Average SEP-Optimal Precoding for Correlated Massive MIMO with ZF Detection: An Asymptotic Analysis
11 Average SEP-Optimal Precoding for CorrelatedMassive MIMO with ZF Detection:An Asymptotic Analysis
Zheng Dong, Jian-Kang Zhang and He Chen
Abstract —This paper investigates the symbol error probabil-ity (SEP) of point-to-point massive multiple-input multiple-output(MIMO) systems using equally likely PAM, PSK, and squareQAM signallings in the presence of transmitter correlation.The receiver has perfect knowledge of the channel coefficients,while the transmitter only knows first- and second-order channelstatistics. With a zero-forcing (ZF) detector implemented at thereceiver side, we design and derive closed-form expressions of theoptimal precoders at the transmitter that minimizes the averageSEP over channel statistics for various modulation schemes.We then unveil some nice structures on the resulting minimumaverage SEP expressions, which naturally motivate us to explorethe use of two useful mathematical tools to systematically studytheir asymptotic behaviors. The first tool is the Szeg¨o’s theoremon large Hermitian Toeplitz matrices and the second tool is thewell-known limit: lim x →∞ (1 + 1 /x ) x = e . The application ofthese two tools enables us to attain very simple expressions ofthe SEP limits as the number of the transmitter antennas goesto infinity. A major advantage of our asymptotic analysis isthat the asymptotic SEP converges to the true SEP when thenumber of antennas is moderately large. As such, the obtainedexpressions can serve as effective SEP approximations for massiveMIMO systems even when the number of antennas is notvery large. For the widely used exponential correlation model,we derive closed-form expressions for the SEP limits of bothoptimally precoded and uniformly precoded systems. Extensivesimulations are provided to demonstrate the effectiveness ofour asymptotic analysis and compare the performance limit ofoptimally precoded and uniformly precoded systems. Index Terms —Massive MIMO, antenna correlation, ZF equal-ization, symbol error probability, Toeplitz matrix and Szeg¨o’stheorem.
I. I
NTRODUCTION
With recent advances in radio frequency (RF) chains andintegrated circuit designs, massive multiple-input multiple-output (MIMO) [2]–[6] has emerged as a key enabler to fulfillthe unprecedented requirement of the upcoming fifth genera-tion (5G) cellular systems on three generic services, includingextreme mobile broadband (eMBB), massive machine-typecommunications (mMTC), and ultra-reliable and low-latencycommunications (URLLC) [7]–[9]. In general, massive MIMO
Zheng Dong was with the Department of Electrical and Computer Engi-neering, McMaster University, Hamilton, ON, Canada, and he is now with theSchool of Electrical and Information Engineering, The University of Sydney,NSW 2006, Australia (Email: [email protected]). Jian-Kang Zhangis with the Department of Electrical and Computer Engineering, McMasterUniversity, Hamilton, ON, Canada (Email:[email protected]).He Chen is with the School of Electrical and Information Engineering,The University of Sydney, NSW, Australia (Email:[email protected]).This paper was presented in part at the IEEE International Conference onCommunications (ICC), London, UK from 8-12, June, 2015 [1]. technology can be categorized into the point-to-point massiveMIMO and the multiuser massive MIMO [4], [10]. In thispaper, we consider a point-to-point massive MIMO where thereceiver has an excess number of antennas than the transmitter( N r − N t (cid:29) ). This model depicts a typical scenario wherea multi-antenna user transmits to its serving base station (BS)equipped with a large number of antennas in the uplink. Toimplement the aforementioned massive MIMO system in a realenvironment, several significant issues should be addressed.The first one is the drastically increased detection complexitythat can restrict the widespread deployment of an extreme largearray. Despite the fact the maximum likelihood (ML) detectoris universally optimal for the uniformly distributed input, itscomplexity is prohibitively high even for a moderately sizedarray. Therefore, for the sake of practicality, we consider theuse of a linear zero-forcing (ZF) detector at the receiver side,which has been proved to have near-optimal performance interms of throughput when the BS array size is large [3], [4],[11].Another major limiting factor arising in the massive MIMOfading channels is the acquisition of channel state information(CSI). In fact, the performance of massive MIMO reliescritically on the availability of the channel knowledge on bothsides in order to harvest the array gain and/or multiplexinggain for increased energy efficiency and data rates. In thispaper, we assume that full CSI is available at the receiverside while only first- and second-order channel statistics areavailable at the transmitter as in [12]. This reason is that theCSI at the receiver can be obtained by letting the transmittersend training pilots, the number of which is proportional tothe number of transmitter antennas [13]. It is worth notingthat, for practical massive MIMO systems, the CSI can beerroneous due to the difficulty of channel estimation causedby the large number of transmitter and/or receiver antennasand the potential low transmission power. It has been shownthat the CSI imperfection will result in lower nominal receivedsignal-to-noise ratio (SNR) in conventional MIMO systemsin [14], which has recently been further verified for massiveMIMO cases in [15]. Since the SNR expressions have thesame structure no matter whether the CSI is perfect or not, ouranalysis and the resultant precoding design can also be appliedto the cases with imperfect CSI, which is omitted due to spacelimitation. On the other hand, the instantaneous CSI at thetransmitter is arguably intractable to attain by either employingdirect training methods relying on the time division duplex(TDD) channel reciprocity or by using quantized channel a r X i v : . [ c s . I T ] J a n feedback in frequency-division duplext (FDD) mode, giventhe large number of receiver antennas in massive MIMOsystems [3]. To make massive MIMO systems more feasible,more practical constraints need to be considered, such aschannel estimation error, pilot contamination, and transceiverhardware impairments. Therefore, proper pilot design andpower control as well as advanced linear precoding or two-layer decoding should be carried out to alleviate these limitingfactors. However, these are beyond the scope of this paper, andwe refer to interested readers to the relevant work [16]–[20]and references therein.More importantly, the effects of fading correlation in mas-sive MIMO systems also need to be better understood. Inpractice, due to the size limitation, the antennas at the trans-mitter side (i.e., user device) can hardly be well separatedand hence tend to be correlated [12], [21]–[25]. This kindof channel correlation model is a typical case which can beverified by using a ‘one-ring’ scattering model as consideredin [26]. For example, when the user device is unobstructedand the BS is surrounded by local scatterers. To show theeffect of fading correlation more explicitly, we specificallyconsider an exponential correlation model [27]–[34], whichis a simplified one-parameter model in practical environmentthat can capture the main phenomenon of spatial correlationbetween antennas, especially for a uniform linear array (ULA).To better understand the channel correlation in such massiveMIMO fading channels, we consider both the precoder designand the corresponding performance analysis.It has been shown that, for a general MIMO system with thefirst- and second-order channel statistics available at the trans-mitter, linear precoding is an efficient and effective scheme tosignificantly alleviate the performance loss caused by the chan-nel fading correlation [35]–[38]. Previously, the performanceof a conventional MIMO system with transmitter correlationand the ZF receiver was first investigated in [23], where itwas showed that the resulting signal-to-noise ratio (SNR) ofeach data stream follows a Chi-squared distribution. Then, thecapacity analysis for MIMO systems with the receiver havingfull CSI and the transmitter having only second-order channelstatistical information was investigated in [12]. The impact oftransmitter correlation in MIMO systems, with full or averageCSI was investigated in [22]. The antenna selection