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Analytical Performance Assessment ofTHz Wireless SystemsALEXANDROS–APOSTOLOS A. BOULOGEORGOS1, (IEEE Member),EVANGELOS N. PAPASOTIRIOU1, AND ANGELIKI ALEXIOU1 (IEEE Member)1 Department of Digital Systems, University of Piraeus Piraeus 18534 Greece (e-mails: [email protected], vangpapasot, [email protected])

Corresponding author: Alexandros-Apostolos A. Boulogeorgos (e-mail: [email protected]).

This work has received funding from the European Commission’s Horizon 2020 research and innovation programme under grantagreement No. 761794.

ABSTRACT This paper is focused on providing the analytical framework for the quantification andevaluation of the joint effect of misalignment fading and hardware imperfections in the presence of multipathfading at terahertz (THz) wireless fiber extenders. In this context, we present the appropriate system modelthat incorporates the different operation, design, and environmental parameters. In more detail, it takes intoaccount the transceivers antenna gains, the operation frequency, the distance between the transmitter (TX)and the receiver (RX), the environmental conditions, i.e., temperature, humidity and pressure, the spatialjitter between the TX and RX antennas that results to antennas misalignment, the level of transceivers’hardware imperfections, and the stochastic characteristics of the wireless channel. Based on this model, weanalyze and quantify the joint impact of misalignment and multipath fading, by providing novel closed-form expressions for the probability and cumulative density functions of the composite channel. Moreover,we derive exact closed-form expressions for the outage probability for both cases of ideal and non-idealradio frequency (RF) front-end. Also, in order to quantify the detrimental effect of misalignment fading, weanalytically obtain the outage probability in the absence of misalignment cases for both cases of ideal andnon-ideal RF front-end. Additionally, we extract novel closed-form expressions for the ergodic capacity forthe case of ideal RF front-end and tight upper bounds for both cases of ideal and non-ideal RF front-end.Finally, an insightful ergodic capacity ceiling for the non-ideal RF front-end case is provided.

INDEX TERMS Beyond 5G systems, Ergodic capacity, Fiber extender, Hardware impairments, Highfrequency communications, Misalignment fading, Outage probability, Performance analysis, Terahertzcommunications, Theoretical framework, α-µ fading.

I. INTRODUCTION

Over the last years, the proliferation of wireless devicesand the increasing number of bandwidth-consuming internetservices have significantly raised the demand for high data-rate transmission with very low latency. While the wirelessworld moves towards the fifth generation (5G), several tech-nological advances, such as high order modulation schemes,massive multiple-input multiple-output systems and full du-plexing, have been presented as promising enablers [1]. How-ever, there is a lack of efficiency and flexibility in handlingthe huge amount of quality of service and experience orienteddata [2], [3]. These realizations have motivated the exploita-tion of higher frequency bands, such as the terraherz (THz)band [4]–[7].

Despite the fact that THz band communications can offer

an unprecented increase in the bandwidth and support ultrahigh data rates, yet they suffer from severe path attenuations,tranceivers antenna misalignment as well as hardware imper-fections [8]. The path attenuation originates from the highfrequencies of this band, which cause interaction and energyabsorption by the molecules of the propagation medium [9].As a result, a great ammount of effort was put in mod-eling the THz channel particularities [10], evaluating theirimpact on the system’s performance [11], [12] and propos-ing countermeasures [13]–[18]. On the other hand, antennamisalignment is caused due to the use of high directiveantennas, which cannot be easily fully-aligned [19]. Finally,the hardware imperfections are the result of componentsmismatch and manufacturing defects in the radio frequency(RF) transceiver chain [20].

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A.–A. A. Boulogeorgos et al.: Analytical Performance Assessment of THz Wireless Systems

A. RELATED WORKIn general, the THz channel particularities were investigatedin several works (see for example [11], [21]–[29] and refer-ences therein). In more detail, in [21], the authors presenteda novel propagation model for electromagnetic nanoscalecommunications in the THz band, based on radiative transfertheory, whereas, in [11], the model was extended in orderto incorporate the effect of molecular relaxation. Similarly,in [22], the authors provided an electromagnetic channelmodel for the body-centric nano-networks operating in theTHz band. Additionally, in [23], a simplified path-loss modelfor the 275− 400 GHz band was introduced, which was em-ployed in [24] in order to evaluate the THz link performancein terms of average signal-to-noise ratio (SNR) and capacity.Furthermore, in [25], a multi-ray THz propagation model waspresented, while, in [26], the authors reported a path-lossmodel for nano-sensor networks operating in the THz bandfor plant foliage applications. Meanwhile, in [27], a propa-gation model for intra-body nano-scale communications wasprovided. Likewise, in [28], a path-loss model, which quan-tifies the total absorption loss assuming that air, natural gasand/or water are the components of the propagation medium,was discussed, whereas, in [29], a multi-ray THz propagationmodel was presented.

Although, all the above mentioned contributions revealedthe particularities of the THz medium, they neglected theimpact of fading, which can be generated due to scatteringon aerosols in the atmosphere [30]. On the contrary, in [31]–[36], the authors presented suitable stochastic models that areable to accommodate the multipath fading effect in the THzband. In particular, in [31], the authors introduced a multi-path THz channel model, where the attenuation factor wasmodeled as a Rayleigh or Nakagami-m distributions underthe non-line-of-sight condition and as a Rician or Nakagami-m distribution under the line-of-sight (LoS) condition. Notethat they supported their claim through experimental results.Meanwhile, in [32], the authors conducted experiments andverified the existence of the shadowing effect in the 300 GHzband. Moreover, in [33] and [34], a two dimensional geomet-rical propagation model for indoor THz communications wasproposed. Based on this model, they developed a parametricreference model for a THz multi-path Rician fading channel.Likewise, in [35], the influence of antenna directivities onthe THz indoor channels for various antenna types wasinvestigated, assuming Rician fading, whereas, in [36], theauthors used a log-distance shadowing path-loss model forTHz nano-sensor networks communications in the vicinityof a plant.

The impact of transmitter (TX) and receiver (RX) beamsmisalignment was discussed in several published works, in-cluding [31], [35], [37], [38]. However, despite the smalltransceiver antenna beam-width, due to the use of highdirective antennas in the THz band, there are only threepapers, which analyze its influence in the THz link [31], [35],[37]. In more detail, in [31] and [37], the authors assumeddeterministic models to incorporate the impact of antennas’

misalignment, while, in [35], it was considered a part of theshadowing effect. The disadvantages of these models are thatthey are unable to accommodate the stochastic characteris-tics of phenomenons like thermal expansion, dynamic windloads and weak earthquakes, which result in the sway ofhigh-rise buildings and cause vibrations of the transceiversantennas; hence, the effect of misalignment between the TXand RX [39].

From the implementation point of view, in the THz band,the direct conversion architecture is of much hype, due to itslow-complexity and cost-effective configuration [40]–[44].The con of this architecture is that it is very sensitive tohardware imperfections, such as in-phase and quadratureimbalance, phase noise and amplifier non-linearities [45],[46]. In general, the effect of hardware imperfections wasquantified in several contributions (see for example [47]–[60]and references therein) and it was concluded that they cansignificantly constrain the system’s performance. However,despite of their paramount importance on the performanceof THz wireless systems, the detrimental effects of hardwareimperfections have been overlooked in the vast majority ofreported contributions. Only very recently, their impact wasexperimentally reported in [41], [44]. Specifically, in [44],the authors estimated the impact of hardware imperfectionsin the 300 GHz band. Finally, in [41], the authors highlightedtheir detrimental effect in THz wireless fiber extender sys-tems. However, to the best of the authors knowledge, there isno analytical study on the effect of hardware imperfectionsin the THz band.

