research article algebraic number precoded ofdm transmission...

Research ArticleAlgebraic Number Precoded OFDM Transmission forAsynchronous Cooperative Multirelay Networks

Hua Jiang1 Lizhen Shen2 Hao Cheng1 and Guoqing Liu1

1 School of Sciences Nanjing Tech University Nanjing 211816 China2 Biotechnology and Pharmaceutical Engineering Nanjing Tech University Nanjing 211816 China

Correspondence should be addressed to Guoqing Liu gqliunj163com

Received 28 July 2014 Accepted 25 October 2014 Published 4 November 2014

Academic Editor Jun-Juh Yan

Copyright copy 2014 Hua Jiang et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a space-time block coding (STBC) transmission scheme for asynchronous cooperative systems By combinationof rotated complex constellations and Hadamard transform these constructed codes are capable of achieving full cooperativediversity with the analysis of the pairwise error probability (PEP) Due to the asynchronous characteristic of cooperative systemsorthogonal frequency division multiplexing (OFDM) technique with cyclic prefix (CP) is adopted for combating timing delaysfrom relay nodesThe total transmit power across the entire network is fixed and appropriate power allocation can be implementedto optimize the network performance The relay nodes do not require decoding and demodulation operation resulting in a lowcomplexity Besides there is no delay for forwarding the OFDM symbols to the destination node At the destination node thereceived signals have the corresponding STBC structure on each subcarrier In order to reduce the decoding complexity the spheredecoder is implemented for fast data decoding Bit error rate (BER) performance demonstrates the effectiveness of the proposedscheme

1 Introduction

Recently cooperative communication networks are known tohave significant potential in increasing network capacity andtransmission reliability In recent years cooperative networkshave attracted substantial interest from the wireless network-ing and communications research [1ndash13] The basic idea isthat intermediate relay nodes act as a virtual distributedantenna array to assist the source node in transmitting itsinformation to the destination node The amplify-and-for-ward (AF) and decode-and-forward (DF) are the mostfamous schemes for cooperative systems [14]

Space-time block code (STBC) [15 16] is an effectiveapproach to achieve spatial diversity for cooperative trans-missions Since the relay nodes are in different locationsand have different oscillators there may exist timing errorsTherefore it is difficult to design proper space-time codingschemes The transmission schemes which are based onorthogonal frequency division multiplexing (OFDM) areexploited for combating the loss of timing phase [2ndash5]In [4] a simple Alamouti scheme is proposed to achieve

cooperative diversity where only a few simple operationssuch as time-reversion and complex conjugation are requiredat the relay nodes the fast ML Alamouti decoding is used atthe destination node However this scheme is only useful forthe case of two relay nodes In [6 7] the clustered orthogonalspace-time block code (OSTBC) schemes for four or morerelay nodes are proposed by clustering the relay nodes Itis shown that limited performance improvement is achievedwhile the number of relay nodes increases

In [17ndash21] a new multidimensional modulation schemeis proposed to increase the modulation diversity It is feasibleto achieve full modulation diversity by applying optimumrotation to signal constellation The algebraic number theoryis introduced in [22] to construct amultidimensional rotationmatrix By properly selecting the algebraic number field agenerator matrix that guarantees full modulation diversityand maximizes the minimum product distance for precodedinformation symbols can be obtained

In this paper we study the diagonal algebraic space-time(DAST) block coding for asynchronous cooperative networkWe consider the OFDM technique with enough cyclic prefix

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 862020 7 pageshttpdxdoiorg1011552014862020

2 Mathematical Problems in Engineering

(CP) [23 24] to combat the timing errors The informationsymbols are precoded by amatrixmultiplication at the sourcenode Throughout the proper operation at the relay nodethe received signals hold the DAST block code structure oneach subcarrier hence it is convenient to decode data byusing ML decoding or fast sphere decoder It is assumed thatthe relay nodes do not have to know any information aboutthe channels but the destination node knows all channelinformation through training Therefore the relay nodes donot need to decode and demodulate signals received from thesource node Only a few simple signal processings are neededat the relay nodes

This paper is organized as follows In Section 2 the relaynetwork model is described and a new transmission schemeis proposed In Section 3 the algebraic number theory isintroduced and the optimal rotation constellation schemeis presented The PEP is analyzed with the optimal powerallocation in Section 4 Section 5 contains the simulationresults Finally the conclusions are given in Section 6

Notation For a vector or matrix 119860 119860119879 119860119867 and 119860 indi-cate the transpose Hermitian and Frobenius norm of 119860respectively 119860 ∘ 119861 denotes the Hadamard product of 119860and 119861 that is the componentwise product lowast indicates theconjugate operation ⊛ denotes the circular convolutiondiag119886

1 119886

119899 is a diagonal matrix with 119886

119894being its 119894th

diagonal entry 119864 indicates the expectation and 119875 indicatesthe probability Gaussian integer is a complex number whosereal and imaginary parts are both integers

2 Relay Network Model andCooperative Protocol

Consider a relay network with one source node one desti-nation node and 119877 (119877 = 2

119902 119902 = 1 2 ) relay nodes as

shown in Figure 1 There is only one antenna at every nodeWe denote the average signal energy 119864

119904= 1 by a normalizd

QAM constellation OFDM technique is used to combat thetime errors It is assumed that the number of subcarriers is119873 and the transmission delay of the signals from 119894th relaynode at the destination node is 120591

119894 which is a multiple of

119879119904with 119879

119904denoting the information symbol duration At

the source node the information bits are modulated intocomplex symbols by normalized QAM constellation Theconsecutive 119873119877 symbols are denoted by 119904

119895119896 119895 = 1 2 119877

119896 = 1 2 119873 Then we let 119904119895= [119904

1198951 119904

1198952 119904

119895119873] represent

the 119895th block consisting of119873 symbolsLet 119872 be an 119877 times 119877 precoding matrix We obtain the

precoded symbols 119909119896= [119909

1119896 119909

119877119896]119879= 119872119904

119896 where

119904119896= [119904

1119896 119904

119877119896]119879 Then the new 119877 consecutive OFDM

block is denoted by 119909119895= [119909

1198951 119909

1198952 119909

119895119873] After 119873-point

IFFT operation and CP of length ℓcp insertion the OFDMsymbols of length 119873 + ℓcp are transmitted to the destinationnode through the relay nodesWe assume that the knowledgeof timing errors can be obtained at the destination node Itis emphasized that the CP length must be larger than themaximum time delay 120591max Besides we make the assumptionthat the channels between any two nodes are quasi-static flat

h1

h2

hRSource

Relays

Destination

R1

R2

RR

f1

f2

fR

Figure 1 Relay network model

fading The fading coefficient from the source node to the119894th relay node is denoted by ℎ

119894 and the fading coefficient

from 119894th relay node to the destination node is denoted by 119891119894

These coefficients are independent and identically distributed(iid) complex Gaussian random variables with zero meanand unit variance

Denote by1198831 119883

119877the 119877 consecutive OFDM symbols

where119883119895consists of IFFT(119909

119895) and the corresponding CP for

119895 = 1 119877The 119896th subcarrier output of 119895th OFDM symbolsin frequency domain can be represented by