strategywith transmitter correlation was studied in [25]. However, allthe results mentioned above were aimed for the conventionalMIMO where the number of antennas at the transmitter andreceiver are relatively small (e.g., several tens of antennas). Tothe best knowledge of the authors, this is the first effort towardsthe asymptotic symbol error probability (SEP) analysis of mas-sive MIMO systems with transmitter correlation and channelstatistics-based precoding. The main purpose of the asymptoticanalysis is to unveil the dominating factors on the averageerror performance of the considered massive MIMO systemwhen the antenna array size is scaled up. The asymptotic errorexpression also serves as a useful system design guidelineeven when the antenna array is moderately large. The potentiallimitation is that the asymptomatic results may diverge fromthe exact values for small or moderate MIMO systems. Wewill show that for our asymptotic analysis, this limitation is rather mild because the asymptotic SEP converges to the trueSEP with tens of transmitter antennas.The main contributions of this paper are summarized asfollows:1) We first derive the average SEP of the considered mas-sive MIMO system with a ZF detector as a function ofthe precoding matrix and the channel covariance matrixfor PAM, PSK, and square QAM modulations, respec-tively. Based on the average SEP expressions, we thenconstruct the optimal precoders in closed-form formatsthat minimize the average SEP of different modulationschemes. Our approach to obtain the optimal precoderis universal, which depends only on the convexity andmonotonic of the average SEP expressions as well asthe ZF equalizer itself.2) We identify some nice structures of the minimal av-erage SEP achieved by the optimal precoders, whichnaturally lead us to explore the use of two useful math-ematical tools for the systematic study of asymptoticbehaviors on their error performance. The first toolis the Szeg¨o’s theorem on large Hermitian Toeplitzmatrices and the second tool is the well-known limit: lim x →∞ (1 + 1 /x ) x = e . By applying these two tools,we obtain neat expressions for the average SEP limits ofPAM, PSK and square QAM constellations, which areshown to have a fast convergence speed as the numberof the transmitter antennas goes to infinity. Our resultcan greatly simplify the error performance evaluationfor massive MIMO systems.3) When the channel covariance matrix is the commonly-used non-symmetric Kac-Murdock-Szeg¨o (KMS) ma-trix, we derive closed-form expressions of the SEP limitsfor both optimally precoded and uniformly precodedsystems. Moreover, a tight approximation to the distri-bution of the individual SNR of each sub-channel is alsodeveloped for the large array, which explicitly reveals thechannel hardening phenomenon of the correlated mas-sive MIMO systems. Specifically, for given transmittedpower and noise level, the SNR of each sub-channelconverges to some constant value almost surely for eachchannel realization when the array size is scaled up. Notations : Matrices and column vectors are denoted byboldface characters with uppercase (e.g., A ) and lowercase(e.g., b ), respectively. ⊗ denotes the Kronecker product. The ( m, n ) -th entry of A is denoted by [ A ] m,n . The m -th entryof b is denoted by b m . (cid:107) b (cid:107) denotes the Euclidean norm of b . Notation A + stands for the pseudo-inverse of A . Thetranspose of A is denoted by A T . The Hermitian transpose of A (i.e., the conjugate and transpose of A ) is denoted by A H .II. P RECODED T RANSMISSION M ODEL WITH Z ERO -F ORCING D ETECTION
A. System Model
