Non-coherent assive SIO Systems in ISI Channes: Consteation Design and Performance Anaysis Huiqiang Xie, Weiyang Xu, ember, IEEE, Wei Xiang, Senior ember, IEEE, e Shao, Shengbo Xu arxiv:89946v [eesssp] 9 Sep 8 Abstract A massive singe-input mutipe-output SIO system with a singe transmit antenna and a arge number of receive antennas in intersymbo interference ISI channes is considered Contrast to existing energy detection ED-based noncoherent receiver where conventiona puse ampitude moduation PA is empoyed, we propose a consteation design which minimizes the symbo-error rate SER with the knowedge of channe statistics To make a comparison, we derive the SERs of the ED-based receiver with both the proposed consteation and PA, namey P e opt and P e pam Specificay, asymptotic behaviors of the SER in regimes of a arge number of receive antennas and high signa-to-noise ratio SNR are investigated Anaytica resuts demonstrate that the ogarithms of both P e opt and P e pam decrease approximatey ineary with the number of receive antennas, whie P e opt degrades faster It is aso shown that the proposed design is of ess cost, because compared with PA, ess antennas are required to achieve the same error rate Index Terms Energy detection, intersymbo interference ISI channe, massive singe-input mutipe-output SIO, consteation design, symbo-error rate SER I INTRODUCTION ASSIVE mutipe-input mutipe-output IO systems, where a arge number of antennas are depoyed at base stations BSs to serve a sma number of users sharing the same frequency resources, have recenty received a great dea of interest due to its potentia gains [], [] For exampe, massive IO is energy efficient as the transmit power scaes down with the number of antennas eanwhie, channe vectors associated with different users are asymptoticay orthogona, thus both intra- and inter-ce interference can be eiminated with simpe detection or precoding agorithms [3] [5] To reap the benefits that massive IO offers, channe state information CSI is required at BSs in coherent communications However, massive antennas make acquiring CSI much more chaenging than before In fact, the computationa This work was supported by the Program for Innovation Team Buiding at Coeges and Universities in Chongqing, China Grant No CXTDX66 and the ey Program of Natura Science Foundation of Chongqing under Grant CSTC7JCYJBX47 H Q Xie, W Y Xu and S B Xu are with the Coege of Communication Engineering, Chongqing University, Chongqing, 444, P R China E-mais: {huiqiangxie, weiyangxu, 3437}@cqueducn Wei Xiang is with the Coege of Science, Technoogy & Engineering, James Cook University, Cairns, QLD 487, Austraia E-mai: weixiang@jcueduau e Shao is with the Nanchang Institute of Technoogy, Nanchang, Jiangxi, 339, P R China E-mai: shaoke@63com compexity of channe estimation is so high that estimating a channes in a timey manner becomes infeasibe In addition, the issue of piot contamination, attributed to reusing piots among adjacent ces, woud make the probem even worse because channe estimates obtained in a given ce wi be corrupted by piots transmitted by users in the other ces [6] As a promising aternative, non-coherent systems require no knowedge of instantaneous CSI at either the transmitter or receiver [7] [9] Compared with their coherent counterparts, non-coherent receivers enjoy benefits of ow compexity, ow power consumption and simpe structures at the cost of a sub-optima performance [] Thus, non-coherent receivers are attractive in arge-scae antenna systems Based on the non-overapping power-space profie without CSI, an optima decision-feedback differentia detector DFDD provides significant gains over conventiona differentia detection [], [] However, the DFDD reies on a particuar channe mode that cannot be expoited in genera With a arge number of antennas, energy detection ED finds its appication in noncoherent massive singe-input mutipe-output SIO systems [3] With a non-negative puse ampitude moduation PA, the transmit symbos can be decoded by averaging the received signa power across a antennas Since the detection/decoding is performed based on the signa energy, the system shoud use non-negative signa consteations For exampe, non-negative PA consteations have been documented for two different wireess standards for iimeterwave mmwave short-range communication, ECA-387 and IEEE853c [4], [5], respectivey Regarding systems with a arge number of antennas, ED was proposed for mmwave communications in [6] Inspired by the semina work in [3], non-coherent massive SIO systems have attracted a ot of attention from the research community [7] [3] The symbo-error rate SER of the ED-based non-coherent SIO system is derived in [7], based on which a minimum distance consteation is presented An asymptoticay optima consteation is proposed in [8], and its performance gap to the optima design coud be made sma with arge-scae antennas or arge-size consteations In [9], two ED-based receivers are proposed, one of which anayzes the instantaneous channe energy based on Gaussian approximations of the probabiity density function PDF, whie the other anayzes the average channe energy with chisquare cumuative distribution function CDF The authors in [] propose to optimize the consteation with varying eves of uncertainty on channe statistics Aso, it is proved