B. MOTIVATION, NOVELTY AND CONTRIBUTIONTo sum up, since the used frequency spectrum for 5G haslimited capacity, THz wireless systems became an attractivecomplementing technology to the less flexible and moreexpensive optical-fiber connections [6], [13], where they canact as countermeasures to the connectivity gaps that existbetween the radio frequency (RF) access network and thefiber optic based backbone network [4], [41], [42], [61].This application scenario can be employed in developingcountries, where there might not be much of a fiber opticstructure and hence to increase its reach and bandwidth tothe last mile, without requiring a huge amount of economicresources to dig up the current brown-field.

To the best of the authors’ knowledge, the joint effect ofmisalignment and RF hardware imperfections over fadingchannel in the THz band have not been addressed in the openliterature. Motivated by the above, this work is devoted toprovide the generalized theoretical framework for the evalu-ation and quantification of the performance of THz wirelessfiber extenders. The technical contribution of this paper isoutlined below:• We establish an appropriate system and channel model

for the THz wireless fiber extender system, which ac-commodates the different design parameters as well astheir interaction. These parameters include the distancebetween the TX and the RX, the TX and RX antennas’

2 VOLUME 4, 2016


gains, the degree of TX and RX misalignment, the levelof hardware imperfection at the transceivers RF front-end, and the transmission power.

• In order to analyze the stochastic behavior of the wire-less THz channel, we derive novel closed-form expres-sion for the probability density function (PDF) andcumulative density function (CDF) of the distributionthat model the composite channel, which accomodatesthe effect of both the misalignment and the multipathfading. Note that the mutlipath fading is modeled as anα − µ distribution, which is a general form for severalwell-known channel models for multipath fading, suchas Rayleigh, Nakagami-m, Weibull, shadow, and com-posite fading [62], [63], that have been experimentallyverified in the THz systems, whereas, for the elevationand horizontal displacement at the RX’s plane indepen-dent and identical Gaussian distributions are assumed.This is a commonly used approach in high-directivesystems that are employed as wireless fiber extenders.However, this is the first paper that provides closed-form expression for analyzing the stochastic behaviorof the composite (multipath and misalignment fading)THz wireless channel. Additionally, we present closed-form expressions for the PDF and CDF of the channelfor the special cases, in which Rayleigh, Nakagami-m,Weibul and Gamma distributions are used to model themultipath fading effect.

• Based on the new channel model, we provide ex-act closed-form expressions for the outage probabilitythat quantifies the performance of the THz wirelessfiber extender system and take into account the levelof misalignment and hardware imperfection. Simpli-fied expression for the insightful case in which thetransceivers have ideal RF front-ends are also reported.As benchmarks, we present closed-form expressions forthe quantification of outage probabilities in both casesof ideal and non-ideal RF front-end in the absence ofmisalignment fading.

• Additionally, we extract novel closed-form expressionsfor the ergodic capacity in the case of ideal RF front-end and tight upper bounds for both cases of ideal andnon-ideal RF front-end.

• Finally, we present an insightful ergodic capacityhigh SNR ceiling, for the practical case of non-idealRF front-end.

C. ORGANIZATION AND NOTATIONSThe remainder of this paper is organized as follows. Sec-tion II presents the ideal and non-ideal RF front-end systemmodels as well as the wireless THz channel model. Sec-tion III is focused on deriving the distribution of the compos-ite misalignment and multipath fading channel. Moreover,the performance analysis in terms of outage probability andergodic capacity for both cases of ideal and non-ideal RFfront-end is provided in Section IV. Finally, respective nu-merical results and discussions are provided in Section V,

while closing remarks are highlighted in Section VI.

NotationsIn this paper, we use lower case bold letters to denotevectors. The operators E[·] and | · | respectively denotethe statistical expectation and the absolute value, whereasexp (x) and log2 (x) stand for the exponential functionand the logarithmic function with base 2. Additionally,the operator ln (x) refers to the natural logarithm of x,while, the operators

√x and lim

x→a(f(x)) return the square

root of x and the limit of the function f(x) as x tendsto a. Likewise, the set of the complex numbers is rep-resented by C, while CN (x, y) denotes a x-mean com-plex Gaussian process with variance y. Meanwhile, δ (·)stands for the Dirac’s function. The upper and lower in-complete Gamma functions [64, eq. (8.350/2), (8.350/3)]are respectively denoted by Γ (·, ·) and γ (·, ·), while theGamma function is represented by Γ (·) [64, eq. (8.310)].

Finally, 2F1(·, ·; ·; ·) and Gm,np,q

(x

∣∣∣∣ a1, a2, · · · , apb1, b2, · · · , bq

)re-

spectively stand for the Gauss hypergeometric function [65,eq. (4.1.1)] and the Meijer’s G-function [64, eq. (9.301)],

whereas Hm,np,q

[z

∣∣∣∣ (a1, b1), · · · , (ap, bp)(c1, d1), · · · , (cp, dp)

]is the Fox H-

function [66, eq. (8.3.1/1)].

II. SYSTEM & CHANNEL MODELThis section is focused on presenting the system and channelmodel under consideration. In particular, in Section II-A, thesystem model is provided, while, in Section II-B, the channelmodel is given.

A. SYSTEM MODELAs illustrated in Fig. 1.a, we consider a wireless fiber exten-der, in which the wireless link is utilized in the THz band. Inorder to countermeasure the channel attenuation, we assumethat both the TX and RX are equipped with highly directiveantennas. Suppose that the information signal, x ∈ C, isconveyed over a flat fading wireless channel h ∈ C1 withadditive noise n ∈ C, the baseband equivalent received signalcan be expressed as

yi = hx+ n, (1)

where h, x and n are statistically independent. Likewise, nis modeled as a complex zero-mean additive white Gaussianprocess with variance equals No.

Although, the received signal model presented in (1),accommodates the impact of the wireless channel andnoise, it neglects the effect of physical radio-frequency (RF)transceivers impefections, namely in-phase and quadratureimbalance (IQI), phase noise (PHN), as well as amplifiernon-linearities, which, in high data rate systems, is detrimen-tal [3], [45]. These imperfections generate a mismatch be-tween the intended signal x and what is actually emitted and

1This channel can, for example, be one of the subcarriers in a multi-carriersystem based on orthogonal frequency-division multiplexing (OFDM).

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OLT

OLT

THz link

(a)

(b)

FIGURE 1: (a) System model under consideration. In thisfigure OLT stands for the optical line terminal. (b) RX’seffective area and TX’s beam footprint with misalignment onthe RX’s plane.

distort the received signal during the reception processing. Inparticular, IQI causes mirror-interference, while local oscilla-tor PHN results to adjacent carrier interference and amplifiersnon-linearities to distributions that, according to Bussganttheorem, can be modeled as a complex Gaussian process [3].Hence, in order to model their combined influence at agiven flat fading subcarrier, we employ a generalized signalmodel [45], [67], which has been both theoretically andexperimentally validated [68]–[71]. According to this model,the baseband equivalent received signal can be obtained as

y = h(x+ nt) + nr + n. (2)

In (2), the parameters nt and nr are repsectively distortionnoises from impairments in the TX and RX [67], and theycan be defined as [67], [72]

nt ∼ CN(0, κ2tP ), and nr ∼ CN(0, κ2

rP |h|2), (3)

where κt and κr are non negative design parameters thatcharacterize the level of imperfections in the TX and RXhardware, respectively, while P stands for the average

transmitted signal power. Finally, note that the κt and κrparameters are interpreted as the error vector magnitudes(EVMs) [73], [74], which is a common quality measure ofRF transceivers and is the ratio of the average distortionmagnitude to the average signal magnitude.