Consider the complex baseband-equivalent model of anarrow-band massive MIMO communication system, with N t transmitting antennas and N r receiving antennas. The receiver is assumed to has an excess number of antennas than the trans-mitter ( N r − N t (cid:29) ). The serially transmitted symbols is firstde-multiplexed into a vector signal s = [ s , s , · · · , s N t ] T ,and then this vector signal s is transformed by an N t × N t full-rank square precoding matrix F into another vector signal x = [ x , x , · · · , x N t ] T = Fs . Then, each element x k of x is fed to the k -th transmitter antenna for transmission. In thespace-time communication, N t transmitting antennas each cantransmit uncoded M -ary PAM, PSK or QAM symbols usingthe same waveform during the same time interval, but weassume that the same constellation is used for each antenna.At the receiver array, the discrete received N r × signal vector r can be written as r = HFs + ξ , (1)where H is the N r × N t complex channel matrix and ξ is the N r × additive complex noise vector. Each of the h mn = [ H ] mn represents the subchannel connecting the n -thtransmitter antenna to the m -th receiver antenna. We also let h Tm = [ h m , · · · , h mN t ] denote the m -th row of H . We assumethat only the channels at the transmitter side are correlatedwhile the channels at the receiver side are uncorrelated [12],[23], since the transmitter antennas of mobile users can hardlybe well separated and tend to be correlated while the BS canhave a large space where the correlation can be made to benegligible. That is, we assume E [( h T(cid:96) ) H h Tm ] = (cid:26) Σ (cid:96) = m, (cid:96) (cid:54) = m. (2)The second-order channel statistic matrix Σ is mainlyaffected by the large-scale fading, which changes much moreslowly than the instantaneous channel coefficients. In practice, Σ can be tracked by the base station with a desired accuracyand is assumed to be known as in [39]. Throughout this paper,we adopt the following assumptions:1) The perfect channel estimates are available at the re-ceiver to allow coherent detection, while the transmitteronly knows first- and second-order channel statistics Σ [12] ;2) The channel H is complex Gaussian distributed, withzero-mean, and covariance matrix I ⊗ Σ ;3) ξ is circularly-symmetric complex Gaussian noise withcovariance σ I ;4) Each element of s is independently and equally likelychosen from PAM, PSK or QAM constellations of thesame size with the covariance matrix of s being I [40] ;5) The total power budget of the transmitter array is unifiedto one, and as a consequence the system SNR is definedas η (cid:44) /σ . B. Zero-Forcing Equalization
Suppose that we use zero-forcing equalization to recover theinformation symbols. To this end, first we obtain the pseudoinverse of super-channel matrix HF , i.e. ( HF ) + = (cid:0) F H H H HF (cid:1) − F H H H . (3)Here we need to explain why the inverse in (3) exists.Under Assumption 2 above, the matrix H H H is the Wishart distribution and as result, F H H H HF is also subject to theWishart distribution [41], [42]. Therefore, the inverse (3)almost exists if the number of receiving antennas is not lessthan that of transmitting antennas [41], [42]. In practice, thecomplexity of ZF receiver can also be very high when N t is large [43], [44]. However, ZF receiver, which needs muchless antennas than the matched filter (MF) based approach forachieving the same error performance in moderate and highSNR regime, is also popular for massive MIMO in order tostrike a balance between performance and complexity [44],[45]. To reduce the complexity of ZF receiver, some low-complexity approximation methods can be employed [6], [46].The ZF detection is captured by the following two steps:1) Perform ZF equalization. Multiplying both sides ofequation (1) by the pseudo-inverse ( HF ) + , we get r (cid:48) = s + ξ (cid:48) , (4)where r (cid:48) = ( HF ) + r and ξ (cid:48) = ( HF ) + ξ . UnderAssumption 3, ξ (cid:48) is the circularly-symmetric complexGaussian noise with covariance σ ( F H H H HF ) − ;2) Perform a hard decision to obtain an estimate ˆ s k of s k ,i.e., ˆ s k = arg min s k ∈S | r (cid:48) k − s k | . Since ZF equalizer is a memoryless detection, i.e., thedecision on the current symbol does not affect the decisionon the next symbol, the average SEP over one vector signal s is the arithmetic mean of all SEPs.III. O PTIMAL P RECODER TO M INIMIZE THE A VERAGE S YMBOL E RROR P ROBABILITY
Our primary purpose of this section is to first give an explicitconvex region for which the optimally precoding matrix F that minimizes the average SEP of the ZF detector can beobtained and then to uncover some nice structures for theoptimally precoded system, which naturally leads us to takingfull advantage of the Szeg¨o’s theorem for the systematic studyof the asymptotic behaviour on the resulting error performanceof massive MIMO systems. A. SEP Expressions for M-ary Signals and Explicit ConvexRegions of Objective Functions