IFFT (119909119895) (119896) =

1

radic119873

119873

sum

119906=1

1199091198951199061198902120587radicminus1(119906minus1)(119896minus1)119873

(1)

The 1radic119873multiplier in terms of IFFT guarantees the powerof signal symbols invariant after IFFT operation We assumethe channel coefficients remain unchanged during the trans-mission of 119877OFDM symbols At the 119894th relay node in the 119895thOFDM symbol duration the received signals are

119884119894119895= radic119875

1119883

119895ℎ

119894+ 119899

119894119895 (2)

where 1198751is the average transmit power at the source node

and 119899119894119895

is the corresponding additive white Gaussian noise(AWGN) with zero mean and unit variance at the 119895th relaynode in the 119894th OFDM symbol duration Note that due to theadditive white Gaussian noise the average power of receivedsignal is 119875

1+ 1

The relay nodes would simply process and transmit thereceived noisy signals Only unary positive andnegative oper-ations are needed For instance in the case of 119877 = 4 theprocessed signal matrix is given by

119884 = 120582[[[

[

11988411

11988421

11988431

11988441

11988412

minus11988422

11988432

minus11988442

11988413

11988423

minus11988433

minus11988443

11988414

minus11988424

minus11988434

11988444

]]]

]

(3)

We can find that it holds the structure similar to the Had-amard matrix Apart from the orthogonal STBC schemewhich is required to switch the OFDM symbols and hasto wait to start process and transmit until the next severalOFDM symbols arrive at the relay nodes [4 5] in our pro-posed scheme the relay nodes can process and broadcastthe received signals immediately without waiting the other

Mathematical Problems in Engineering 3

symbols arriving As a result the proposed schemewould notinduce the transmission delay

Let 119884119894119895

= 120582119878119894119895119884

119894119895be the transmit signal from the 119894th

relay node in the 119895th OFDM symbol duration where 119878119894119895

belonging to +1 minus1 is the entry of Hadamard matrix 119878 of119877 dimensions and scalar 120582 = radic119875

2 (119875

1+ 1) guarantees the

average transmit power is 1198752for one transmission at every

relay node At the destination node the received signal in the119895th OFDM symbol duration can be written as

119885119895=

119877

sum

119894=1

(119884119894119895⊛ 119865

119894) 119891

119894+119882

119895 (4)

where119882 is the correspondingAWGNat the destination nodeand 119865

119894is an 119873 point vector whose (120591

119894+ 1)th element is one

and the others are zero Time delays in the time domain areexpressed by circular convolution with 119865

119894 After CP removal

and119873-point FFT transformation the received signals can berewritten as

119885119895= 120582

119877

sum

119894=1

119878119894119895(radic119875

1119909

119895∘ 119891

120591119894ℎ119894119891

119894+ 119873

119894119895∘ 119891

120591119894119891119894) +119882

119895 (5)

where 119891120591119894 = [1 119890minus2120587120591119894radicminus1119873

119890minus2120587120591119894radicminus1(119873minus1)119873

]119879 with 119891 =

[1 119890minus2120587radicminus1119873


]119879 meaning the phase change

in frequency domain corresponding to the sample time delay120591119894in time domain119873

119894119895= FFT(119899

119894119895) and119882

119895= FFT(119882

119895) For

every subcarrier 119896 1 le 119896 le 119873 we have

119911119896= 120582radic119875

1(diag (119909

1119896 119909

2119896 119909

119877119896) 119878) (119891

120591

119896∘ ℎ ∘ 119891)

+ 120582 (119873119896∘ 119878) (119891

120591

119896∘ 119891) + 119908

119896

(6)

where we have defined

119911119896=[[

[

1199111119896

119911119877119896

]]

]

119891120591

119896=[[

[

1198911205911

119896

119891120591119877

119896

]]

]

ℎ =[[

[

ℎ1

ℎ119877

]]

]

119891 =[[

[

1198911

119891119877

]]

]

119908119896=[[

[

1199081119896

119908119877119896

]]

]

(7)

119873119896is an 119877 times 119877 matrix with the entries 119873

119894119895119896 1 le 119894 le 119877

1 le 119895 le 119877 The Hadamard transform is useful for reducingthe high peak-to-average power ratio over different transmitantennas resulting in power amplification

3 Rotated Constellations UsingAlgebraic Number Theory

In order to achieve full modulation diversity over the Ray-leigh fading channel and Gaussian fading channel the rota-tion of a multidimensional signal symbol vector is discussed

in this section The minimum product distance of the con-stellation considered is defined as

119889min = min119909=119872(119904minus119904

1015840)119904 =1199041015840

119877

prod

119895=1

10038161003816100381610038161003816119909

119895

10038161003816100381610038161003816 (8)

where 119904 and 1199041015840 belong to an 119877-dimensional constellation

(QAM or PAM)The algebraic number theory was employedto construct a proper precoding rotation matrix 119872 whichmaximizes theminimumproduct distances in certain dimen-sionsThe diversity order is theminimumHamming distancebetween any two coordinate vectors of constellation points Itis emphasized that a rotation matrix constructed by algebraicnumber theory is a Vandermondematrix which can simplifythe encoding and reduce the computational complexity in asimilar way of fast Fourier transformation Furthermore thisalgebraic construction of rotations is useful for reduction ofpeak-to-mean envelope power ratio (PMEPR)

It can be considered that the algebraic constellation hasfull spatial diversity if the associated minimum productdistance119889min is strictly positive equivalently the componentsof vectors 119909 = 119872119904 and 1199091015840

= 1198721199041015840 (with 119904 = 119904

1015840) in the rotatedconstellation are all different To construct a precodingmatrix119872 of dimension119877 (119877 = 2119902

119902 = 1 2 ) with fullmodulationdiversity we apply the canonical embedding to some totallycomplex cyclotomic number fields The reader can refer to[25 26] for more comprehensive details about cyclotomicnumber fields and canonical embedding

The algebraic norm for real integer in number fields atfirst appears in solving problems such as the integer solutionsof finding all 119909119899

+ 119910119899= 119911

119899 for 119899 = 2 which is stated inthe Fermat theorem It is emphasized that119873(120572) is an integerand 119873(120572) = 0 if and only if 120572 = 0 hence 119873(120572) ge 1 for120572 = 0 The algebraic norm should be compared with thecanonical embedding With the application of the canonicalembedding 120590

119894to each element of basis [1 120579 1205792

120579119873minus1

] of119870 the generator matrix is given by

119872 =1

radic119877

[[[[

[

1 1205791sdot sdot sdot 120579

119877minus1

1


119877minus1

2

d

1 120579119877sdot sdot sdot 120579

119877minus1

119877

]]]]

]

= [1205851 120585

2 120585

119877] (9)

where 120585119896is the corresponding column of 119872 for 119896 =

1 2 119877 Due to the characteristic of Vandermonde matrixand 120579

119906minus 120579V = 0 for 1 le 119906 = V le 119877 the matrix 119872

has full rank This means that matrix 119872 is eligible to bea generator complex matrix for multidimensional rotatedconstellation It is important to select the roots 120579