B. CHANNEL MODELThe channel coefficient, h, can be obtained as

h = hlhphf , (4)

where hl, hp and hf respectively model the path gain, themisalignment fading, which results in pointing errors, and thefading hf .

1) Path-gain coefficientThe path gain coefficient can be expressed as

hl = hflhal, (5)

where hfl models the propagation gain and, according toFriis equation, can be modeled as [75]

hfl =c√GtGr

4πfd(6)

where Gt and Gr respectively represent the antenna orien-tation dependent transmission and reception gains, c standsfor the speed of light, f is the operating frequency and d isthe distance between the TX and the RX. Additionally, haldenotes the molecular absorption gain and can be evaluatedas [23], [24]

hal = exp

(−1

2κα(f)d

), (7)

where κα(f) denotes the absorption coefficient describingthe relative area per unit of volume, in which the moleculesof the medium are capable of absorbing the electromagneticwave energy. Note that the main cause of absorption loss inmillimeter and THz frequencies is the water vapor [11], [21],[24] that causes discrete, yet deterministic loss to the signalsin the frequency domain. Other atmospheric molecules suchas oxygen, also cause some level of loss to the signals, but itis minor when compared to the loss due to water vapor [24],[25]. The water vapor dominates the overall molecular ab-sorption loss above 0.2 THz frequencies.

In order to evaluate the molecular absorption coefficient,the radiative transfer theory [76] as well the informationprovided by the high resolution transmission molecular ab-sorption (HITRAN) database [77] are widely used. However,since THz fiber extenders are expected to operate in the275 − 400 GHz band, we utilize a simplified model for themolecular absorption loss in this band, which was initiallypresented in [23]. According to this model the absorptioncoefficient can be evaluated as

κα(f) = y1(f, v) + y2(f, v) + g(f), (8)

4 VOLUME 4, 2016


where the parameters y1(f, v), y2(f, v) and g(f) are definedas [24]

y1(f, v) =A(v)

B(v) + ( f100c − c1)2

, (9)

y2(f, v) =C(v)

D(v) + ( f100c − c2)2

(10)

and

g(f) = p1f3 + p2f

2 + p3f + p4 (11)

where c stands for the speed of light and c1 = 10.835 cm−1,c2 = 12.664 cm−1, p1 = 5.54 × 10−37 Hz−3, p2 =−3.94 × 10−25 Hz−2, p3 = 9.06 × 10−14 Hz−1, p4 =−6.36× 10−3 and

A(v) = g1v(g2v + g3), (12)

B(v) = (g4v + g5)2, (13)C(v) = g5v(g6v + g7), (14)

D(v) = (g8v + g9)2 (15)

with g1 = 0.2205, g2 = 0.1303, g3 = 0.0294, g4 = 0.4093,g5 = 0.0925, g5 = 2.014, g6 = 0.1702, g7 = 0.0303,g8 = 0.537 and g9 = 0.0956. Moreover, v stands for thevolume mixing ratio of the water vapor (not to be confusedwith relative humidity) and can be evaluated as [23]–[25]

v =φ

100

pw(T, p)

p, (16)

where φ, p respectively stand for the relative humidity and theatmospheric pressure, while pw(T, p) is the saturated watervapor partial pressure in temperature T , and, according toBuck’s equation [78], can be calculated as

pw(T, p) = q1 (q2 + q3φh) exp

(q4 (T − q5)

T − q6

), (17)

with q1 = 6.1121, q2 = 1.0007, q3 = 3.46 × 10−6 hPa−1,q4 = 17.502, q5 = 273.15 oK, q6 = 32.18 oK and φh beingthe pressure in hPa. Note that the model presented in (8) wasshown to have great accuracy for up to 1 Km links in standardatmospheric conditions, i.e temperature of 296 oK, pressureof 101325 Pa and relative humidity of 0.5 [23].

2) Misalignment fading coefficientAs illustrated in Fig. 1.b, we assume that the RX has acircular detection beam of radius α, covering an area A.Moreover, the TX has also a circular beam, which at distanced has a radius ρ, which belongs in the interval 0 ≤ ρ ≤ wd,where wd is the maximum radius of the beam at distanced. Furthermore, both beams are considered on the positivex− y plane and r is the pointing error expressed as the radialdistance of the transmission and reception beams. Due to thesymmetry of the beam shapes, hp depends only on the radialdistance r = |r|. Therefore, without loss of generality, it isassumed that the radial distance is located along the x axis.As a consequence and according to [79], the misalignmentfading coefficient, hp, which represents the fraction of the

power collected by the RX, covering an area A at distance dcan be approximated as

hp(r; d) ≈ Ao exp

(− 2r2

w2eq

), (18)

where weq is the equivalent beam-width, whereas Ao isthe fraction of the collected power at r = 0 and can becalculated as

u =

√πa√2wd

, (19)

with a being the radius of the RX effective area and wd is theradius of the TX beam footprint at distance d. Moreover, notethat the equivalent beam-width, w2

eq , is related to w2d through

w2eq = w2

d

√π erf (u)

2u exp (−u2). (20)

By considering independent identical Gaussian distribu-tions for the elevation and horizontal displacement [79],[80], the radial displacement at the RX follows a Rayleighdistribution with probability density function (PDF), whichcan be expressed as

fr(r) =r

σ2s

exp

(r2

2σ2s

), (21)

where σ2s is the variance of the pointing error displacement

at the RX. By combining (18) and (21), the PDF of |hp| canbe rewritten as [79]

fhp(x) =γ2

Aγ2

o

xγ2−1, 0 ≤ x ≤ Ao, (22)

where

γ =weq2σs

(23)

is the ratio between the equivalent beam width radius at theRX. Note that this model was extensively used in severalstudies in free space optical systems (see e.g., [81], [82] andreferences therein).

3) Multipath fadingIn order to accomodate the multipath fading effect, we model|hf | as a generalized α − µ distribution [83], with PDF thatcan be expressed as

fhf (x) =αµµ

hαµf Γ(µ)xαµ−1 exp

(−µx

α

hαf

), (24)

where α > 0 is a fading parameter, µ is the normalizedvariance of the fading channel envelope, and hf is the α-root mean value of the fading channel envelop. Note thatthis distribution is a general form for many well-knowndistributions, such as Rayleigh (α = 2, µ = 1), Nakagami-m(α = 2 and µ is the fading parameter), Weibull (µ = 1 and αis the fading parameter), etc. [84].

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III. STATISTICAL ANALYSIS OF THE STOCHASTICCHANNEL COEFFICIENTThe following theorems return the probability density func-tion (PDF) and cumulative density function (CDF) of therandom process that is used to model the stochastic behaviorof the channel, i.e.,

|hfp| = |hf ||hp|. (25)

Theorem 1. The PDF of |hfp| can be analytically evalu-ated as

f|hfp|(x) = γ2A−γ2

0

µγ2

α

hαfΓ (µ)xγ

2−1

× Γ

(αµ− γ2

α, µxα

hαfA−α0

). (26)

Proof: Please see Appendix A.

Theorem 2. The CDF of |hfp| can be obtained as

F|hfp|(x) = 1− 1

α

xγ2

hγ2

f

γ2

Aγ2

o

×µ−1∑k=0

µγ2

α

k!Γ

(αk − γ2

α, µxα

hαfA−αo

). (27)

Proof: Please see Appendix B.From (26) and (27), it is evident that the presented distribu-

tion depends on the multipath fading channels characteristics,which are modeled through the parameters α and µ, as wellas the level of misalignment fading that is described via theparameters γ2 and Ao.