that the non-coherent massive SIO system satisfies the same scaing aw as its coherent counterpart [3] However, the aforementioned studies focus on intersymbo interference ISI free scenarios [7] [], [3] Athough an ISI channe can be transformed into mutipe fat-fading channes by using OFD, the inherent high peak-to-average-ratio and sensitivity to the carrier frequency offset present new chaenges Different from the above, the authors in [] consider the use of the ED-based receiver in mutipath environments, where a zeroforcing ZF equaizer is empoyed to remove ISI In non-coherent massive SIO systems, optimizing the consteation design can provide significant performance enhancement over conventiona PA Accordingy, anoakos et a propose an optima consteation to improve the error performance in fat-fading channes [] Athough both anaytica and numerica resuts show the potentia of consteation optimization in ISI-free communications, whether it coud reduce the error rate and what the optima design woud be in mutipath scenarios are not cear Towards this end, this paper focuses on designing a consteation that is abe to minimize the SER for the ED-based SIO system with ISI The main contributions of our study can be summarized as foows: In the presence of mutipath channes, the generic SER P e of the ED-based SIO receiver, which reates to the consteation and decision threshods, for the case of a finite number of receive antennas is derived; Based upon the derived cosed-form expression of P e, we present a consteation design and decoding scheme with the objective of minimizing the error probabiity Then, the SERs of both the proposed design and PA, namey P e opt and P e pam, are derived for comparison; Asymptotic behaviors of P e opt and P e pam in regimes of a arge number of receive antennas and high signa-tonoise ratios SNRs are investigated in detai It is shown that the ogarithm of P e opt can be approximated as a ineary decreasing function of the number of antennas, and decreases at a faster rate than the ogarithm of P e pam when more antennas are equipped Due to the mutipath effect and a finite number of antennas, both P e opt and P e pam exhibit an irreducibe error foor at high SNRs However, P e opt converges to a rate much ower than P e pam under the same condition The remainder of this paper is organized as foows The system mode is presented in Section II The derivation of SER, optima consteation design and threshod setting are detaied in Section III Section IV presents a thorough SER performance anaysis under different scenarios Numerica simuation resuts are presented to show the effectiveness of our agorithm in Section V Finay, Section VI concudes this paper Notation: C n m indicates a matrix composed of compex numbers of size n m Bod-font variabes represent matrices or vectors For a random variabe x, x CN µ, σ means it foows a compex Gaussian distribution with mean µ and P e, P e pam and P e opt refer to the error probabiities corresponding to genera non-negative consteations, PA and the proposed consteation, respectivey covariance σ E[ ], Var[ ], Cov[ ] and denote the expectation, variance, covariance and L norm of the argument, respectivey H denotes the Hermitian transpose R{ } and I{ } separatey refer to the rea and imaginary parts of a compex number sup is the east upper bound Finay, erf and erfc are taken to indicate the Gaussian error function and compementary Gaussian error function, respectivey II SYSTE ODEL Consider a SIO network consisting of a singe-antenna transmitter and a receiver with antennas [] The channe between the transmitter and each receive antenna is modeed as a finite impuse response FIR fiter with L taps [4] We assume an independent channe reaization from one bock to another The received signa at time t can be represented by L yt = h st + nt = where yt C, nt C indicates a compex Gaussian noise vector with eements n i CN, σ n, h C refers to the channe reaization of the th path with h,i CN, σ h, st denotes the transmit symbo drawn from a certain non-negative consteation, and is the number of receive antennas Our study considers the transmit SNR, which is defined as SNR = E[ st ]/σ n = /σ n We focus on the foowing encoding and decoding scheme It is assumed that both the receiver and transmitter possess no knowedge of instantaneous channe and noise, but the channe and noise statistics are avaiabe, ie, means and variances The non-negative transmit symbo st is seected from a consteation set P = { p, p,, p }, subject to the average power constraint p i i= where p i denotes the ith consteation point and is the consteation size Based on the ED principe, after the received signa having been fitered, squared and integrated, the average power across a antennas can be written as zt = yt 3 In fat-fading channe, zt is taken as the decision metric for symbo detection [3] Accordingy, the positive ine is partitioned into mutipe decoding regions to decide which symbo was transmitted according to the observation of zt In fact, zt can be approximated to one of the Gaussian variabes depending on a priori information of transmitted symbos For exampe, with a PA consteation of = 4, the PDF of zt over an additive white Gaussian noise AWGN channe is shown in Fig a, where = and SNR = 4 db Ceary, four distinct Gaussian-ike curves fzt p i i=,,3,4 can be observed, corresponding to four consteation points As can be seen from this figure, there is a notabe overap region between fzt p and fzt p, caused by additive noise and a finite number of antennas Furthermore, Figs a and b indicate that the overap region enarges when the number
fztjpi 3 AWGN Channe 4 fztjpi L 5 5 5 3 35 4 a 4-Path ISI Interference, = 4 p p p3 p4 fztjpi number of antennas can be expanded as [] p p p3 p4 5 5 5 3 35 4 b 4-Path ISI Interference, = 4 5 5 5 3 35 4 c Fig Conditiona PDFs of zt in different scenarios, where SNR = 4 db for a -bit non-negative PA consteation a AWGN channe, = ; b 4-path ISI channe, = ; and c 4-path ISI channe, = RX- Energy coection RX- ZF equaizer X kh k st + kh k st = {z } ISI X + < st st hh h 6= {z } ISI p p p3 p4 zt = Decoding RX- Fig Configuration of a non-coherent massive SIO receiver with nonnegative consteations of channe taps increases This overap wi make it difficut to separate these two decoding regions, and thus decision-making between p and p is prone to errors Athough depoying more antennas heps reduce this overap, as shown in Figs b and c, it incurs an extra cost Therefore, an optima consteation is essentia to reducing the error probabiity + < In this section, a cosed-form expression of SER of EDbased receivers with ISI is derived Accordingy, our consteation design and threshod optimization is proposed to minimize the error probabiity A SER of the ED-based Receiver in ISI Channes We assume that the knowedge of channe and noise statistics is avaiabe at the receiver First, zt in 3 with a finite hh ntst + = {z noise component NC } ISI3 kntk {z } 4 where the first component contains the desired signa When, both ISI and ISI3 converge to zero and the noise component NC σn, eaving ISI the