119896 119896 =

1 2 119877 for generating full modulation diversity rotationsandmaximizing theminimumproduct distanceWith proper120579119896the matrix 119872 becomes an orthogonal matrix that is

119872119867119872 = 119868 According to the property of algebraic norm the

minimum product distance is a nonzero integer It is revealedthat 119889min is related to the special properties of algebraicnumber field when the full modulation diversity is obtainedIn fact the diversity product is 1 for the optimal cyclotomicrotation matrix no matter what the space-time code size andconstellation size are which have been proved in [26]


We consider any two columns 120585119906and 120585V of 119872 119906 V =

1 2 119877 without loss of generality we assume V lt 119906 thenthe complex inner product of 120585

119906and 120585V is

⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119906minus1

(120579lowast

119896)Vminus1

=1

119877

119877

sum

119896=1

(120579119896120579

lowast

119896)V(120579

119896)119906minusV

(10)

The problem transforms to the 119877 minus 1 power symmetric func-tions with 119877 complex roots with the constraints of theproperties of the minimal polynomial 120583

120579(119909) Due to the fact

that the complex roots 120579119896are on the unit circle we hold

120579119896120579

lowast

119896= 1 and119872 is an orthogonal matrix so that we obtain

⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119898= 0 119898 = 1 2 119877 minus 1 (11)

We assume the minimal polynomial

120583120579 (119909) =

119877

prod

119896=1

(119909 minus 120579119896) =

119877

sum

119898=0

119886119898119909

119898 (12)

where 119886119877= 1 119886

119877minus1= sum

1le119896le119877120579119896 119886

119877minus2= sum

1le119906ltVle119877120579119906120579V

and1198860= 120579

11205792sdot sdot sdot 120579

119877 Notice that it is an elementary symmetric

polynomial According to Newtonrsquos identities we have

119886119898=

1

119877 minus 119898

119877minus119898

sum

119895=1

((minus1)119895minus1119886119898+119895

119877

sum

119896=1

(120579119896)119895) (13)

Since sum119877

119896=1(120579

119896)119898= 0 for 119898 = 1 2 119877 minus 1 we have 119886

119896= 0

for 119896 = 1 2 119877 minus 1 Notice the coefficients of the minimalpolynomial 120583

120579(119909) are a Gaussian integer and 120579119877

1= 120579

119877= 119894

where 119894 = radicminus1 which yields

120583120579 (119909) = 119909

119877minus 119894 (14)

Using the above results it is easy to achieve the complex roots120579119896= exp((4119896 minus 3)1205871198942119877) for 119896 = 1 119877

4 PEP Analysis and Optimal Power Allocation

The optimum power allocation between the source node andthe relay nodes is discussed in [10] to minimize the pair-wise error probability (PEP) which is well known to be animportant measure for performance analysis In this sectionwe focus on the high power regime and the upper boundof PEP and provide the theoretical bases for reducing theupper bound of PEP with the precoding matrix At first forsimplicityrsquos sake we rewrite (6) as follows

119911119896= 120582radic119875

1119860119872119904

119896+ 119908

1015840

119896 (15)

where 119860 = diag((119891120591

119896∘ ℎ ∘ 119891)

119879119878) and 119908

1015840

119896= 120582(119873

119896∘

119878)(119891120591

119896∘ 119891) + 119908

119896 The expression (15) implies that equivalently

the rotated constellation symbols are transmitted over thediagonal channel matrix in space and time Since |119891120591119894

119896| =

1 for any 119894 it is obvious to see that 1199081015840

119896is an independent

Gaussian random vector so that we can obtain119864(1199081015840

119896) = 0 and

Var(1199081015840

119896) = (1 + 120582

2sum

119877

119894=1|119891

119894|2)119868

119877 Assuming ideal channel state

information (CSI) the maximum-likelihood (ML) decodingis implemented The PEP of mistaking 119904

119896by 1199041015840

119896is given by

119875 (119904119896997888rarr 119904

1015840

119896) =

1

2119864ℎ119891

119890minus11987511198752(119867

119867(1198621015840minus119862)119867(1198621015840minus119862)119867)4(1+1198751+1198752 sum

119877

119894=1|119891119894|2)

(16)

where 119862 and 1198621015840 are corresponding code matrix of 119904119896and

1199041015840

119896 respectively When 119877 goes to infinity by the law of large

numbers it shows that (1119877)sum119877

119894=1|119891

119894|2 converges to 1 since the

variance ofsum119877

119894=1|119891

119894|2 tends to zerosThus we havesum119877

119894=1|119891

119894|2asymp

119877 It is also implied that the fading has little effect when 119877 islargeWe assume the total transmit power in thewhole systemis 119875 per symbol transmission then we have 119875 = 119875

1+119877119875

2and

1198751119875

2

2 (1 + 1198751+ 119875

2119877)

=119875

1(119875 minus 119875

1)

2119877 (1 + 119875)le

1198752

8119877 (1 + 119875) (17)

It is not hard to see that the probability can be minimizedwhen 119875

1= 1198752 and 119875

2= 1198752119877 that is it is optimal to allocate

half of the total power to the transmit node and the otherhalf to the relay nodes It is significant that every relay nodeonly consumes a little amount of power to contribute to thetransmission

To obtain the upper bound of PEP we have to computethe expectation over ℎ 119891 Note that 119867 = 119865

1015840ℎ where 1198651015840

=

diag(1198911205911

119896119891

1 119891

120591119877

119896119891

119877) Since |119891120591119894

119896| = 1 we have 119865 = 1198651015840119867

1198651015840=

diag(|1198911|2 |119891

119877|2) By using the tight upper bound of

Gaussian tail function we have the following approximateinequality

119875 (119904119896997888rarr 119904

1015840

119896)

≲1

2119864ℎ119891

119890minus(119875216119877(1+119875))ℎ

1198671198651015840119867

(1198621015840minus119862)119867(1198621015840minus119862)1198651015840ℎ

=1

2119864119891

int120587minus119877 exp(minus

1198751119875

2(ℎ

119867119865

1015840119867119866119865

1015840ℎ)

4 (1198751+ 1)119873

0

minus ℎ119867ℎ)119889ℎ

=1

2119864119891

detminus1[119868

119877+

1198752

16119877 (1 + 119875)(119862

1015840minus 119862)

119867

(1198621015840minus 119862)119865]

(18)

Due to the fact that (1198621015840minus 119862)

119867(119862

1015840minus 119862) = 119878 diag(|119909(1)

119896minus

1199091015840(1)

119896|2 |119909

(119877)

119896minus 119909

1015840(119877)

119896|2)119878 and 119878119878 = 119877119868

119877 it is not hard to

see det[(1198621015840minus 119862)

119867(119862

1015840minus 119862)] ge 119877

119877119889

2

min = (119877119864)119877 With the

knowledge that 120575119894= |119891

119894|2 is 1205942 distribution with 2 degrees of

freedom and the corresponding probability density functionis 119891(120575

119894) = 119890

minus120575119894 we obtain

119875 (119904119896997888rarr 119904

1015840

119896) ≲

1

2

119877

prod

119894=1

int

infin

0

(1 +120575

119894

120573)

minus1

119890minus120575119894119889120575

119894

=1

2[120573 exp (120573) 119864119894 (minus120573)]119877

(19)


where

120573 =16119864 (1 + 119875)