A. SPECIAL CASE 1In the special case in which the multipath fading can bemodeled as a Rayleigh distribution, the PDF and CDF of|hfp| can be expressed, by setting α = 2 and µ = 1 in (26)and (27), as

fR|hfp|(x) = γ2A−γ2

0

1

hfxγ

2−1Γ

(2− γ2

2,x2

hfA−2

0

)(28)

and

FR|hfp|(x) = 1− 1

2

xγ2

hγ2

f

γ2

Aγ2

o

Γ

(−γ2

2,x2

h2f

A−2o

), (29)

respectively.

B. SPECIAL CASE 2In the special case in which the multipath fading can bemodeled as a Nakagami-m distribution, the PDF and CDFof |hfp| can be expressed, by setting α = 2 and µ = min (26) and (27), as

fN|hfp|(x) = γ2A−γ2

0

mm− 2m−γ22

h2fΓ (m)

xγ2−1

× Γ

(2m− γ2

α,m

x2

hfA−2

0

)(30)

and

FN|hfp|(x) = 1− 1

2

xγ2

hγ2

f

γ2

Aγ2

o

×m−1∑k=0

mk+ γ2−2k2

k!Γ

(2k − γ2

2,m

x2

h2f

A−2o

),

(31)

respectively.

C. SPECIAL CASE 3In the special case in which the multipath fading can bemodeled as a Weibull distribution, the PDF and CDF of |hfp|can be expressed, by setting µ = 1 in (26) and (27), as

fW|hfp|(x) = γ2A−γ2

0

1

hαfxγ

2−1Γ

(α− γ2

α,xα

hfA−α0

)(32)

and

FW|hfp|(x) = 1− 1

α

xγ2

hγ2

f

γ2

Aγ2

o

Γ

(−γ

2

α,xα

hαfA−αo

), (33)

respectively.

D. SPECIAL CASE 4In the special case in which the THz wireless link suffersonly from shadowing, then according to [63] and [85], |hf |can be modeled as a Gamma distribution, i.e., α = 1, µ = ms

and hf = Ω. Hence, the CDF and the PDF of |hfp| can berespectively expressed as

fG|hfp|(x) = γ2A−γ2

0

mγ2

s

ΩΓ (ms)xγ

2−1

× Γ

(ms − γ2,

msx

ΩA0

)(34)

and

FG|hfp|(x) = 1− xγ2

Ωγ2

γ2

Aγ2

o

×ms−1∑k=0

mγ2

s

k!Γ

(k − γ2,ms

x

ΩAo

), (35)

where Ω stands for the average power of shadowing in thearea of interest and ms is the shadowing severity such that asms increases the shadowing severity decreases. In the limitcase in which ms → ∞, the distribution of |hf | tends to theDirac’s distribution as

f∞|hf |(x) = δ (x− Ω) . (36)

In other words, in the limit case, there is no shadowing effect.

6 VOLUME 4, 2016


IV. PERFORMANCE ANALYSISIn this section, the instantaneous SNR and signal to noiseplus distortion ratio (SNDR) are derived and accordinglythe system’s performance is quantified, in terms of outageprobability and ergodic capacity. In more detail, in Sec-tion IV-A, the instantaneous SNR and SNDR for the cases ofideal and non-ideal RF front-end are respectively provided,whereas, in Section IV-B novel closed-form expressions forthe evaluation of the corresponding outage probabilities aregiven. Additionally, in Section IV-C, an exact closed-formexpression and a simplified upper bound for the ergodiccapacity in the case of ideal RF front-end as well as a tightupper bound and a ceiling for the ergodic capacity in the caseof non-ideal RF front-end are extracted.

A. INSTANTENOUS SNR & SNDRThis section is focused on presenting the instantaneous SNRin the case of ideal RF front-end and the SNDR in the case ofnon-ideal RF front-end.

1) Ideal RF front-endIn the case of ideal RF front-end, based on (1), the instanta-neous SNR of the received signal can be obtained as

ρi =|h|2PNo

. (37)

2) Non-ideal RF front-endThe instantaneous SNDR can be expressed as

ρ =|h|2P

κ2|h|2P +No, (38)

where κ2 = κ2t + κ2

r .

B. OUTAGE PROBABILITY1) Ideal RF front-endIn the case of ideal RF front-end, Proposition 1 providesa novel closed-form expression for the evaluation of theoutage probability in the presence of misalignment fading,while Lemma 1 returns the corresponding expressions in theabsence of misalignment fading.

Proposition 1. In the case of ideal RF front-end, the outageprobability, in the presence of misalignment fading, can beanalytically evaluated as

P ido (γth) = F|hfp|

(√γthNo|hl|2P

), (39)

where γth stands for the SNR threshold.

Proof: The outage probability can be defined as

P ido (γth) = Pr (ρi ≤ γth) , (40)

which, based on (37), can be rewritten as

P ido (γth) = Pr

(|h|2PNo

≤ γth), (41)

or, after some algebraic manipulations as

P ido (γth) = Pr

(|hfp| ≤

√γthNo|hl|2P

), (42)

which can be rewritten as (39). This concludes the proof.

Lemma 1. In the case of ideal RF front-end, the outageprobability, in the absence of misalignment fading, can beobtained as

P id,wmo =

γ

(µ, µ

γα2thN

α2o

|hl|αhαf Pα2

)Γ(µ)

. (43)

Proof: Similar to (42), in the absence of misalignmentfading, the outage probability probability can be expressed as

Po (γth) = Pr

(|hf | ≤

√γthNo|hl|2P

), (44)

or equivalently

Po (γth) = F|hf |

(√γthNo|hl|2P

), (45)

which, by using (66), can be rewritten as in (43). Thisconcludes the proof.

2) Non-ideal RF front-endIn the case of non-ideal RF front-end, Proposition 2 providesa novel closed-form expression for the evaluation of the out-age probability, whereas Lemma 2 returns the correspondingexpressions in the absence of misalignment fading.

Proposition 2. The outage probability can be analyticallyevaluated as

Po (γth) =

F|hfp|

(√γthNo

P |hl|2(1−γthκ2)

), for γth ≤ 1

κ2

1, otherwise(46)

Proof: Please see Appendix C.

Lemma 2. In the case of non-ideal RF front-end, the outageprobability, in the absence of misalignment fading, can beobtained as

Pwmo =

γ

µ,µ γ

α2thN

α2o

|hl|αhαfPα2 (1−γthκ2)

α2

Γ(µ) , for γth ≤ 1

κ2

1, otherwise(47)

Proof: Please see Appendix D.From (43) and (46), we observe that the outage probability

is an increasing function of γth. Moreover, by comparing (39)with (46) and (43) with (47), it becomes evident that themaximum spectrum efficiency of the selected transmissionscheme, which is an increasing function of γth, is limited bythe level of hardware imperfections. In more detail, if a trans-mission scheme is selected that requires a SNR threshold thatis higher than 1

κ2 , then the system is always in an outage state.

VOLUME 4, 2016 7


C. ERGODIC CAPACITY1) Ideal RF front-endIn the case of ideal RF front-end, Theorem 3 returns a novelclosed-form expression for the evaluation of the ergodiccapacity, whereas Theorem 4 provides a simplified upperbound.

Theorem 3. The ergodic capacity can be analytically evalu-ated as in (48), given in the top of the next page. In (48),

P =|hl|2PNo

, (49)

B = γ2 − 1, (50)

K =αµ− γ2

α, (51)

L =µ

hαfAa0

(52)

and

Ξ = γ2A−γ2

0

µµ−µα−γ2α

hαfΓ (µ). (53)

Proof: Please see Appendix E.

Theorem 4. The ergodic capacity can be upper bounded as

Cid ≤ log2

(1 +

P |hl|2

No

γ2A2o

2 + γ2

h2f

µ2α

Γ(

2α + µ

)Γ(µ)

). (54)

Proof: Please see Appendix F.