ony component that affects the SER However, since can never be infinite, the non-zero ISI, ISI3 and NC woud adversey affect the error performance Hence, a cosed-form expression of SER, which is the basis for our consteation design, can ony be accuratey derived when taking into consideration of a finite number of receive antennas The derivation of the PDF of zt, which is a non-trivia task, is required to cacuate the error rate Towards this end, we resort to the foowing emma Lemma : If the number of receive antennas grows arge, then the foowing approximations are attainabe thanks to the Centra Limit Theorem CLT zt N µz, σz 5 with µz and variance σz shown as foows µz = III T HE P ROPOSED C ONSTELLATION D ESIGN AND T HRESHOLD O PTIIZATION L X σh st + L X σh st + σn = σz = L X! σh st + σn = where N µ, σ indicates a rea Gaussian variabe Proof: The proof can be found in Appendix A Speciay, the Gaussian approximation is accurate since tends to be arge, even when SNR is ow According to Lemma, zt can represented as foows z t = µz + N, σz = σh st + L X σh st + σn + δ t 6 = " dr; p + <n dl; dr; p + <n "4 "3 " dl;3 dr;3 p3 + <n dl;4 p4 + <n Fig 3 Decoding regions for a consteation size of = 4 where δ t is a nondeterministic item and δ t N, σz In mutipath channes, a ZF equaizer is empoyed to remove ISI in zt before symbo detection [] The ZF equaization matrix can be computed by [] T wzf = ed GT G G 7
4 σ ψp i = w σh 4 p i + σh σn + L σ h σh = p i + L = σ 4 h + σh σh + L σ h σn + σn 4 = where e d is an a zero vector with the d-th entry being unity, which how to compute d can find in [], and σh σh σ h L σh σh σh L G = σh σh σh L where G R J J+L, J is the ength of equaizer The anaytica formua for the decision metric ψt is defined as J ψ t = wj ZF z t j j= J = s t + wσn + wj ZF δ t j j= where wj ZF is the j-th eements in w ZF The equaizer wi work we when, the third item in 8 wi converge to zeros However, since can never be infinite, the third item in 8 woud adversey affect the error performance The configuration of a non-coherent massive SIO receiver is shown in Fig It is worth noting that zt after equaization, denoted by ψt, sti foows the Gaussian distribution because of the inear ZF equaizer For ease of derivation, we denote the previous symbos st by average power s = p + + p When the current transmit symbo st = p i, the decision metric ψt foows the Gaussian distribution, ie 8 ψt N µ ψ p i, σ ψp i 9 where the mean and variance of ψt is shown as foowing and in on the top of next page µ ψ p i = p i + wσ n where w is a constant computed by mutipying the equaizer coefficients with an a-one vector [] The aforementioned resuts make the derivation of SER straightforward Proposition : With a finite number of receive antennas, the SER of the ED-based receiver in ISI channes is given beow P e = P p i = i= erf i= d L,i σψ p i + erf d R,i σψ p i where d L,i and d R,i are vaues of threshods shown in Fig 3, P p i denotes the probabiity of correct decision on p i Specificay, d L, = and d R, = + The shaded area in this figure, which is decided by d L,i and d R,i, indicates the decision region for p i Proof: The proof can be found in Appendix B It is worth noting that the SER in is a generaized resut suitabe for a variety of non-negative consteations Given variance σ ψ p i and decoding threshods, one can obtain the corresponding error probabiity Intuitivey, some interesting remarks are made as foows: The SER over fat-fading channes can be obtained by setting L = in oreover, the mutipath effect of ISI channes coud incur performance degradation compared with case of the fat-fading channe; Depoying more receive antennas is a straightforward and effective approach to reduce the error probabiity, because when grows unimited, we have σ ψ p i, dr,i σψ p i erf P e ;, erf dl,i σψ p i, and finay If and are fixed and SNR, σ ψ p i converges to a steady state independent of σ n This is equivaent to that an error foor appears when the SNR is arger than a certain vaue Therefore, a requirement of high SNRs is not critica in this scenario Before further anaysis, the infuence of a ZF equaizer on SER needs to be carified Athough it is designed to remove the infuence of ISI, the ZF equaizer causes another probem of noise enhancement This is even worse in our study since the equaization increases the variance of ISI components in 4 at the same time, making the decision between neighboring PDFs more prone to errors This performance degradation can be minimized by consteation design B Proposed Consteation Design It is readiy observed from that the error probabiity decreases with the decrease of σ ψ p i, which reates to p i This observation, couped with the reationship between SER and d L,i or d R,i, ceary demonstrates the potentia to improve the error performance via optimizing the consteation Obviousy, minimizing the average symbo error probabiity equas to maximizing the probabiity of correct decision, ie maximize {P,,, } Subject to P p i, i= 3 p i, p i p i+ i= where k represents the decision region [p i +wσ n d L,i, p i + wσ n + d R,i, which is subject to the constraint of transmit power However, soving 3 is not straightforward First, the Cauchy-Schwarz inequaity is utiized to simpify the probem soving process
5 Lemma : In Eucidean space R n with standard inner product, the Cauchy-Schwarz inequaity states that for a sequences of rea numbers a i and b i, we have n n n a i b i 4 i= i= a i i= where the equaity hods if and ony if a i = kb i for a certain constant k R + If we set b i =, the Cauchy-Schwarz inequaity can be rewritten as n n a i i= dl, σψ p n i= a i n b i 5 where the equaity hods if and ony if a = a = = a n As a resut, the maximum of 3, or equivaenty the maximum dl,i dr,i of i= erf σψ + erf σψ is achieved p i p i if P p = P p = = P p, which can be expanded as dl, dr, erf σψ + erf σψ = p p dl,i dr,i erf σψ + erf σψ = 6 p i p i = erf + erf dr, σψ p Proposition : The average symbo error probabiity P e is convex in the space spanned by P Proof: The proof can be found in [5] According to Proposition, P p i is convex with respect to p i in region d L,i, d R,i ] Thus, P p i can be maximized if erf d L,i σψ p i = erf d R,i σψ p i 7 Submitting 7 into 