1198752 119864119894 (119909) = int

119909

minusinfin

119890119905

119905119889119905 (20)

is the exponential integral functionThis upper bound is suf-ficient to derive the optimization criteria On the high powerregime of log119875 ≫ 1 we have

11989016119864(1+119875)119875

2

= 1 + 119874(1

119875) asymp 1

119864119894 (16119864 (1 + 119875)

1198752) = minus log119875 + 119874 (1) asymp minus log119875

(21)

thus

119875 (119904 997888rarr 1199041015840) ≲

1

2[16119864(1 + 119875)

119875]

119877

(log119875119875

)

119877

asymp1

2(16119864)

119877119875

minus119877(1minusloglog119875 log119875)

(22)

Therefore the achieved diversity gain is119877(1minusloglog119875 log119875)which is linear in the number of relay nodes If 119875 increasesgreatly (log119875 ≫ loglog119875) full diversity of 119877 is obtained thesame as the multiple-input multiple-output (MIMO) systemwith 119877 transmit antennas and one receive antenna

5 Simulation Results

In this section we present some simulated performance ofDAST codes forwireless asynchronous cooperative networksThe MATLAB 80 sumulation tool was used for the simu-lation (on a Core i7-3770 34GHz PC) The different valuesof the number of relays 119877 and total transmit power 119875 areconsidered We assume that the length of OFDM subcarriersis 119873 = 64 and the length of cyclic prefix ℓcp = 16 with thetotal bandwidth of 10MHz and the OFDM symbol durationis 119879

119904= 64 120583s To satisfy the conditions that the time delay

must be less than the CP length 120591119894is randomly chosen from

0 to 15 with the uniform distribution and 1205911is assumed

to be 0 for the first relay node The information symbol ismodulated by normalized 4-QAM We fix the total transmitpower 119875 which is measured in decibel Using the optimalpower allocation strategy the transmit power of the sourcenode is 1198752 and the relay nodes share the other power Theaverage SNR at the destination node can be calculated tobe 1198752

4(119875 + 1) When 119875 ≫ 1 the SNR becomes 1198754 Atthe destination node the ML decoding of the DAST blockcode can be implemented by the sphere decoder to obtainalmost the same performance at a moderate complexity Theperformances of the unrotated constellation for the relaynetwork and the DAST block code for the multiple-inputmultiple-output (MIMO) system are given For the sake ofcomparison the performances of Alamouti code scheme [4]are also shown

In Figure 2 we show the decoding performance of thenetwork system equipped with two relay nodes Assume allthe systems have the same total transmit power and all thenodes in the relay network systems have the same power

100

10minus1

10minus2

10minus3

10minus4

BER

Power (dB)5 10 15 20 25 30

Unrotated schemeAlamouti scheme

DAST schemeDAST (2 times 1 MIMO)

Figure 2 BER comparison of relay network with 119877 = 2

constraint As expected we can see the performance for unro-tated constellation is poor For 119877 = 2 since the Alam-outi scheme is the unique complex orthogonal design at atransmission rate of 1 it seems hard for the DAST schemeto outperform it However there are some drawbacks forAlamouti scheme such that the relay nodes have to stackevery two OFDM symbols and then exchange the transmitorder of the two OFDM symbols which leads to one OFDMsymbol time slot delay The advantage of DAST scheme willbe immediate and significant when more relay nodes areemployed

Figures 3(a) and 3(b) show the decoding performance ofthe four-relay network systems and the MIMO system with 4transmit antennas and 1 receive antenna Figure 3(a) showsthe BER performances with respect to the total transmitpower Figure 3(b) shows the BER performances with respectto the receive SNR In the MIMO system the receive SNRis assumed to be 119875 We observe that the slopes of the BERcurves of the DAST scheme approache the slopes of the BERof the DAST MIMO systems when the total transmit power119875 or the receive SNR increases which indicates that the newscheme can achieve diversity degree of 4 We can see that thedistance between the curves of the MIMO system and DASTrelay system is large because the transmit antennas can fullycooperate in MIMO system However at the same receiveSNR the gap of the two curves is diminishing which canbe seen clearly in Figure 3(b) Since OSTBC design cannotachieve full transmission rate for more than two antennas itis not applicable for the system with more than 2 relay nodesTherefore the cluster Alamouti code method [5] with rate12 is used for the network with 4 relay nodes 16QAM isconsidered to maintain the same transmission rate We cansee that DAST scheme has a gain of about 3 dB at 10minus6 in


Power (dB)5 10 15 20 25 30

Unrotated schemeAlamouti clustered scheme


100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

(a) The same total transmit power

SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7

(b) The same receive SNR


Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


Figure 4 BER performance with 119877 = 8

Figure 3(b) It should be noted that this gain will be enhancedwhen increasing the size of the constellation In addition thecluster Alamouti code scheme has one OFDM symbol timedelay to perform signal processing at the relay nodes

Finally the example for 119877 = 8 is given to show the BERperformance of the DAST scheme and the MIMO systemwith respect to the total transmit power in Figure 4 It also

achieves a diversity order similar to that of DAST MIMOsystems When more relay nodes are utilized the curvesdescendmuch faster at the high SNR regimeThis implies thebetter BER performance is achieved

6 Conclusions

In this paper we propose the use of DAST block codes forasynchronous cooperative relay networks OFDM techniqueis implemented at the source node and only +minus opera-tions are required at the relay nodes without decoding andtransmission delay By using this method the received signalsymbols at the destination node hold the STBC structure afterremoving CP and IFFT operationThis structure is useful fordecoding It can be considered efficient to adopt fast decodingalgorithms such as sphere decoder to maintain the MLdecoding performance in a polynomial time It is not requiredwith the transmitted signal symbols and channel informationat the relay nodes We analyze the PEP and observe that theproposed scheme is capable of achieving full spatial diversityfor high total transmit power Simulation results on theDASTscheme are demonstrated which verifies our results

For possible future works it would be interesting toinvestigate further of imperfect CSI In the existing works thechannels are assumed to be perfectly estimated However thechannel estimation errors are inevitable in practice and affectthe BER performance Therefore the investigation of impactof imperfect CSI on the proposed schemes would be ourfurther work Moreover the symbol time offset and carrierfrequency offset in OFDM technique can also be explored


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Thisworkwas supported in part by theNatural Science Foun-dation of Jiangsu Province under Grant no BK2011793

References

[1] SWei D L Goeckel andMCValenti ldquoAsynchronous cooper-ative diversityrdquo IEEE Transactions onWireless Communicationsvol 5 no 6 pp 1547ndash1557 2006

[2] HWang Q Yin and X-G Xia ldquoFull diversity space-frequencycodes for frequency asynchronous cooperative relay networkswith linear receiversrdquo IEEE Transactions on Communicationsvol 59 no 1 pp 236ndash247 2011

[3] X Li F Ng and T Han ldquoCarrier frequency offset mitigation inasynchronous cooperative OFDM transmissionsrdquo IEEE Trans-actions on Signal Processing vol 56 no 2 pp 675ndash685 2008