2) Non-ideal RF front-endIn the case of non-ideal RF front-end, the following theoremreturns an upper bound of the ergodic capacity.

Theorem 5. The ergodic capacity can be upper bounded as

C ≤ log2

1 +P |hl|2 γ

2A2o

2+γ2

h2f

µ2α

Γ( 2α+µ)

Γ(µ)

κ2P |hl|2 γ2A2

o

2+γ2

h2f

µ2α

Γ( 2α+µ)

Γ(µ) +No

. (55)

Proof: In the case of non-ideal RF front-end, the ergodiccapacity can be defined as

C = E[log2

(1 +

p

κ2p+No

)], (56)

where

p = |h|2P. (57)

By applying Jensen’s inequality [86] into (56), we canupper-bound the ergodic capacity as

C ≤ log2

(1 +

E [p]

κ2E [p] +No

), (58)

which, by employing (106) gives (55). This concludes theproof.

The following lemma returns a capacity ceiling in the highSNR regime.

Lemma 3. As the SNR tends to infinity, the ergodic capacityis constrained by

Cc = log2

(1 +

1

κ2

). (59)

Proof: The capacity ceiling can be defined as

Cc = limρ→∞

C, (60)

where ρ represents the average SNR. Based on (55), (60) canbe equivalently expressed as

Cc = limNo→0

log2

1 +P |hl|2 γ

2A2o

2+γ2

h2f

µ2α

Γ( 2α+µ)

Γ(µ)

κ2P |hl|2 γ2A2

o

2+γ2

h2f

µ2α

Γ( 2α+µ)

Γ(µ) +No

,

(61)

which, after some algebraic manipulations, can be rewrittenas (59). This concludes the proof.

From (59), it is evident that maximum achievable ergodiccapacity is constrained by the transceivers level of hardwareimperfections and not by the multipath and/or misalignmentfading characteristics.

V. RESULTS AND DISCUSSIONIn this section, we investigate the joint effects of the de-terministic and stochastic path-gain, i.e., misalignment andmultipath fading, components as well as the impact oftransceivers hardware imperfections in the outage and er-godic capacity performance of the THz wireless fiber exten-der by illustrating analytical and simulation results. In par-ticular, we consider the following insightful scenario. Unlessotherwise is stated, it is assumed that Gt = Gr = 55 dBi2,α = 2, and µ = 4. Moreover, standard environmentalconditions, i.e., φ = 50%, p = 101325 Pa, and T = 296 oK,are assumed. Furthermore, we consider that the transmissiondistance ranges between 10 and 100 m. Finally, unless oth-erwise is stated, in all the illustrations,the solid curves repre-sent analytical values obtained throughthe derived formulas,while the markers represent Monte-Carlo simulation results.

A. IMPACT OF THE DETERMINISTIC PATH-GAINFig. 2 depicts the outage probability as a function of thetransmission distance, d, for different values of f and P

No,

assuming ideal RF front-end and σs = 0.01 m. As expected,for fixed f and P

No, as d increases, the outage probability

also increases. For instance, for f = 275 GHz and PNo

=10 dB, the outage probability increases for about 96%, asthe distance alters from 10 to 15 m. This indicates that evenrelatively small changes in the transmission distance can

2According to [14], [41] and [87]–[89], this antenna gain can be prac-tically achieved by employing high-gain Cassegrain antennas, which arewidely used for wireless fiber extender appliances in the millimeter andTHz bands.

8 VOLUME 4, 2016


Cid =ΞP−B+1

2

2 ln (2)H4,1

3,4

[L

P α2

∣∣∣∣ (−B+1

2 , α2),(1− B+1

2 , α2), (1, 1)

(0, 1) , (K, 1) ,(−B+1

2 , α2),(−B+1

2 , α2) ] (48)

10 20 30 40 50 60 70 80 90 10010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

P/No =25 dB

Out

age

Prob

abili

ty

d (m)

f = 275 GHz f = 300 GHz f = 350 GHz f = 375 GHz f = 400 GHz

P/No =10 dB

FIGURE 2: Outage probability vs transmission distance, fordifferent values of f and normalized transmission powerequals 10 and 25 dB.

FIGURE 3: Outage probability vs relative humidity andtemperature, assuming ideal RF front-end, f = 300 GHz,d = 10 m and P

No= 50 dB, γth = 0.5, and σs = 0.01 m.

result in significant alterations on the system’s outage perfor-mance. Additionally, for a given d and P

No, we observe that

as the frequency increases, the propagation loss increases, orequivalently the deterministic path-gain decreases; hence, theoutage probability increases. Finally, for a fixed d and f , asPNo

increases, the outage probability decreases. For example,for d = 15 m and f = 300 GHz, by changing the P

Nofrom 10 to 25 dB, the outage probability is decreased from1.66786× 10−5 to 1.8633× 10−11.

Fig. 3 illustrates the impact of atmospheric conditionson the system’s outage performance. In particular, the out-

FIGURE 4: Capacity vs distance and relative humidity, as-suming ideal RF front-end, f = 380 GHz, P

No= 25 dB,

and σs = 0.01 m.

age probability is plotted as a function of the relative hu-midity and the temperature, assuming ideal RF front-end,f = 300 GHz, d = 10 m and P

No= 50 dB, γth = 0.5,

and σs = 0.01 m. From this figure, it is evident that,for a given relative humidity, as the temperature increases,the molecular absorption also increases; hence, the outageprobability increases. For instance, for φ = 50%, a tem-perature alteration from 280 oK to 290 oK results in about1.8% outage performance degradation. Moreover, we ob-serve that for a fixed temperature, as the relative humidityincreases, the outage probability also increases. For example,for T = 280 oK, when the relative humidity shifts from 50%to 60%, a 0.39% outage probability increase occurs. In otherwords, we observe that the temperature alteration can resultto a slightly more significant effect on the system’s outageperformance compared to the humidity alteration. Moreover,these results reveal the importance of taking into account theatmospheric conditions, when designing and establishing aTHz wireless link.

Fig. 4 shows the ergodic capacity as a function of thetransmission distance and the relative humidity, assumingideal RF front-end, f = 380 GHz, P

No= 25 dB and σs =

0.01 m. As expected, for a fixed relative humidity, as thetransmission distance increases, the deterministic path-lossincreases; hence, the ergodic capacity decreases. Similarly,for a given transmission distance, as the relative humidityincreases, the absorption loss also increases; therefore, theergodic capacity decreases. Finally, we observe that the im-pact of propagation loss is more severe compared to the oneof the molecular absorption loss, since the ergodic capacitydegradation due to a transmission distance increase is much

VOLUME 4, 2016 9


FIGURE 5: Capacity vs frequency and relative humidity,assuming ideal RF front-end, d = 30 m, P

No= 25 dB, and

σs = 0.01 m.

more severe than the corresponding degradation due to arelative humidity increase.

Fig. 5 depicts the ergodic capacity as a function of thefrequency and the relative humidity, assuming ideal RF front-end, d = 30 GHz, P

No= 25 dB and σs = 0.01 m. From this

figure, it is evident that the impact of the level of humidityon the ergodic capacity depends on the operation frequency.In more detail, we observe that up to 320 GHz the effect ofrelative humidity alteration is relatively low, whereas in the380 GHz region, it is detrimental. For example, in 300 GHz,a relative humidity alteration from 30% to 60% results toa 0.1% ergodic capacity degradation, whereas the samealteration in the 380 GHz causes a corresponding 98.9%decrease. Moreover, in the 400 GHz region, although the freespace path loss is higher compared to the one in the 380 GHzregion, the achievable ergodic capacity is much higher. Thisis the impact of the high molecular in the 380 GHz, whichis one of the water vapor resonance frequencies [23], [90].These observations indicate the importance of taking intoaccount the variation of the environmental conditions in ageographical area as well as the molecular absorption phe-nomenon, when selecting the operation frequency.