6, the foowing resut is obtained dr, dl, erf σψ = erf σψ = p p 8 erf = = erf dr, σψ p dl, σψ p This equation indicates that to minimize the overa SER, the number of errors with respect to each consteation point shoud be the same This observation is quite different from the case of PA consteations As can be observed from 8 and Fig 3, two important resuts can be obtained, ie d R,i σ ψ p i = d R,i = d L,i, d L,i+ σ ψ p i+ = p i+ p i σ ψ p i+ + σ ψ p i = T, i =,,, 9 where T is defined for the ease of anaysis Since erf is a monotonicay increasing function of its argument, maximizing T is equivaent to maximizing the probabiity of correct decision Thus, the optimization probem in 3 can be transformed into maximize {P,,, } Subject to T, p i+ p i σ ψ p i+ + σ ψ p i = T, p i, p i p i+ i= The first constraint in can be rewritten as p i+ p i = T σ ψ p i+ + σ ψ p i Given a known p i, σ ψ p i can be cacuated by Thus is transformed into p i+ p i T σ ψ p i = T σ ψp i+ Afterwards, with a fixed T and an initia vaue of p, the probem can be converted to the foowing quadratic equation where AT p i+ + BT, p i p i+ + CT, p i = 3 AT = w σh 4 T BT, p i = p i T + σψ p i T = + w σ h σ n + w σ h CT, p i = C C T, p i L C = w σh 4 + σh σh = + L σh σn + σn 4 C T, p i = pi T + σ ψ p i L σh = 4 If we set the initia vaue of p =, p i+ can be cacuated iterativey For exampe, 4 shows that both BT, p i and CT, p i reate to σ ψ p i, thus the constraint condition p i+ with T can be soved as foows σψ T p i + wσn + w L σh + p i = p i+ = T wσ h 5 Notice that the range of T, which are the bounds of bisection, < T < wσ h, guarantee the soution is rea positive and are reated with the number of antennas Because the arge number of antennas at the receiver, the range is enough arge to find out optimize consteation soution The resut can be found in Appendix C With the same approach, one can obtain p, p 3,, p Then, the optima probem can be represent as maximize {P,,, } T, Subject to 5, p i, p i p i+ i= Proposition 3: i p i is an increasing function wrt T Proof: The proof can be found in Appendix C In genera, p is initiaized to be zero 6
Consteation Set Consteation Set 6 3 5 3 5 SNR = 9 db In case of any change of parameters that woud resut in a arger σ ψ p i+, the distance between p and p wi enarge to make the decision between p and p ess prone to errors; 5 Proposed Consteation Non-negative PA 5 Proposed Consteation Non-negative PA As a resut, the distances between other neighboring p i woud decrease accordingy to keep the same constraint of transmit power In this way, the proposed design is capabe of adaptivey optimizing p i according to the channe and noise statistics 5 5 5 5 SNR db 5 6 5 4 3 Number of channe taps C Threshod Optimization Based on 9, the decision metric can be decoded according to the maximum ikeihood or other rues [3] Given the right and eft distances of d R,i and d L,i shown in Fig 3, the decision threshod for st = p i can be obtained as Fig 4 Proposed consteation with = and = 4 a Consteation versus the SNR; and b Consteation versus the number of channe taps The red dashed ine indicates the energy eve of non-negative PA If i p i satisfies the power constraint, an optima consteation is obtained If not, T needs to be adjusted accordingy If the average power is ess than power constraint, T needs to be increased If not, T wi be decreased At ast, a simpe method of bisection can be empoyed to cacuate the optima T max In concusion, the soution to 6 can be summarized in a step-by-step manner in Agorithm, where T ower and T upper indicate the ower and upper bounds of bisection range, respectivey Agorithm The soution to : Parameter initiaization: p =, T =, i= p i =, T ower =, T upper = //wσh ; : whie T upper T ower 3 and i= p i 3 do 3: T = Tupper+T ower ; 4: Utiizing 5 and 4 to compute p, p 3,, p ; 5: if i= p i then 6: T upper = T ; 7: ese 8: T ower = T ; 9: end if : end whie : return p, p,, p and T Fig 4 a compares the proposed consteation and a nonnegative PA at various SNRs It is shown that the distance between p and p of the proposed design is arger than that of PA at ow SNRs However, the distances between other neighboring p i of our design are smaer compared to the case of PA This is reasonabe because the error decision between p and p pays the most important roe in computing the SER oreover, our design converges to PA graduay with the increase of SNR Simiar resuts is obtained if we anayze the behavior of p i when the number of channe taps varies, as shown in Fig 4 b Thus, the foowing remarks are made: d L,i = d R,i = T max σ ψ p i 7 The optimized decision boundaries between two neighboring consteation points is then denoted as trel i = p i + wσ n d L,i trer i = p i + wσ n + d R,i 8 With the optimized threshod, a transmit symbo can be decoded as foows p, ψt, trer ], ŝt = pi, ψt trel i, trer i ], i =, 3,, p, ψt trel, + 9 In the ight of 8, the probabiity of correct decision is consisted of the same vaue P p i = erf T max, i =,,, 3 Then, according to the maximized T and decision threshods, the error probabiity in can be approximated as P e opt erf T max + 3 Therefore, finding the minimum P e opt is equivaent to maximizing T, because erf is a monotonicay increasing function of its argument D Reation with the Rate Function Scheme Among the existing pubications, the one that is most reated to our proposed scheme is the one presented in [3], which is based on the rate function This scheme was shown to be the first consteation design for non-coherent massive SIO based on ED However, it ony expores the scenario of fatfading channe otivated by this, we wi next compare this scheme with the proposed design First, we briefy review the key idea of the rate function scheme, which is represented by the foowing emma Lemma 3: For any d > and zero mean iid random variabes u, u,, u n, we have [3] i= P u i d e Id 3
