba

bilit

y













x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


05

1

15

2

25

3

35

4

45Ac

hiev

able

thro

ughp

ut (p

acke

tsslo

t)



RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


05

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2

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3

35

4

45

Achi

evab

le th

roug

hput

(pac

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slot)



RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

05

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Achi

evab

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(pac

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slot)




119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



0

01

02

03

04

05

06

07

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09

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utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



0

01

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age p

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






x119898119897 = (ℎ119905119898)119867 ( sum

119868119905119895=1119895 =119898

ℎ119905119898 (ℎ119905

119898)119867 + 1205901198682) y (33)








Appendix

A Proof of Lemma 1


119891119903 (V) = 2V11987721 minus 1198772

2

1198772 le V le 1198771 (A1)


1198911199032 (119910) = 119891119903 (119910minus12) (119910minus12)1015840

119910= 211990912

11987721 minus R2

2

(12119909minus12)= 11198772

1 minus 11987722

11987722 le 119910 le 1198772

1(A2)


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RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


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RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



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RAMA 12 = 4

RAMA 12 = 3

RAMA 12 = 6


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RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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119891|119892|2 (119909) = 12119890minus1199092 (A3)


119891|ℎ|2 (119911) = intinfin

0119891|119892|2 (119911119910) 1198911199032 (119910) 119910 119889119910

= int11988921

11988922

12119890minus1199111199102 111987721 minus 1198772

2

119910119889119910 = 21199112 (11987721 minus 1198772

2)

sdot int11988921

11988922

119890minus1199111199102 (1199111199102 ) 119889(1199111199102 )= 11199112 (1198772

1 minus 11987722) (119890minus119877221199112 (1198772

2119911 + 2)minus 119890minus119877211199112 (1198772

1119911 + 2)) (A4)



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





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Advances in

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0

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1O

utag

e pro

babi

lity

conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6



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age p

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conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6




conv-GFRAMA 12 = 2

RAMA 12 = 3

RAMA 12 = 4

RAMA 12 = 6

0

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age p

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1 minus 11987722) (A5)


119865|ℎ|2 (119911) = ℎ (119911) minus ℎ (0) = 1 + ℎ (119911) (A6)

119875 (119864conv119905119898119872ac


0119891|ℎ119905|2 (119911119872ac

) int+infin

119911119872acminus1=119911119872ac


119911119898+1=119911119898+2

119891|ℎ119905|2 (119911119898+1)times int120601(sum

119872ac119895=119898+1 119911119895+1205902119875)

119911119898=119911119898+1

119891|ℎ119905|2 (119911119898) times int+infin

119911119898minus1=max(119911119898120601(sum119872ac119895=119898 119911119895+1205902119875))




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




1199113=max(1199114 120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=max(1199113(120601(1minus120601))(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112) int+infin

1199111=1199112

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I1

+ int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=max(1199113 120601(sum119872ac119895=3 +1205902119875))

119891|ℎ119905|2 (1199112)int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟I2

]]]]]1198891199111 sdot sdot sdot 119911119872ac

(A7)

1198681 = int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113) int+infin

1199112=(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112)int+infin

1199111=1199112

119891|ℎ119905|2 (1199111) (A8)

1198682 = int+infin

1199113=1199114

119891|ℎ119905|2 (1199113) int(120601(1minus120601))(sum119872ac119895=3 +1205902119875)

1199112=1199113

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) + int+infin

1199113=max(1199114120601(sum119872ac119895=4 119911119895+1205902119875))

119891|ℎ119905|2 (1199113)


1199112=120601(sum119872ac119895=3 +1205902119875)

119891|ℎ119905|2 (1199112) int+infin

1199111=120601(sum119872ac119895=2 119911119895+1205902119875)

119891|ℎ119905|2 (1199111) (A9)













119875 (119864conv11 ) = int1206011205902119875

1199111=0119891|ℎ119905|2 (1199111) 1198891199111 = 119865|ℎ119905|2 (1206011205902

119875 )= 1 + 2119890minus(11987721120601120590

2119875)2 minus 2119890minus(119877221206011205902119875)2

1206011205902 (11987721 minus 1198772

2) 119875 (C1)


22 ) are givenby

119875 (119864conv12 ) = 2 int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot int120601(1199112+1205902119875)

1199111=1199112

119891|ℎ119905|2 (1199112) 11988911991111198891199112 = 2int+infin

1199112=0119891|ℎ119905|2 (1199112)

sdot (119865|ℎ119905|2 (120601 (1199112 + 1205902119875)) minus 119865|ℎ119905|2 (1199112)) 1198891199112


(C2)

119875 (119864conv22 ) = 2 int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)

sdot int+infin

1199111=120601(1199112+1205902119875)119891|ℎ119905|2 (1199111) 1198891199111 1198891199112

= 2int1206011205902119875

1199112=0119891|ℎ119905|2 (1199112)


119875 )))1198891199112

= 2(119865|ℎ119905|2 (120601(1205902

119875 )) minusF(120601(1205902

119875 ))+F (0))

(C3)


respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



respectively where

F (119911) = int119891|ℎ119905|2 (119911) 119865|ℎ119905|2 (120601(119911 + 1205902

119875 ))119889119911 (C4)


When 119872ac = 3 119875(119864conv13 ) 119875(119864conv

23 ) and 119875(119864conv33 ) can be




Acknowledgments


References





























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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