Fig. 6 demonstrates the outage probability as a function ofthe operation frequency for different values of γth and P

No,

assuming ideal RF fron-end, d = 20 m and σs = 0.01 m. Asexpected, for a fixed frequency, as γth increases, the outageprobability also increases. For instance, for f = 380 GHzand P

No= 20 dB, an 89.38% increase of the outage proba-

bility is observed, as the γth alters from 0.5 to 2. Moreover,this figure reveals the frequency dependency of the system’soutage performance as well as that the worst operation fre-quency is 380 GHz.

B. IMPACT OF MULTIPATH FADINGIn Fig. 7, the outage probability is depicted as a function ofP |hl|2No

for different values of µ and γth, assuming ideal RF

280 290 300 310 320 330 340 350 360 370 380 390 4001E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

Out

age

Prob

abili

ty

f (GHz)

th = 0.5

th = 1

th = 1.5

th =2

P/No = 10 dB

P/No = 20 dB

FIGURE 6: Outage probability vs the operation frequency fordifferent values of γth and P

No, assuming ideal RF front-end

and d = 20 m.

FIGURE 7: Outage probability vs P |hl|2No

for different valuesof µ and γth, assuming ideal RF front-end and d = 30 m.

front-end, d = 30 m and γ2 = 1. As a benchmark, the worstcase scenario of Rayleigh fading is plotted. From this figure,we observe that, for a given µ and γth, as P |hl|

2

Noincreases, the

outage probability decreases. Moreover, for a given P |hl|2No

and µ, as γth increases, the outage probability also increases.For instance, for P |hl|2

No= 40 dB and µ = 8, the outage

probability increases for about 287.4%, as γth alters from 1

to 15. Finally, for a fixed P |hl|2No

and γth, as µ increases, i.e.,as the strength of the LoS component increases, the system’soutage behavior improves.

Fig. 8 shows the joint impact of multipath and misalign-ment fading on the ergodic capacity. Specifically, the ergodiccapacity is plotted as a function of P

Nofor different values

of µ and σs, assuming ideal RF front-end and d = 20 m.Note that, in this figure, the numerical results are shown with

10 VOLUME 4, 2016


FIGURE 8: Capacity vs PNo

for different values of µ and σs,assuming ideal RF front-end, α = 2 and d = 30 m.

continuous lines, while markers are employed to illustrate thesimulation results. The analytical results coincide with thesimulations, verifying the derived expressions. Moreover, theergodic capacity for the special case in which the multipathfading follows Rayleigh distribution is plotted as a bench-mark. This figure reveals that the impact of misalignmentfading is somewhat more detrimental compared to the one ofthe multipath fading. Moreover, we observe that the increaseof the transmission power can act as a countermeasure tothe joint impact of multipath and misalignment fading in theergodic capacity.

C. IMPACT OF MISALIGNMENT FADING

Fig. 9 illustrates the impact of misalignment fading on thesystem’s outage performance. In more detail, the outageprobability is provided as a function of the spatial jitter,σs, for different values of P

No, assuming ideal RF front-end,

d = 30 m, γth = 1, α = 2 and µ = 4, for the cases in whichthe system suffers from misalignment fading (continuouslines) or not (dashed lines). As expected, for a given P

No,

as σs increases, the outage probability also increases. Forexample, for P

No= 10 dB, a σs alteration from 0.02 to 0.05

causes a 1298.1% outage performance degradation, whereas,for P

No= 20 dB, the same σs alteration results to a 264040%

outage performance degradation. This indicates that the im-pact of misalignment increases as the P

Noincreases.

Fig. 10 depicts the ergodic capacity versus σs for differentvalues of P

No, assuming ideal RF front-end, d = 30 m,

α = 2 and µ = 4. In this figure, the numerical results areshown with continuous lines, while markers are employed toillustrate the simulation results. It becomes evident from thisfigure that the analytical results are identical with simulationresults; thus, verifying the presented analytical framework.Moreover, it is observed that the spatial jitter causes a signifi-

FIGURE 9: Outage probability vs σs for different values ofPNo

, assuming ideal RF front-end, d = 30 m, γth = 1, α = 2and µ = 4, for the cases in which the system suffers frommisalignment fading (continuous lines) or not (dashed lines).

FIGURE 10: Capacity vs σs for different values of PNo

,assuming ideal RF front-end, d = 30 m, α = 2 and µ = 4.

cant ergodic capacity degradation. In more detail, we observethat, for a given P

No, as σs increases, the ergodic capacity

decreases. Furthermore, for a given σs, as PNo

increases, theRX antenna is able to collect more of the emitted power;hence, the ergodic capacity also increases. For instance, forσs = 0.05, a P

Noincrease from 10 to 20 dB results to a

134.63% ergodic capacity improvement.

D. IMPACT OF HARDWARE IMPERFECTIONSFig. 11 illustrates the impact of hardware imperfections onthe system’s outage performance. Specifically, the outageprobability is plotted as a function of P |hl|2

Nofor different

VOLUME 4, 2016 11


FIGURE 11: Outage probability vs P |hl|2

Nofor different values

of κtr = κt = κr and γth, assuming α = 2, µ = 4 andd = 20 m.

values of κtr = κt = κr and γth, assuming α = 2, µ = 4and d = 20 m. As a benchmark, the corresponding outageprobability in the case of ideal RF front-end is depicted.From this figure, it is evident that, for a given P |hl|2

Noand γth,

as the level of hardware imperfection increases, the outageprobability also increases. Moreover, for a fixed P |hl|2

No, as

γth increases, the impact of hardware imperfections becomemore severe. For example, for P |hl|2

No= 30 dB and γth = 1,

the outage probability increases for about 9.3%, when κtralters from 0.10 to 0.30, whereas, for the same P |hl|2

Noand κtr

alteration, the outage probability increases for about 200%,when γth = 5. This indicates that by using transmissionschemes with lower spectral efficiency, we can significantlyconstrain the impact of hardware RF imperfections. Finally,in the worst case scenario in which κtr = 0.4 and γth = 5,the condition γth ≥ 1

κ2 is satisfied, and the outage probabilityis always 1.

Fig. 12 demonstrates the joint effect of hardware imper-fections and misalignment fading on the system’s outageperformance. In particular, the outage probability is givenas a function of κtr, for different values of σs, assumingd = 30 m, P

No= 25 dB, γth = 5, α = 2 and µ = 4.

As a benchmark, the corresponding outage probability inthe case in which there is no misalignment fading is alsoplotted. From this figure, we observe that for a given κtr,as σs increases, i.e., as the impact of misalignment becomesmore severe, the outage probability also increases. For exam-ple, for κtr = 0.10, the outage probability increases from10−5 to 1.25 × 10−3, as σs shifts from 0.01 to 0.04 m.Moreover, in the low σs regime, it is shown that the outageperformance degradation created by the RF imperfections issomewhat more severe compared with the detrimental effectsof misalignment fading. For instance, for the case of κth =

FIGURE 12: Outage probability vs κtr, for different valuesof σs, assuming d = 30 m, P

No= 25 dB, γth = 5, α = 2

and µ = 4.

0.10, the doubling of the level of misalignment from 0.01to 0.02 m causes a 54.58% outage probability degradation,whereas, for σs = 0.01 m, the doubling of the level ofhardware imperfections from κtr = 0.10 to 0.20 results to a381.01% outage probability degradation. On the other hand,in the relatively high σs regime, the outage performancedegradation created by the RF imperfections is somewhat lesssevere compared with the detrimental effects of misalign-ment fading. Finally, in the case of ideal RF front-end, i.e.,κtr = 0, the assumption of perfect transceivers alignmentresults to around 402.19% error in the corresponding outageprobability for σs = 0.01 m. These results highlight the im-portance of both accurate misalignment characterization andaccounting for RF impairments in the realistic performanceanalysis and design of THz wireless fiber extender systems.