7 where Id = sup θd og E[e θu ] is the rate function θ> Based on emma 3 and moment generating function of u i, the upper bound of SER can be obtained, ie P U e d R,i kp i + e d L,i kp i 33 i= where kp i = E[u m] = σ r mt, u m = r m t µ rmt and r m t = h m st + n m with r m t denoting the signa received by the mth antenna at time instant t, and h m being the channe between the transmitter and the mth antenna in the scenario of fat-fading channe To have a deep understanding of the difference between these two schemes, we consider the appication of the proposed consteation design in fat-fading channe Hence, the received signa and decision metric can be rewritten as rt = hst + nt 34 ψ fat t = rt 35 where rt = [r t, r t,, r t] H and h = [h, h,, h ] H Based on the CLT shown in Appendix A, ψ fat t aso foows the Gaussian distribution The reationship between kp i and the variance of ψ fat t, which is denoted by σ ψ,fat p i, is σ ψ,fatp i = σ r mt = kp i 36 As mentioned earier, as ong as the threshod and variance are known, a cosed-form expression of SER can be obtained According to, the SER of a non-coherent SIO system in fat-fading channes can be written as dl,i dr,i P e,fat = erfc + erfc kpi kpi i= 37 Lemma 4: The compementary error function erfcx approaches its imit when x + as foows [6] erfcx e x, if x + πx Using Lemma 4, 37 is abe to be approximated as P e,fat d R,i d L,i kpi e kp i + e kp i π i= d R,i d L,i 38 The foowing inequaities are easiy obtained when is arge kp i πd R,i <, kp i πd L,i < 39 As a consequence, the upper bound of P e in 38 is P e,fat < e d R,i kp i + e d L,i kp i = P U 4 i= From 4, an interesting concusion can be obtained Through the scae of 38, the same upper bound of SER is observed as in [3] Therefore, the effectiveness of our proposed scheme in fat-fading channe is vaidated In concusion, this paper provides a genera framework for the consteation design in ED-based non-coherent massive SIO systems The resuts are appicabe in both fat-fading and mutipath-fading channes IV PERFORANCE ANALYSIS AND DISCUSSION In this section, we discuss the infuence of key parameters on the error probabiity, incuding the number of receive antennas, SNR and consteation size The non-negative PA and our proposed consteation design are incuded for comparative performance study In Section III-A, we have derived the SER of the EDbased receiver with the proposed consteation design For comparative purposes, the SER expression for a non-negative PA consteation is given as foows P e pam = + erfc i= erfc i= d R pam,i σψ p i pam d L pam,i σψ p i pam 4 where d R pam,i = d L pam,i+ = i+ ε, ε = 6 The consteation of a non-negative PA scheme is denoted by p i pam = [, ε,, i ε,, ε] The resut in 4 can be obtained straightforwardy by using the same approaches as in Appendix A, with the PA consteation and decision threshods from [] A SER Approximation By appying Lemma 4 to 3 and 4, it can be shown that SERs of the proposed consteation design and PA are expressed as P e opt P e pam + i= e T max πtmax π i= π σ ψ p i pam e d R pam,i σ ψ p i pam e d R pam,i σ ψ p i pam d L pam,i σ ψ p i pam d L pam,i 4 When the consteation size of the non-negative PA is sma, eg, = 4, P e pam is dominated by the component of i = in 4 [9] For exampe, simuation resuts indicate that the error decision between p and p accounts for up to 99% of the overa errors when = 4 and = 4 at SNR = 6 db oreover, this phenomenon is aso observed in Fig Therefore, the SER of PA reduces to P e pam + π π σ ψ p pam e d R pam, σ ψ p pam e d R pam, σ ψ p pam d L pam, σ ψ p pam d L pam, 43
8 It foows from 4 that d R pam,i = d L pam,i+ and σ ψ p pam > σ ψ p pam Simuation resuts indicate that the error decision on p accounts for up to 7764% of the overa errors when = 4 and = 4 at SNR = 6 db For convenience of anaysis, the first term in 43 is removed 3 Therefore, we can obtain the foowing ogarithmic SER d L pam, og P e pam σψ p og e + og pam σ ψ p pam π d L pam, 44 Simiary, the ogarithmic operation is appied to the SER of the proposed design, ie og P e opt T max og e + og πtmax 45 Based upon the resuts in 44 and 45, we now can compare the error performances between the proposed consteation design and PA in the event of a arger number of receive antennas and high SNRs B Infuence of a Finite Number of Receive Antennas SER as a Function of a Finite Number of Antennas: In order to study the infuence of a finite number of receive antennas on SER, we fix the SNR and consteation size Therefore, the variance in can be expressed as a function of σ ψ p i, = w ζp i 46 where ζp i = σh 4 p i + σh σ + L n σ h = σ h p i + L = σ4 h + σ h σh + L = σ h σn + σn 4 Since the SNR is constant in this scenario and σh is a propagation-reated parameter, ζp i ony reates to p i It foows from 9 that T max = d opt w 47 ζp opt Through substituting 47 into 45, the ogarithm of P e opt is expressed as a function of d opt og P e opt = w ζp opt og e og + og π ζp w opt d opt 48 where the third term is a constant irreevant of In addition, if grows arge, the second component in 48 pays a much ess significant roe compared with the first one As a resut, og P e opt approximates to a inear decreasing function of This confirms that depoying more antennas is an effective way to reduce decoding errors 3 This approximation in fact decreases the SER of the ED receiver empoying non-negative PA, and thus it represents the worse-case scenario for our consteation design in terms of performance comparison Non-negative PA versus the Proposed Optima Consteation: Appying the same approach in 46-48, the error performance of non-negative PA can be represented as d L pam, og P e pam = w ζp pam og e og ζp pam w + og π d L pam, 49 which can aso approximate to a inear decreasing function of when, the same as in 48 It is readiy observed that the key to performance comparison between og P e opt and og P e pam ies in their sopes with respect to After removing constants in common, it is equivaent to comparing d L pam, /ζp pam and d opt/ζp opt First of a, it proves that d L pam, /ζp pam is the minimum of d i,pam /ζp i pam for non-negative PA consteations, and monotonicay increases or increases first and then decreases refer to Appendix D for a detaied mathematica treatment According to the Cauchy-Schwarz inequaity, d i,opt /ζp i,opt is found to be equa to d opt/ζp opt for a optima consteation points Accordingy, we set a baseine of d opt/ζp opt to compare with d L pam, /ζp pam Specificay, there exist three conditions isted beow, as shown in Fig 5 Case : If d L pam,i /ζp i pam i is ess