[4] Z Li and X-G Xia ldquoA simple Alamouti space-time trans-mission scheme for asynchronous cooperative systemsrdquo IEEESignal Processing Letters vol 14 no 11 pp 804ndash807 2007

[5] Z Li X-G Xia and M H Lee ldquoA simple orthogonal space-time coding scheme for asynchronous cooperative systems forfrequency selective fading channelsrdquo IEEE Transactions onCommunications vol 58 no 8 pp 2219ndash2224 2010

[6] Z Li X-G Xia and B Li ldquoAchieving full diversity and fast MLdecoding via simple analog network coding for asynchronoustwo-way relay networksrdquo IEEE Transactions on Communica-tions vol 57 no 12 pp 3672ndash3681 2009

[7] XGuo andN-G Xia ldquoA distributed space-time coding in asyn-chronous wireless relay networksrdquo IEEE Transactions on Wire-less Communications vol 7 no 5 pp 1812ndash1816 2008

[8] S Talwar Y Jing and S Shahbazpanahi ldquoJoint relay selectionand power allocation for two-way relay networksrdquo IEEE SignalProcessing Letters vol 18 no 2 pp 91ndash94 2011

[9] Y Jing andH Jafarkhani ldquoRelay power allocation in distributedspace-time coded networks with channel statistical informa-tionrdquo IEEE Transactions on Wireless Communications vol 10no 2 pp 443ndash449 2011

[10] Y Jing and B Hassibi ldquoDistributed space-time coding in wire-less relay networksrdquo IEEE Transactions on Wireless Communi-cations vol 5 no 12 pp 3524ndash3536 2006

[11] B Sirkeci-Mergen and A Scaglione ldquoRandomized space-time coding for distributed cooperative communicationrdquo IEEETransactions on Signal Processing vol 55 no 10 pp 5003ndash50172007

[12] F Oggier and B Hassibi ldquoCyclic distributed space-time codesfor wireless relay networks with no channel informationrdquo IEEETransactions on Information Theory vol 56 no 1 pp 250ndash2652010

[13] Y Jing ldquoCombination of MRC and distributed space-timecoding in networks with multiple-antenna relaysrdquo IEEE Trans-actions onWireless Communications vol 9 no 8 pp 2550ndash25592010

[14] R U Nabar H Bolcskei and F W Kneubuhler ldquoFading relaychannels Performance limits and space-time signal designrdquo

IEEE Journal on Selected Areas in Communications vol 22 no6 pp 1099ndash1109 2004

[15] SM Alamouti ldquoA simple transmit diversity technique for wire-less communicationsrdquo IEEE Journal on Selected Areas in Com-munications vol 16 no 8 pp 1451ndash1458 1998

[16] V Tarokh H Jafarkhani and A R Calderbank ldquoSpace-time block coding for wireless communications performanceresultsrdquo IEEE Journal on Selected Areas in Communications vol17 no 3 pp 451ndash460 1999

[17] H El Gamal and M O Damen ldquoUniversal space-time codingrdquoIEEE Transactions on Information Theory vol 49 no 5 pp1097ndash1119 2003

[18] X Giraud E Boutillon and J Belfiore ldquoAlgebraic tools to buildmodulation schemes for fading channelsrdquo IEEE Transactions onInformation Theory vol 43 no 3 pp 938ndash952 1997

[19] P Dayal and M K Varanasi ldquoAn algebraic family of complexlattices for fading channels with application to space-timecodesrdquo IEEE Transactions on InformationTheory vol 51 no 12pp 4184ndash4202 2005

[20] BA Sethuraman B S Rajan andV Shashidhar ldquoFull-diversityhigh-rate space-time block codes from division algebrasrdquo IEEETransactions on Information Theory vol 49 no 10 pp 2596ndash2616 2003

[21] M O Damen K Abed-Meraim and J-C Belfiore ldquoDiagonalalgebraic space-time block codesrdquo IEEE Transactions on Infor-mation Theory vol 48 no 3 pp 628ndash636 2002

[22] J Boutros and E Viterbo ldquoSignal space diversity a power- andbandwidth-efficient diversity technique for the Rayleigh fadingchannelrdquo IEEE Transactions on Information Theory vol 44 no4 pp 1453ndash1467 1998

[23] H Abdzadeh-Ziabari M G Shayesteh and M Manaffar ldquoAnimproved timing estimation method for OFDM systemsrdquo IEEETransactions on Consumer Electronics vol 56 no 4 pp 2098ndash2105 2010

[24] Y Jie C Lei L Quan and C De ldquoAmodified selected mappingtechnique to reduce the peak-to-average power ratio of OFDMsignalrdquo IEEE Transactions on Consumer Electronics vol 53 no3 pp 846ndash851 2007

[25] J Boutros E Viterbo C Rastello and J-C Belfiore ldquoGood lat-tice constellations for both rayleig h fading and gaussian chan-nelsrdquo IEEE Transactions on Information Theory vol 42 no 2pp 502ndash518 1996

[26] G Wang and X-G Xia ldquoOn optimal multilayer cyclotomicspace-time code designsrdquo IEEE Transactions on InformationTheory vol 51 no 3 pp 1102ndash1135 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(CP) [23 24] to combat the timing errors The informationsymbols are precoded by amatrixmultiplication at the sourcenode Throughout the proper operation at the relay nodethe received signals hold the DAST block code structure oneach subcarrier hence it is convenient to decode data byusing ML decoding or fast sphere decoder It is assumed thatthe relay nodes do not have to know any information aboutthe channels but the destination node knows all channelinformation through training Therefore the relay nodes donot need to decode and demodulate signals received from thesource node Only a few simple signal processings are neededat the relay nodes

This paper is organized as follows In Section 2 the relaynetwork model is described and a new transmission schemeis proposed In Section 3 the algebraic number theory isintroduced and the optimal rotation constellation schemeis presented The PEP is analyzed with the optimal powerallocation in Section 4 Section 5 contains the simulationresults Finally the conclusions are given in Section 6

Notation For a vector or matrix 119860 119860119879 119860119867 and 119860 indi-cate the transpose Hermitian and Frobenius norm of 119860respectively 119860 ∘ 119861 denotes the Hadamard product of 119860and 119861 that is the componentwise product lowast indicates theconjugate operation ⊛ denotes the circular convolutiondiag119886

1 119886

119899 is a diagonal matrix with 119886

119894being its 119894th

diagonal entry 119864 indicates the expectation and 119875 indicatesthe probability Gaussian integer is a complex number whosereal and imaginary parts are both integers

2 Relay Network Model andCooperative Protocol

Consider a relay network with one source node one desti-nation node and 119877 (119877 = 2

119902 119902 = 1 2 ) relay nodes as

shown in Figure 1 There is only one antenna at every nodeWe denote the average signal energy 119864

119904= 1 by a normalizd

QAM constellation OFDM technique is used to combat thetime errors It is assumed that the number of subcarriers is119873 and the transmission delay of the signals from 119894th relaynode at the destination node is 120591