In Fig. 13, the ergodic capacity is plotted as a function PNo

,for different values of κtr, assuming d = 30 m, σs = 0.01 m,α = 2 and µ = 4. In this figure, the continuous linesrepresent Monte Carlo simulation results, while the dashedlines stand for the capacity upper bound, which is derivedby (54) and (55). Additionally, the dashed-dotted lines denotethe capacity ceiling, given by (59). Note that, as a benchmark,the capacity versus P

Nofor the ideal RF chains case is also

plotted. As expected, as PNo

increases, the ergodic capacityalso increases. However, in the case of non-ideal RF chains,the ergodic capacity saturates as it approaches the capacityceiling, as proven from Lemma 3. As a result, since thecapacity ceiling is determined by the level of imperfections,it increases as κtr decreases. Moreover, we observe that thehardware imperfections have a small effect on the ergodiccapacity at the low P

Noregime, whereas, at the high P

Noregime their impact is detrimental. Finally, from this figure, itis evident that the upper bounds derived by Theorems 4 and 5

12 VOLUME 4, 2016


FIGURE 13: Capacity vs PNo

, for different values of κtr,assuming d = 30 m, σs = 0.01 m, α = 2 and µ = 4.

are relatively tight; especially, in the high PNo

regime.

VI. CONCLUSIONSWe studied the performance of wireless THz fiber extenders,when the RF front-end is impaired by hardware imperfec-tions and the transceivers antennas are not fully-aligned. Inparticular, by assuming α − µ fading and Gaussian distri-butions for the elevation and horizontal displacement, weprovided a general analytical framework for evaluating theoutage probability for both cases of ideal and non-ideal RFfront-end. Next, we presented novel closed-form expressionsfor the ergodic capacity in the case of ideal RF front-end aswell as upper bounds for both cases of ideal and non-idealRF front-end. Likewise, a simple capacity ceiling for the non-ideal RF front-end case was extracted. Our results illustratedthe degradation due to the joint effect of misalignment fadingand RF imperfections on the outage and capacity perfor-mance of the THz wireless extender. Among other, it wasrevealed that the impact of misalignment fading is somewhatmore detrimental compared to the one of the multipath fad-ing, whereas the performance degradation created by the RFimperfections is more severe compared with the detrimentaleffect of misalignment fading. Finally, the importance of bothaccurate misalignment characterization and accounting forRF impairments in the realistic performance analysis anddesign of THz wireless fiber extenders was highlighted.

APPENDICESAPPENDIX APROOF OF THEOREM 1According to [91], the PDF of |hfp| can be obtained as

f|hfp|(x) =

ˆ Ao

0

1

yf|hf |

(x

y

)f|hm|(y)dy. (62)

By substituting (22) and (24) into (62) and after some alge-braic manipulations, the PDF of |hfp| can be equivalentlyexpressed as

f|hfp|(x) =αµµxαµ−1(hf

)αµΓ(µ)

γ2

Aγ2

o

I(x), (63)

where

I(x) =

ˆ Ao

0

yγ2−αµ−1 exp

(−µx

αy−α

hα

)dy, (64)

which, by employing [64, eq. (8/350/1)], can be rewritten asin (26). This concludes the proof.

APPENDIX BPROOF OF THEOREM 2Since |hf | and |hp| are independent random variables, theCDF of |hfp| can be derived as

F|hfp|(x) =

ˆ Ao

0

F|hf |

(x

y

)f|hp|(y)dy, (65)

where f|hp|(x) is the PDF of |hp|, while F|hf |(x) stand forthe CDF of |hf |. Since |hf | follows α-µ distribution, its CDFcan be expressed as [92, eq. (8)]

F|hf |(x) = 1−Γ

(µ, µx

α

hαf

)Γ(µ)

, (66)

which, by assuming that µ is an integer and employing [64,eq. (8.352/2)], (66) can be rewritten as

F|hf |(x) = 1−µ−1∑k=0

1

k!

(µxα

hαf

)kexp

(−µx

α

hαf

). (67)

Moreover, by substituting (22) and (67) into (65), the CDF of|hfp| can be equivalently expressed as

F|hfp|(x) = K −µ−1∑k=0

Jk(x), (68)

where

K =

ˆ Ao

0

f|hp|(y)dy (69)

and

Jk(x) =µk

k!

xαk(hf

)αk γ2

Aγ2

o

×ˆ Ao

0

yγ2−αk−1 exp

(−µx

α

hαfy−α

)dy. (70)

Next, we evaluate each term that composes (68). Sincef|hp|(y) represents the PDF of |hp|,

K = 1. (71)

Moreover, by setting

z = A(x) y−α, (72)

VOLUME 4, 2016 13


with

A(x) = µxα

hαf, (73)

(70) can be rewritten as

Jk(x) =µk

k!

xαk(hf

)αk γ2

Aγ2

o

(A(x))γ2−αkκ

α

×ˆ ∞A(x)Aαo

z−γ2−αkα −1 exp (−z) dz,

(74)

which, according to [64, eq. (8.350/2)], can be evaluated as

Jk(x) =µk

k!

xαk(hf

)αk γ2

Aγ2

o

(A(x))γ2−αkα

α

× Γ

(αk − γ2

α,A(x)

Aαo

). (75)

Additionally, by employing (73), (75) can be expressed as

Jk(x) =1

α

µk+ γ2−αkα

k!

xγ2

hγ2

f

γ2

Aγ2

o

Γ

αk − γ2

α,µx

α

hαf

Aαo

.

(76)

Finally, by substituting (71) and (76) into (68), and aftersome algebraic manipulations, we get (27). This concludesthe proof.

APPENDIX CPROOF OF PROPOSITION 2In the case of non-ideal RF front-end, the outage probabilitycan be defined as

Po (γth) = Pr (ρ ≤ γth) , (77)

which, based on (38), can be rewritten as

Po (γth) = Pr

(|h|2P

κ2|h|2P +No≤ γth

), (78)

or equivalently

Po (γth) = Pr(|h|2P

(1− κ2γth

)≤ No

). (79)

By employing (4), (79) can be expressed as

Po (γth) = Pr

(|hfp|2

(1− κ2γth

)≤ NoP |hl|2

). (80)

Moreover, by assuming that 1 − κ2γth ≥ 0, i.e., γth ≤ 1κ2 ,

the outage probability can be derived as

Po (γth) = Pr

(|hfp|2 ≤

NoP |hl|2 (1− κ2γth)

), (81)

or

Po (γth) = F|hfp|

(√No

P |hl|2 (1− κ2γth)

). (82)

Finally, if 1− κ2γth < 0, then the condition

|hfp|2(1− κ2γth

)≤ 0 ≤ No

P |hl|2(83)

is always valid and the outage probability equals 1. Thisconcludes the proof.

APPENDIX DPROOF OF LEMMA 2Similar to (82), in the absence of misalignment fading, if1 − κ2γth ≥ 0 the outage probability probability can beexpressed as

Pwmo (γth) = Pr

(|hf | ≤

√No

P |hl|2 (1− κ2γth)

), (84)

or equivalently

Pwmo (γth) = F|hf |

(√No

P |hl|2 (1− κ2γth)

), (85)

which, by using (66), can be rewritten as (47). Moreover, if1 − κ2γth < 0, then, similarly to (83), it can be proven thatthe outage probability equals 1. This concludes the proof.