than the baseine Case 3 in Fig 5 a and b, then d L pam, /ζp pam < d opt/ζp opt because d L pam, /ζp pam is the minimum of d L pam,i /ζp i pam i=,,, ; Case : If some d L pam,i /ζp i pam i are greater than the baseine Case in Fig 5 a and b, we can have d L pam, /ζp pam < d opt/ζp opt according to the power constraint; Case 3: If a d L pam,i /ζp i pam i=,,, are greater than the baseine Case in Fig 5 a and b, thus the SER of non-negative PA is smaer than P e opt However, thanks to Cauchy-Schwarz inequaity, P e opt is the smaest error probabiity Therefore, the presumption that a d L pam,i /ζp i pam i=,,, are greater than the baseine is unfounded In fact, our extensive numerica simuations revea that ony Case wi occur As a consequence, d L pam, /ζp pam are aways beow the baseine To summarize, the resut of comparing the first components of 48 and 49 is given by d opt d L pam, w ζp opt og e < w og e 5 ζp pam Athough the SER of the non-negative PA maybe ower than that of the proposed design when is sma, P e opt decreases at a much faster rate than P e pam with massive antennas according to 5 Hence, to maintain the same SER, our consteation design requires ess antennas than PA, which is of great benefits for practica appications C Infuence of SNR Non-negative PA versus the Proposed Optima Consteation: In this section, the number of receive antennas and
9 d i;pam pi;pam Case a d i;pam p i;pam Baseine d i;pam pi;pam Case Case Case b d i;pam p i;pam Baseine Dx x = og e + x n 56 Equation 55 indicates that Dx is a monotonousy decreasing function of x Thanks to 56, Dx/ x monotonousy increases if < x < / og e n 77 When x > 77, Dx/ x monotonousy decreases, which means the rate of descent at x is arger than x if x > x > 77 In Section IV-B, it has aready been proved that d L pam, / ζp pam < d opt / ζp opt The same concusion is aso appicabe to this situation, ie Case 3 Case 3 d L pam, σψ p pam, σn < d opt σψ p opt, σn 57 i-th Consteation Point i =! i-th Consteation Point i =! Fig 5 Three situations about how the baseine crosses the curve of d i,pam /ζp i,pam a The curve monotonicay increases; and b The curve monotonicay increases and then decreases consteation size are fixed so as to investigate the reationship between the error performance and SNR Specificay, we wi investigate the rate of descent of the SER against the SNR In this scenario, the variance in can be written as a function of noise variance, ie w σ ψ p i, σn = where W p i, σ n = σ n + σ h p i + W p i, σn 5 L = σ h Obviousy, σ ψ p i, σ n decreases when the SNR grows It foows from 9 that T max = Substituting 5 to 45 gives rise to og p e opt d opt σψ p opt, σ n 5 d opt σ ψ p opt, σ n og e og d opt σψ p opt, σn + og π Simiary, 44 can be transformed into og p e pam d L pam, σ ψ p pam, σ n og e og d L pam, σψ p pam, σn + og π 53 54 As can be observed from 53 and 54, the first two terms foow the same mode Dx = x og e og x, whie the ast term is a constant independent of the SNR Therefore, comparing the rates of descent of 53 and 54 amounts to comparing the rates of descent of x opt = d opt / σ ψ p opt, σn and x pam = d L pam, / σ ψ p pam, σn in Dx = x og e og x with the same SNR The first and second derivatives of Dx with respect to x are Dx x = x og e x n 55 For ease of exposition, we use x opt = d opt / σ ψ p opt, σn and x pam = d L pam, / σ ψ p pam, σn in the foowing anaysis Specificay, there exist three cases in consideration of the distribution of x opt and x pam Case : x pam < x opt < 77, shown in Fig 6 a This case arises ony if the number of receive antennas is very sma and the SNR is ow Fig 6 a demonstrates that the rate of descent of og p e pam is faster than that of og p e opt ; Case : x pam < 77 < x opt, shown in Fig 6 b The comparison between og p e pam and og p e opt depends on two factors, ie, the distance to 77 of x pam and x opt, and the rate of change of the descent of Dx Detais on this case are beyond the scope of this study; and Case 3: 77 < x pam < x opt, shown in Fig 6 c Obviousy, the descent rate of og p e opt is greater than that of og p e opt if x > 77 With the increase of SNR, the rates of descent of both og p e pam and og p e opt wi become even greater According to our extensive simuations, Cases and occur when < 5 and SNR < 6 db However, in the presence of massive receive antenna array, Case 3 woud be the case Consequenty, it can be inferred that to meet the same SNR requirement, our design requires a smaer transmit power than PA consteations High SNR Anaysis: When the SNR becomes arge enough, we have σn and the noise variance can be safey removed Then, the variance in reduces to 58 The second and third terms in 58 are constants For a fixed and consteation size, the consteation design is aso fixed Since we have aready introduced a baseine, T max can be expressed as the vaue of the first consteation p =, ie d R, T max = L w σh 4 + 59 σh σh = where d R, is the right threshod of p, as shown in Fig 3 Equation 59 shows that for a fixed, T max converges to a constant in the high SNR region Therefore, an error foor woud arise so that no matter how arge the transmit power becomes, the error performance cannot be further improved
Dx Dx Dx σ ψp i w σh 4 p i + L σ h = σh p i + L = σ 4 h + σ h σ h 58 5 a 5 b 5 c - Non-negative PA-ZF with SNR= db Non-negative PA-ZF with SNR=6 db Proposed-ZF with SNR= db Proposed-ZF with SNR=6 db x = xpam x = xpam x = xpam 5 x = xopt 5 5 x = xopt - x = xopt SER -3-5 -5-5 - - - -4-5 -5-5 -5-77 44 x - 77 44 x - 77 44 x Fig 6 The three situations about x opt and x pam distribution in Dx = x og e og x a x pam < x opt < 77, b x pam < 77 < x opt, c 77 < x pam < x opt The red-ine is Dx = x og e og x -6 5 5 5 3 35 4 Number of antennas Fig 7 SER versus the number of receive antennas at various SNRs with = 4 according to 3 There are two ways to aeviate but not eiminate the error foor The first approach is to empoy more receive antennas, which increases the numerater of T max, and then owers the SER when the error foor appears The second one is to aeviate the frequency-seectivity of the mutipath channe, such as empoying the OFD technique In this way, the denominator of T max can be decreased, and the error probabiity converges to a