119894 which is a multiple of

119879119904with 119879

119904denoting the information symbol duration At

the source node the information bits are modulated intocomplex symbols by normalized QAM constellation Theconsecutive 119873119877 symbols are denoted by 119904

119895119896 119895 = 1 2 119877

119896 = 1 2 119873 Then we let 119904119895= [119904

1198951 119904

1198952 119904

119895119873] represent

the 119895th block consisting of119873 symbolsLet 119872 be an 119877 times 119877 precoding matrix We obtain the

precoded symbols 119909119896= [119909

1119896 119909

119877119896]119879= 119872119904

119896 where

119904119896= [119904

1119896 119904

119877119896]119879 Then the new 119877 consecutive OFDM

block is denoted by 119909119895= [119909

1198951 119909

1198952 119909

119895119873] After 119873-point

IFFT operation and CP of length ℓcp insertion the OFDMsymbols of length 119873 + ℓcp are transmitted to the destinationnode through the relay nodesWe assume that the knowledgeof timing errors can be obtained at the destination node Itis emphasized that the CP length must be larger than themaximum time delay 120591max Besides we make the assumptionthat the channels between any two nodes are quasi-static flat

h1

h2

hRSource

Relays

Destination

R1

R2

RR

f1

f2

fR

Figure 1 Relay network model

fading The fading coefficient from the source node to the119894th relay node is denoted by ℎ

119894 and the fading coefficient

from 119894th relay node to the destination node is denoted by 119891119894

These coefficients are independent and identically distributed(iid) complex Gaussian random variables with zero meanand unit variance

Denote by1198831 119883

119877the 119877 consecutive OFDM symbols

where119883119895consists of IFFT(119909

119895) and the corresponding CP for

119895 = 1 119877The 119896th subcarrier output of 119895th OFDM symbolsin frequency domain can be represented by

IFFT (119909119895) (119896) =

1

radic119873

119873

sum

119906=1

1199091198951199061198902120587radicminus1(119906minus1)(119896minus1)119873

(1)

The 1radic119873multiplier in terms of IFFT guarantees the powerof signal symbols invariant after IFFT operation We assumethe channel coefficients remain unchanged during the trans-mission of 119877OFDM symbols At the 119894th relay node in the 119895thOFDM symbol duration the received signals are

119884119894119895= radic119875

1119883

119895ℎ

119894+ 119899

119894119895 (2)

where 1198751is the average transmit power at the source node

and 119899119894119895

is the corresponding additive white Gaussian noise(AWGN) with zero mean and unit variance at the 119895th relaynode in the 119894th OFDM symbol duration Note that due to theadditive white Gaussian noise the average power of receivedsignal is 119875

1+ 1

The relay nodes would simply process and transmit thereceived noisy signals Only unary positive andnegative oper-ations are needed For instance in the case of 119877 = 4 theprocessed signal matrix is given by

119884 = 120582[[[

[

11988411

11988421

11988431

11988441

11988412

minus11988422

11988432

minus11988442

11988413

11988423

minus11988433

minus11988443

11988414

minus11988424

minus11988434

11988444

]]]

]

(3)

We can find that it holds the structure similar to the Had-amard matrix Apart from the orthogonal STBC schemewhich is required to switch the OFDM symbols and hasto wait to start process and transmit until the next severalOFDM symbols arrive at the relay nodes [4 5] in our pro-posed scheme the relay nodes can process and broadcastthe received signals immediately without waiting the other



Let 119884119894119895

= 120582119878119894119895119884




2 (119875




119885119895=

119877

sum

119894=1

(119884119894119895⊛ 119865

119894) 119891

119894+119882

119895 (4)







119885119895= 120582

119877

sum

119894=1

119878119894119895(radic119875

1119909

119895∘ 119891

120591119894ℎ119894119891

119894+ 119873

119894119895∘ 119891

120591119894119891119894) +119882

119895 (5)



]119879 with 119891 =





119894119895= FFT(119899

119894119895) and119882

119895= FFT(119882

119895) For


119911119896= 120582radic119875

1(diag (119909

1119896 119909

2119896 119909

119877119896) 119878) (119891

120591

119896∘ ℎ ∘ 119891)

+ 120582 (119873119896∘ 119878) (119891

120591

119896∘ 119891) + 119908

119896

(6)


119911119896=[[

[

1199111119896

119911119877119896

]]

]

119891120591

119896=[[

[

1198911205911

119896

119891120591119877

119896

]]

]

ℎ =[[

[

ℎ1

ℎ119877

]]

]

119891 =[[

[

1198911

119891119877

]]

]

119908119896=[[

[

1199081119896

119908119877119896

]]

]

(7)


119894119895119896 1 le 119894 le 119877





119889min = min119909=119872(119904minus119904

1015840)119904 =1199041015840

119877

prod

119895=1

10038161003816100381610038161003816119909

119895

10038161003816100381610038161003816 (8)




= 1198721199041015840 (with 119904 = 119904




+ 119910119899= 119911



120579119873minus1


119872 =1

radic119877

[[[[

[


119877minus1

1


119877minus1

2

d

1 120579119877sdot sdot sdot 120579

119877minus1

119877

]]]]

]

= [1205851 120585

2 120585

119877] (9)





119896 119896 =







119906and 120585V is

⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119906minus1

(120579lowast

119896)Vminus1

=1

119877

119877

sum

119896=1

(120579119896120579

lowast

119896)V(120579

119896)119906minusV

(10)




120579119896120579

lowast


⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119898= 0 119898 = 1 2 119877 minus 1 (11)


120583120579 (119909) =

119877

prod

119896=1

(119909 minus 120579119896) =

119877

sum

119898=0

119886119898119909

119898 (12)

where 119886119877= 1 119886

119877minus1= sum

1le119896le119877120579119896 119886

119877minus2= sum

1le119906ltVle119877120579119906120579V

and1198860= 120579




119886119898=

1

119877 minus 119898

119877minus119898

sum

119895=1

((minus1)119895minus1119886119898+119895

119877

sum

119896=1

(120579119896)119895) (13)

Since sum119877

119896=1(120579

119896)119898= 0 for 119898 = 1 2 119877 minus 1 we have 119886

119896= 0



1= 120579

119877= 119894


120583120579 (119909) = 119909

119877minus 119894 (14)




119911119896= 120582radic119875

1119860119872119904

119896+ 119908

1015840

119896 (15)

where 119860 = diag((119891120591

119896∘ ℎ ∘ 119891)

119879119878) and 119908

1015840

119896= 120582(119873

119896∘

119878)(119891120591

119896∘ 119891) + 119908



119896| =




119896) = 0 and

Var(1199081015840

119896) = (1 + 120582

2sum

119877

119894=1|119891

119894|2)119868



119896by 1199041015840

119896is given by

119875 (119904119896997888rarr 119904

1015840

119896) =

1

2119864ℎ119891

119890minus11987511198752(119867

119867(1198621015840minus119862)119867(1198621015840minus119862)119867)4(1+1198751+1198752 sum

119877

119894=1|119891119894|2)

(16)


1199041015840



119894=1|119891



119894=1|119891


119894=1|119891

119894|2asymp


1+119877119875

2and

1198751119875

2

2 (1 + 1198751+ 119875

2119877)