APPENDIX EPROOF OF THEOREM 3In the case of ideal RF front-end, the ergodic capacity can bedefined as

Cid = E [log2 (1 + ρ)] , (86)

or

Cid = E[log2

(1 + P|hfp|2

)], (87)

which can be evaluated as

Cid =

ˆ ∞0

log2

(1 + Px2

)f|hfp|(x)dx. (88)

By substituting (26) into (88), and after some algebraicmanipulations, the ergodic capacity can be expressed as

Cid =Ξ

ln (2)

ˆ ∞0

xB ln(1 + Px2

)Γ (K,Lxα) dx, (89)

or, by using [65, eq. (15.1.1)], as

Cid =Ξ

ln (2)

ˆ ∞0

xB+12F1 [1, 1; 2;−x] Γ (K,Lxα) dx.

(90)

Moreover, by employing [93, eq. (11)] and [64, eq.(9.34/7)], (90) can be written as

Cid =Ξ

ln (2)

ˆ ∞0

xBG1,22,2

[Px2

∣∣∣∣ 1, 11, 0

]×G2,0

1,2

[Lxα

∣∣∣∣ 10,K

]dx, (91)

14 VOLUME 4, 2016


which, by setting z = x2, can be expressed as

Cid =Ξ

2 ln (2)

ˆ ∞0

zB+12 −1G1,2

2,2

[Pz∣∣∣∣ 1, 1

1, 0

]×G2,0

1,2

[Lz

α2

∣∣∣∣ 10,K

]dz. (92)

Finally, by using [94] into (92), the ergodic capacity can berewritten as (48). This concludes the proof.

APPENDIX FPROOF OF THEOREM 4Based on (86), the ergodic capacity can be rewritten as

Cid = E[log2

(1 +

p

No

)], (93)

where

p = |h|2P. (94)

By applying Jensen’s inequality [86] into (93), we canupper-bound the ergodic capacity as

C ≤ log2

(1 +

E [p]

No

). (95)

Next, we analytically evaluate E [p]. Since |hp| and |hf |are independent random variables, |hp|2 and |hf |2 are alsoindependent random variables; hence,

E [p] = PE[|hl|2|hf |2|hp|2

], (96)

can be rewritten as

E [p] = P |hl|2E[|hf |2

]E[|hp|2

]. (97)

The expected value of |hf |2 can be evaluated as

E[|hf |2

]=

ˆ ∞0

x2f|hf |(x)dx, (98)

which, by employing (24), can be equivalently written as

E[|hf |2

]=

αµµ

hαµf Γ(µ)L, (99)

where

L =

ˆ ∞0

xαµ+1 exp

(−µx

α

hαf

)dx. (100)

By setting u = µxα

hαfand employing [64, eq. (8.350/2)], (100)

can be analytically evaluated as

L =1

α

(µ

hαf

)−αµ+2α

Γ

(2

α+ µ

). (101)

By substituting (101) into (99), we can express the expectedvalue of |hf |2 as

E[|hf |2

]=

h2f

µ2α

Γ(

2α + µ

)Γ(µ)

. (102)

Similarly, the expected value of |hp|2 can be evaluated as

E[|hp|2

]=

ˆ Ao

0

x2f|hp|(x)dx, (103)

which, by employing (22), can be rewritten as

E[|hp|2

]=

γ2

Aγ2

o

ˆ Ao

0

xγ2+1dx, (104)

or

E[|hp|2

]=

γ2

2 + γ2A2o. (105)

Moreover, by using (105) and (102), (97) can be written as

E [p] = P |hl|2γ2A2

o

2 + γ2

h2f

µ2α

Γ(

2α + µ

)Γ(µ)

. (106)

Finally, by substituting (106) into (95), we get (54). Thisconcludes the proof.

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VOLUME 4, 2016 17


ALEXANDROS-APOSTOLOS A. BOULOGE-ORGOS (S’11–M’16) was born in Trikala,Greece in 1988. He received the Electrical andComputer Engineering (ECE) diploma degree andPh.D. degree in Wireless Communications fromthe Aristotle University of Thessaloniki (AUTh) in2012 and 2016, respectively.

From November 2012, he has been a memberof the wireless communications system group ofAUTh, working as a research assistant/project en-

gineer in various telecommunication and networks projects. During 2017, hejoined the information technologies institute, while from November 2017, hehas joined the Department of Digital Systems, University of Piraeus, wherehe conducts research in the area of wireless communications. Moreover,from October 2012 until September 2016, he was a teaching assistant at thedepartment of ECE of AUTh, whereas, from February 2017, he serves as anadjacent lecturer at the Department of Informatics and TelecommunicationsEngineering of the University of Western Macedonia and as an visitinglecturer at the Department of Computer Science and Biomedical Informaticsof the University of Thessaly.

Dr. Boulogeorgos has authored and co-authored more than 35 techni-cal papers, which were published in scientific journals and presented atprestigious international conferences. Furthermore, he has submitted two(one national and one European) patents. Likewise, he has been involvedas member of Technical Program Committees in several IEEE and non-IEEE conferences and served as a reviewer in various IEEE journals andconferences. Dr. Boulogeorgos was awarded with the “Distinction Scholar-ship Award” of the Research Committee of AUTh for the year 2014 and wasrecognized as an exemplary reviewer for IEEE Communication Letters for2016 (top 3% of reviewers). His current research interests spans in the areaof wireless communications and networks with emphasis in high frequencycommunications, optical wireless communications and communications forbiomedical applications. He is a member of the IEEE and the TechnicalChamber of Greece.

EVANGELOS N. PAPASOTIRIOU was born inAthens, Greece in 1992. He received the bachelordegree of Computer Science from the Departmentof Computer Science and Biomedical Informaticsfrom the University of Thessaly in 2016 and themaster’s degree in Digital Communications andNetworks from the Department of Digital Systemsof the University of Piraeus in 2018. From October2017 he has joined the Department of DigitalSystems, University of Piraeus, where he conducts

research in the area of wireless communications. Moreover since May 2018he is pursuing a Ph.D. degree in wireless Communications.

ANGELIKI ALEXIOU received the Diploma inElectrical and Computer Engineering from the Na-tional Technical University of Athens in 1994 andthe PhD in Electrical Engineering from ImperialCollege of Science, Technology and Medicine,University of London in 2000. Since May 2009she is faculty member at the Department of Dig-ital Systems, University of Piraeus, where sheconducts research and teaches undergraduate andpostgraduate courses in the area of Broadband

Communications and Advanced Wireless Technologies. Prior to this ap-pointment she was with Bell Laboratories, Wireless Research, LucentTechnologies, now Alcatel-Lucent, in Swindon, UK, first as a member oftechnical staff (January 1999- February 2006) and later as a TechnicalManager (March 2006-April 2009). Prof Alexiou is a co-recipient of BellLabs Presidents Gold Award in 2002 for contributions to Bell Labs LayeredSpace-Time (BLAST) project and the Central Bell Labs Teamwork Awardin 2004 for role model teamwork and technical achievements in the ISTFITNESS project. Prof Alexiou is the elected Chair of the Working Group onRadio Communication Technologies of the Wireless World Research Forum.She is a member of the IEEE and the Technical Chamber of Greece. Hercurrent research interests include radio interface, MIMO and high frequen-cies (mmWave and THz wireless) technologies, cooperation, coordinationand efficient resource management for Ultra Dense wireless networks andmachine-to-machine communications, and ‘cell-less’ architectures basedon softwarization, virtualization and extreme resources sharing. She is theproject coordinator of the H2020 TERRANOVA project (ict-terranova.eu).

18 VOLUME 4, 2016

analytical performance assessment of thz ... - terranova

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