steady state with ower vaues oreover, since an error foor is inevitabe in this situation, high SNRs may not be required SER - - -3-4 -5 Non-negative PA-ZF with = Non-negative PA-ZF with =4 Proposed-ZF with = Proposed-ZF with =4 V SIULATION RESULTS To demonstrate the performance of the proposed consteation design, this section presents numerica resuts obtained via onte Caro simuations The receiver structure shown in Fig is appied Non-negative PA is considered as a benchmark [9] Throughout simuations, a 4-tap ISI channe with an exponentia decay mode is empoyed [4] For both scenarios with the proposed consteation and PA, the energy is first coected by the massive antenna array and then a ZF equaizer is utiized to remove the ISI Finay, the transmit symbo is decoded with the decision metric Fig 7 pots SERs with different numbers of receive antennas at SNR = db and 6 db, where a consteation size of = 4 is empoyed First of a, a remarkabe performance gap between P e opt and P e pam exhibits the benefit of our proposed consteation design As expected, the ogarithmic SER decreases amost ineary with for both consteations This observation proves the effectiveness of the approximations in 48 and 49, as we as demonstrating the huge potentia of massive receiver array in non-coherent SIO systems Besides, as the number of antennas increases, the error probabiity of the proposed consteation decreases at a much faster rate than the PA, which verifies the resut in -6-3 3 6 9 SNR db Fig 8 SER versus SNR with = and = 4, where = 4 5 Furthermore, 5 indicates that the sope of SER versus woud become arger at higher SNRs, which is aso ceary demonstrated in Fig 7 ore importanty, the required number of antennas for our design to meet a predefined SER can be decreased compared with PA To be more specific, Tabe I ists the number of receive antennas required in a variety of scenarios For instance, antennas is needed to achieve SER = 5 at SNR = db for the proposed consteation design, whereas the number of antennas for non-negative PA to achieve the same performance is 4 Therefore, a great dea of cost in hardware impementation can be saved This saving becomes more pronounced if the SNR increases or a ower SER is required, as Tabe I shows the required numbers of antennas in the other two settings can be reduced by 57% and 667%, respectivey Fig 8 reports the simuated SER versus the SNR with
8 7 6 - Non-negative PA-ZF with =4 Non-negative PA-ZF with =5 Proposed-ZF with =4 Proposed-ZF with =5 Number of antennas 5 4 3 Non-negative PA-ZF with SNR=6 db Non-negative PA-ZF with SNR= db Proposed-ZF with SNR=6 db Proposed-ZF with SNR= db 3 4 5 6 7 8 Consteation Size SER - -3-4 -5-6 3 6 9 SNR db Fig 9 Number of receive antennas versus the consteation size to achieve SER = 4 at various SNRs TABLE I THE NUBERS OF RECEIVING ANTENNAS REQUIRED TO ACHIEVE DIFFERENT SERS SER = 5 SER = SER = 3 SNR = db SNR = 6 db SNR = 6 db Non-negative PA 4 4 375 Our Consteation 6 5 Reduced by 5% 57% 667% various numbers of antennas Obviousy, the rate of descent of our consteation design is arger than that of non-negative PA, the same as the anaytica resuts in 57 oreover, both P e opt and P e pam converge to non-zero constants as the SNR further grows This error foor effect is caused by a finite and ISI channe As a resut, simpy raising the transmit power cannot make the probem go away Athough empoying more antennas can ead to a better performance, carefu attention shoud be paid to address the baance between system performance and cost It is aso worth noting that P e opt converges to a error probabiity much ower than that of P e pam Fig 9 pots numbers of receive antennas needed to achieve SER = 4 for these two consteations In genera, a greater consteation size resuts in a arger data rate but ess reiabe transmission As can be observed from this figure, the proposed consteation design performs significanty better than non-negative PA For exampe, with a consteation of size = 4, our design needs approximatey one third of the number of antennas to achieve the same SER performance compared with PA On the other hand, both consteation designs require more antennas to maintain SER = 4 if the consteation size increases, but the proposed one needs far ess antennas than non-negative PA Fig draws the simuated SER versus SNR for different consteations Requiring the same number of antennas = 4, the proposed consteation design with a consteation size = 5 performs even better than a non-negative PA consteation with = 4, athough a higher consteation Fig SER versus SNR with = 4 for various consteations size means ess reiabe transmission This observation again confirms the huge potentia of our proposed consteation design to improve the error performance VI CONCLUSION This paper proposed a consteation design for a noncoherent massive SIO system using the ED receiver in ISI channes, aiming at minimizing the symbo error probabiity ore specificay, a cosed-form expression of the SER is derived, given a finite number of receiver antennas, and known channe and noise statistics, based on which an optimization probem reating to the consteation design was formuated and soved To provide a deeper insight, we further compared the proposed consteation design with non-negative PA in different aspects The performance anaysis indicates that the ogarithms of P e opt and P e pam both ineary decrease with the number of receive antennas However, the proposed consteation design causes a much faster decine of SER versus On the other hand, the SNR anaysis demonstrates that the proposed design is abe to offer the same error probabiity with ess transmit power or receive antennas than non-negative PA This is especiay important from the perspective of saving energy and impementation costs Finay, an error foor for P e opt wi appear at high SNRs, indicating that the performance of non-coherent massive SIO systems is interference-imited rather than noise-imited APPENDIX A PROOF OF LEA Suppose {X, X,, X n } is a sequence of iid random variabes with E[X i ] = µ and Var[X i ] = σ < Then according to the Lindeberg-Lévy CLT [7], as n approaches infinity, the random variabes n n n i= X i µ converge in distribution to a norma N, σ, ie n d n X i µ N, σ 6 n i= The Lindeberg-Lévy CLT can be transformed into n d X i µ + n N, σ = N µ, σ 6 n n i=