=119875

1(119875 minus 119875

1)

2119877 (1 + 119875)le

1198752

8119877 (1 + 119875) (17)


1= 1198752 and 119875




1015840ℎ where 1198651015840

=

diag(1198911205911

119896119891

1 119891

120591119877

119896119891

119877) Since |119891120591119894

119896| = 1 we have 119865 = 1198651015840119867

1198651015840=

diag(|1198911|2 |119891



119875 (119904119896997888rarr 119904

1015840

119896)

≲1

2119864ℎ119891

119890minus(119875216119877(1+119875))ℎ

1198671198651015840119867

(1198621015840minus119862)119867(1198621015840minus119862)1198651015840ℎ

=1

2119864119891


1198751119875

2(ℎ

119867119865

1015840119867119866119865

1015840ℎ)

4 (1198751+ 1)119873

0

minus ℎ119867ℎ)119889ℎ

=1

2119864119891

detminus1[119868

119877+

1198752

16119877 (1 + 119875)(119862

1015840minus 119862)

119867

(1198621015840minus 119862)119865]

(18)


119867(119862

1015840minus 119862) = 119878 diag(|119909(1)

119896minus

1199091015840(1)

119896|2 |119909

(119877)

119896minus 119909

1015840(119877)

119896|2)119878 and 119878119878 = 119877119868


see det[(1198621015840minus 119862)

119867(119862

1015840minus 119862)] ge 119877

119877119889

2

min = (119877119864)119877 With the

knowledge that 120575119894= |119891



119894) = 119890


119875 (119904119896997888rarr 119904

1015840

119896) ≲

1

2

119877

prod

119894=1

int

infin

0

(1 +120575

119894

120573)

minus1

119890minus120575119894119889120575

119894

=1

2[120573 exp (120573) 119864119894 (minus120573)]119877

(19)


where

120573 =16119864 (1 + 119875)

1198752 119864119894 (119909) = int

119909

minusinfin

119890119905

119905119889119905 (20)


11989016119864(1+119875)119875

2

= 1 + 119874(1

119875) asymp 1

119864119894 (16119864 (1 + 119875)


(21)

thus

119875 (119904 997888rarr 1199041015840) ≲

1

2[16119864(1 + 119875)

119875]

119877

(log119875119875

)

119877

asymp1

2(16119864)

119877119875


(22)










100

10minus1

10minus2

10minus3

10minus4

BER

Power (dB)5 10 15 20 25 30







Power (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7



Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6






6 Conclusions






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Let 119884119894119895

= 120582119878119894119895119884




2 (119875




119885119895=

119877

sum

119894=1

(119884119894119895⊛ 119865

119894) 119891

119894+119882

119895 (4)







119885119895= 120582

119877

sum

119894=1

119878119894119895(radic119875

1119909

119895∘ 119891

120591119894ℎ119894119891

119894+ 119873

119894119895∘ 119891

120591119894119891119894) +119882

119895 (5)



]119879 with 119891 =





119894119895= FFT(119899

119894119895) and119882

119895= FFT(119882

119895) For


119911119896= 120582radic119875

1(diag (119909

1119896 119909

2119896 119909

119877119896) 119878) (119891

120591

119896∘ ℎ ∘ 119891)

+ 120582 (119873119896∘ 119878) (119891

120591

119896∘ 119891) + 119908

119896

(6)


119911119896=[[

[

1199111119896

119911119877119896

]]

]

119891120591

119896=[[

[

1198911205911

119896

119891120591119877

119896

]]

]

ℎ =[[

[

ℎ1

ℎ119877

]]

]

119891 =[[

[

1198911

119891119877

]]

]

119908119896=[[

[

1199081119896

119908119877119896

]]

]

(7)


119894119895119896 1 le 119894 le 119877





119889min = min119909=119872(119904minus119904

1015840)119904 =1199041015840

119877

prod

119895=1

10038161003816100381610038161003816119909

119895

10038161003816100381610038161003816 (8)




= 1198721199041015840 (with 119904 = 119904




+ 119910119899= 119911



120579119873minus1


119872 =1

radic119877

[[[[

[


119877minus1

1


119877minus1

2

d

1 120579119877sdot sdot sdot 120579

119877minus1

119877

]]]]

]

= [1205851 120585

2 120585

119877] (9)





119896 119896 =







119906and 120585V is

⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119906minus1

(120579lowast

119896)Vminus1

=1

119877

119877

sum

119896=1

(120579119896120579

lowast

119896)V(120579

119896)119906minusV

(10)




120579119896120579

lowast


⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119898= 0 119898 = 1 2 119877 minus 1 (11)


120583120579 (119909) =

119877

prod

119896=1

(119909 minus 120579119896) =

119877

sum

119898=0

119886119898119909

119898 (12)

where 119886119877= 1 119886

119877minus1= sum

1le119896le119877120579119896 119886

119877minus2= sum

1le119906ltVle119877120579119906120579V

and1198860= 120579




119886119898=

1

119877 minus 119898

119877minus119898

sum

119895=1

((minus1)119895minus1119886119898+119895

119877

sum

119896=1

(120579119896)119895) (13)

Since sum119877

119896=1(120579

119896)119898= 0 for 119898 = 1 2 119877 minus 1 we have 119886

119896= 0



1= 120579

119877= 119894


120583120579 (119909) = 119909

119877minus 119894 (14)




119911119896= 120582radic119875

1119860119872119904

119896+ 119908

1015840

119896 (15)

where 119860 = diag((119891120591

119896∘ ℎ ∘ 119891)

119879119878) and 119908

1015840

119896= 120582(119873

119896∘

119878)(119891120591

119896∘ 119891) + 119908



119896| =




119896) = 0 and

Var(1199081015840

119896) = (1 + 120582

2sum

119877

119894=1|119891

119894|2)119868



119896by 1199041015840

119896is given by

119875 (119904119896997888rarr 119904

1015840

119896) =

1

2119864ℎ119891

119890minus11987511198752(119867

119867(1198621015840minus119862)119867(1198621015840minus119862)119867)4(1+1198751+1198752 sum

119877

119894=1|119891119894|2)

(16)


1199041015840



119894=1|119891



119894=1|119891


119894=1|119891

119894|2asymp


1+119877119875

2and

1198751119875

2

2 (1 + 1198751+ 119875

2119877)

=119875

1(119875 minus 119875

1)

2119877 (1 + 119875)le

1198752

8119877 (1 + 119875) (17)


1= 1198752 and 119875




1015840ℎ where 1198651015840

=

diag(1198911205911

119896119891

1 119891

120591119877

119896119891

119877) Since |119891120591119894

119896| = 1 we have 119865 = 1198651015840119867

1198651015840=

diag(|1198911|2 |119891



119875 (119904119896997888rarr 119904

1015840

119896)

≲1

2119864ℎ119891

119890minus(119875216119877(1+119875))ℎ

1198671198651015840119867

(1198621015840minus119862)119867(1198621015840minus119862)1198651015840ℎ

=1

2119864119891


1198751119875

2(ℎ

119867119865

1015840119867119866119865

1015840ℎ)