First, the received signa at the mth antenna is L y m t = h,m st + n m t 6 = Due to the mutua independence among a h,m, the received signas at different antennas are mutuay independent It can be derived that y m t is a compex Gaussian variabe ie, L y m t CN, σh s t + σn 63 = Furthermore, it is shown that yt is a sum of iid y m t m=,,, with [ E y m t ] [ = E R{y m t} ] + E [I{y m t} ] [ Var L = σh st + σn = y m t ] = Var = L [ R{y m t} ] + Var [I{y m t} ] σh st + σn = 64 According to 6 and 64, the resut in 5 is proved Hence, the proof of Lemma is concuded APPENDIX B PROOF OF PROPOSITION From Fig 3, the range of p i is decoded by d L,i and d R,i As for a singe transmit symbo p i, the error probabiity is P e p i = P r {p i i } 65 where i = [p i + wσn d L,i, p i + wσn + d R,i Through invoking the Gaussian approximations, the probabiity of correct decision for each transmit symbo can be written as P p i = erf d L,i σψ p i + erf d R,i σψ p i 66 where σp i denotes the variance in Based on above, the average probabiity of errors is given by P e = P p i i= = d L,i d erf + erf R,i σψ p i= i σψ p i 67 Therefore, the proof of Proposition is concuded APPENDIX C SOLUTION OF THE QUADRATIC EQUATION 3 AND PROOF OF PROPOSITION 3 A Soution of the quadratic equation 3 The soution of 3 wi be proved in beow At the first, B and C can be represented as BT, p i = B T, p i T + B, 68 B CT, p i = w σh 4 B T, p i T 69 where B T, p i = p i + T σ ψ p i and B = w σ h σn + L σh = Then, BT, p i 4ACT, p i wi be soved as BT, p i 4ACT, p i = B T, p i T 4 + 4B T, p i B T + 4B w σh 4 4 T = T B T, p i B + w σh 4 B T, p i = T B w σh 4 B T, p i T + B w σ 4 h B wσ h + wσ h B T, p i 7 Based on 68 and 7, the p i+ = BT, p i ± BT, p i 4ACT, p i A B T, p i B T + B ± T wσh + wσ h B T, p i = T w σh 4 σψ T p i + wσn + w L σh + p i = = p i or T wσ h 7 From 7, through contro the range of T, < T < wσ h, obviousy, both the soution of quadratic 3 are rea positive From above equation, the soution a is positive, The process of choose the soution can be expained as that, p i coud not be chosen because if it wi cause that the a consteation point are equa The other soution is our choice, thus p i+ can be represent as 5 B Proof of Proposition 3 Then it can be represent as p i+ is function of T p i+ = T σψ p i + wσ n + w = a T p i + b T T wσ h L = σ h + p i 7
3 where T wσ a T = h + T wσ h T w b T = σ T wσ n + L σh h In this paper, the i= i= = p i = at i p + i b T at i 73 i= In order to sove this function increase or decrease, it can be divided into two part, at i and b T at i, to sove First, for the at i derivative of T, = i wσ h T wσ h + i i + i wσ h T wσ h + i i+ T wσ h T wσ h 74 Second, for the b T at i derivative of T at i T bt at i T = + w σn + + w σn + σn + L L = L = = σ h σh T wσ h + i i T wσ h σh From 74 and 75,for < T < 4w σ h T T wσ h + i T wσ h i+ i T wσ h T wσ h + i T wσ, wσh h i 75 at i T and bt at i T are arger than zero, it means that at i and b T at i are monotone increasing to T, thus is monotone increasing with T APPENDIX D PROOF OF d L pam, /ζp pam IS INIU OF d i,pam /ζp i pam p i i= At first, d i,pam / ζp i pam can be divided into two parts, namey d R pam,i / ζp i pam and /d L pam,i / ζp i pam As can be inferred from 4 and 46, d R pam,i = d L pam,i+ and ζp i pam increase with p i pam Thus, the foowing reationship hods d R pam,i ζpi pam > d L pam,i+ ζpi+ pam 76 Given 76, we need to prove that d L pam, / ζp pam is the minimum vaue of d L pam,i+ / ζp i+ pam, which can be expressed as a function of i αi = w D i + i + 4 i 4 + E Dε i + F, i =,,, 77 Dε where D = σh 4 E = σh σn + L σ h = F = L σh 4 + ε = = 6 σ h σh σh + L σ h σn + σn 4 To find the minimum vaue, we take the first derivative of αi with respect to i αi i = w D = gi fi i4 + E Dε i + F, i =,,, 78 Dε where gi = F Dε i+ F Dε, fi = i 5 +3i 4 +i 3 + E Dε i + E Dε i Equation 78 indicates that both gi and fi are monotonicay increasing functions There are three cases concerning the monotonicity of function αi, namey, increase first and then decrease, monotonicay increase, and monotonicay decrease In either case, one can find the minimum vaue by comparing the vaues of endpoints As a resut, the ratio of endpoints α and α is α + α = G 79 3 F +6E +36D 36D 4 +F +6E where G = It is evident from 79 that α is greater than α Therefore, d L pam,i / ζp i pam is an increasing function or increases initiay and then decreases Therefore, α is the minimum vaue and we have proved that d L pam, /ζp pam is the minimum vaue of d i,pam /ζp i pam REFERENCES [] L Lu, G Y Li, A L Swindehurst, A Ashikhmin, and R Zhang, An overview of massive mimo: Benefits and chaenges, IEEE Journa of Seected Topics in Signa Processing, vo 8, no 5, pp 74 758, 4 [] T L arzetta, Noncooperative ceuar wireess with unimited numbers of base station antennas, IEEE Transactions on Wireess Communications, vo 9, no, pp 359 36, [3] F Rusek, D Persson, B Lau, E G Larsson, T L arzetta, O Edfors, and F Tufvesson, Scaing up mimo: Opportunities and chaenges with very arge arrays, IEEE Signa Processing agazine, vo 3, no, pp 4 6, [4] T S Rappaport, S Sun, R ayzus, H Zhao, Y Azar, Wang, G N Wong, J Schuz, Samimi, and F Gutierrez, iimeter wave mobie communications for 5g ceuar: It wi work! IEEE Access, vo, pp 335 349, 3 [5] Y Han, H Zhang, S Jin, X Li, R Yu, and Y Zhang, Investigation of transmission schemes for miimeter-wave massive mu-mimo systems, IEEE Systems Journa, vo, no, pp 7 83, 7 [6] T Truong and R W Heath, Effects of channe aging in massive mimo systems, Journa of Communications and Networks, vo 5, no 4, pp 338 35, 3 [7] Witrisa, G Leus, G J Janssen, Pausini, F Trösch, T Zasowski, and J Romme, Noncoherent utra-wideband systems, IEEE Signa Processing agazine, vo 6, no 4, 9 [8] H Urkowitz, Energy detection of unknown deterministic signas, Proceedings of the IEEE, vo 55, no 4, pp 53 53, 967 [9] R oorfed and A Finger, utieve pam with optima ampitudes for non-coherent energy detection, in Proc 9 IEEE Internat Conf on Wireess Commun and Signa Processing WCSP, Nanjing, China, Nov 9, pp 5
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