4 (1198751+ 1)119873

0

minus ℎ119867ℎ)119889ℎ

=1

2119864119891

detminus1[119868

119877+

1198752

16119877 (1 + 119875)(119862

1015840minus 119862)

119867

(1198621015840minus 119862)119865]

(18)


119867(119862

1015840minus 119862) = 119878 diag(|119909(1)

119896minus

1199091015840(1)

119896|2 |119909

(119877)

119896minus 119909

1015840(119877)

119896|2)119878 and 119878119878 = 119877119868


see det[(1198621015840minus 119862)

119867(119862

1015840minus 119862)] ge 119877

119877119889

2

min = (119877119864)119877 With the

knowledge that 120575119894= |119891



119894) = 119890


119875 (119904119896997888rarr 119904

1015840

119896) ≲

1

2

119877

prod

119894=1

int

infin

0

(1 +120575

119894

120573)

minus1

119890minus120575119894119889120575

119894

=1

2[120573 exp (120573) 119864119894 (minus120573)]119877

(19)


where

120573 =16119864 (1 + 119875)

1198752 119864119894 (119909) = int

119909

minusinfin

119890119905

119905119889119905 (20)


11989016119864(1+119875)119875

2

= 1 + 119874(1

119875) asymp 1

119864119894 (16119864 (1 + 119875)


(21)

thus

119875 (119904 997888rarr 1199041015840) ≲

1

2[16119864(1 + 119875)

119875]

119877

(log119875119875

)

119877

asymp1

2(16119864)

119877119875


(22)










100

10minus1

10minus2

10minus3

10minus4

BER

Power (dB)5 10 15 20 25 30







Power (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7



Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6






6 Conclusions






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906and 120585V is

⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119906minus1

(120579lowast

119896)Vminus1

=1

119877

119877

sum

119896=1

(120579119896120579

lowast

119896)V(120579

119896)119906minusV

(10)




120579119896120579

lowast


⟨120585119906 120585V⟩ =

1

119877

119877

sum

119896=1

(120579119896)119898= 0 119898 = 1 2 119877 minus 1 (11)


120583120579 (119909) =

119877

prod

119896=1

(119909 minus 120579119896) =

119877

sum

119898=0

119886119898119909

119898 (12)

where 119886119877= 1 119886

119877minus1= sum

1le119896le119877120579119896 119886

119877minus2= sum

1le119906ltVle119877120579119906120579V

and1198860= 120579




119886119898=

1

119877 minus 119898

119877minus119898

sum

119895=1

((minus1)119895minus1119886119898+119895

119877

sum

119896=1

(120579119896)119895) (13)

Since sum119877

119896=1(120579

119896)119898= 0 for 119898 = 1 2 119877 minus 1 we have 119886

119896= 0



1= 120579

119877= 119894


120583120579 (119909) = 119909

119877minus 119894 (14)




119911119896= 120582radic119875

1119860119872119904

119896+ 119908

1015840

119896 (15)

where 119860 = diag((119891120591

119896∘ ℎ ∘ 119891)

119879119878) and 119908

1015840

119896= 120582(119873

119896∘

119878)(119891120591

119896∘ 119891) + 119908



119896| =




119896) = 0 and

Var(1199081015840

119896) = (1 + 120582

2sum

119877

119894=1|119891

119894|2)119868



119896by 1199041015840

119896is given by

119875 (119904119896997888rarr 119904

1015840

119896) =

1

2119864ℎ119891

119890minus11987511198752(119867

119867(1198621015840minus119862)119867(1198621015840minus119862)119867)4(1+1198751+1198752 sum

119877

119894=1|119891119894|2)

(16)


1199041015840



119894=1|119891



119894=1|119891


119894=1|119891

119894|2asymp


1+119877119875

2and

1198751119875

2

2 (1 + 1198751+ 119875

2119877)

=119875

1(119875 minus 119875

1)

2119877 (1 + 119875)le

1198752

8119877 (1 + 119875) (17)


1= 1198752 and 119875




1015840ℎ where 1198651015840

=

diag(1198911205911

119896119891

1 119891

120591119877

119896119891

119877) Since |119891120591119894

119896| = 1 we have 119865 = 1198651015840119867

1198651015840=

diag(|1198911|2 |119891



119875 (119904119896997888rarr 119904

1015840

119896)

≲1

2119864ℎ119891

119890minus(119875216119877(1+119875))ℎ

1198671198651015840119867

(1198621015840minus119862)119867(1198621015840minus119862)1198651015840ℎ

=1

2119864119891


1198751119875

2(ℎ

119867119865

1015840119867119866119865

1015840ℎ)

4 (1198751+ 1)119873

0

minus ℎ119867ℎ)119889ℎ

=1

2119864119891

detminus1[119868

119877+

1198752

16119877 (1 + 119875)(119862

1015840minus 119862)

119867

(1198621015840minus 119862)119865]

(18)


119867(119862

1015840minus 119862) = 119878 diag(|119909(1)

119896minus

1199091015840(1)

119896|2 |119909

(119877)

119896minus 119909

1015840(119877)

119896|2)119878 and 119878119878 = 119877119868


see det[(1198621015840minus 119862)

119867(119862

1015840minus 119862)] ge 119877

119877119889

2

min = (119877119864)119877 With the

knowledge that 120575119894= |119891



119894) = 119890


119875 (119904119896997888rarr 119904

1015840

119896) ≲

1

2

119877

prod

119894=1

int

infin

0

(1 +120575

119894

120573)

minus1

119890minus120575119894119889120575

119894

=1

2[120573 exp (120573) 119864119894 (minus120573)]119877

(19)


where

120573 =16119864 (1 + 119875)

1198752 119864119894 (119909) = int

119909

minusinfin

119890119905

119905119889119905 (20)


11989016119864(1+119875)119875

2

= 1 + 119874(1

119875) asymp 1

119864119894 (16119864 (1 + 119875)


(21)

thus

119875 (119904 997888rarr 1199041015840) ≲

1

2[16119864(1 + 119875)

119875]

119877

(log119875119875

)

119877

asymp1

2(16119864)

119877119875


(22)










100

10minus1

10minus2

10minus3

10minus4

BER

Power (dB)5 10 15 20 25 30







Power (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7



Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6






6 Conclusions






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120573 =16119864 (1 + 119875)

1198752 119864119894 (119909) = int

119909

minusinfin

119890119905

119905119889119905 (20)


11989016119864(1+119875)119875

2

= 1 + 119874(1

119875) asymp 1

119864119894 (16119864 (1 + 119875)


(21)

thus

119875 (119904 997888rarr 1199041015840) ≲

1

2[16119864(1 + 119875)

119875]

119877

(log119875119875

)

119877

asymp1

2(16119864)

119877119875


(22)










100

10minus1

10minus2

10minus3

10minus4

BER

Power (dB)5 10 15 20 25 30







Power (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7



Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6






6 Conclusions






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Power (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6


SNR (dB)5 10 15 20 25 30



100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6

10minus7



Power (dB)50 10 15 20 25

100

10minus1

10minus2

10minus3

10minus4

BER

10minus5

10minus6






6 Conclusions






Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article algebraic number precoded ofdm transmission...

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