We first study the average SEP for given preocoding matrix F , where the average is taken over all the fading coefficients.The proposed approach generalizes the optimal transmitterdesign of the ZF equalizer in [47] not only from the deter-ministic model to the statistical MIMO channel model, butalso from the specific BPSK signal constellation to the general M -ary signal constellations including PAM, PSK and QAMsignalling. Lemma 1: SEP expression:
The average SEP for M -aryPAM, PSK and QAM signals with precoding matrix F aregiven by P ( F ) = 1 N t N t (cid:88) k =1 G (cid:16) [ (cid:0) F H ΣF (cid:1) − ] kk (cid:17) , (5) where G ( x ) is convex for < x < T , and we have:1) For PAM signals, G ( x ) = G PAM ( x ) = M − Mπ (cid:82) π/ (cid:16) η ( M − x sin θ (cid:17) − ( N r − N t +1) dθ , T = T PAM = η ( N r − N t )2( M − .2) For PSK signals, G ( x ) = G PAM ( x ) = M − Mπ (cid:82) π/ (cid:16) η ( M − x sin θ (cid:17) − ( N r − N t +1) dθ , T = T PSK = η ( N r − N t ) sin ( π/M )2 .3) For QAM signals, G ( x ) = G QAM ( x ) = √ M − √ Mπ (cid:82) π/ π/ (cid:16) η M − x sin θ (cid:17) − ( N r − N t +1) dθ + √ M − Mπ (cid:82) π/ (cid:16) η M − x sin θ (cid:17) − ( N r − N t +1) dθ , T = T QAM = η ( N r − N t )4( M − .The proof is provided in Appendix-A.To attain a unified expression, we now drop the subscriptsof both G ( x ) and T . Hence, G ( x ) is convex if < x ≤ T . Correspondingly, the noise power associated with F (e.g.,the variable of the objective function of Eq. (5)) satisfies thefollowing condition, < [( F H ΣF ) − ] kk ≤ T for k = 1 , , . . . , N t . (6)To develop an explicit constraint from (6), we have thefollowing proposition: Proposition 1: If F is restricted to be in set (cid:8) F : ζ ( F H F ) ≥ λ T (cid:9) where ζ ( F H F ) is the minimum eigen-value of F H F , then the constraint in (6) is satisfied.The proof is given in Appendix-B. Proposition 1 requiresthat the minimal average transmitting power of the sub-channels, ζ ( F H F ) , must be larger than certain predefinedthreshold that is related to the modulation signals, systemSNR η and channel statistics. It is worth pointing out that,our result here identifies the conditions for the optimality ofthe precoder to a much general and simplified form comparedwith that of [47]. Since T is proportional to system SNR η and N r − N t , constraint in Proposition 1 is easy to satisfyin slightly high SNR regime, especially for the large MIMOsystem considered in this paper. B. SEP-optimal PrecodersTheorem 1:
Let the eigenvalue decomposition of Σ be Σ = WΛW H , where W is a unitary matrix, and Λ =diag( λ , λ , · · · , λ N t ) with λ ≤ λ ≤ · · · λ N t . If F isrestricted to be in set (cid:8) F : ζ ( F H F ) ≥ λ T (cid:9) , where T isthe threshold in (6) and ζ ( F H F ) is the minimum eigenvaluesof F H F . Then, the optimal precoder minimizing the averageSEP is given by (cid:101) F = 1 (cid:112) tr( Λ − / ) WΛ − / (cid:101) V , (7)where (cid:101) V is the N t × N t normalized DFT matrix, and theresulting minimum average SEP is determined byP min ( (cid:101) F ) = G (cid:18)(cid:16) N t (cid:88) k =1 λ − / k (cid:17) /N t (cid:19) . (8) The proof is provided in Appendix-C. Here, we make thefollowing two comments on Theorem 1.1) The optimal precoder design problems with ZF detec-tion were also considered in [35], [47]. However, ourTheorem 1 provides an explicitly sufficient conditionthat guarantees the optimality of the proposed precoder.Note that although the convexity constraints given inProposition 1 are rather mild, the design of precodingmatrix F given in Theorem 1 can be suboptimal whenthese constraints are violated as the objective functionmay not be convex.2) Here, it is highly worth pointing out that the resultingSEP for the optimal precoder exposes a very interestingstructure which motivates us to systematically study theasymptotic SEP performance in massive MIMO systems. C. Extension to Multiuser Cases
The main objective of this subsection is to show how toextend our design to multiuser massive MIMO systems. Weconsider the case where K users each with N t transmitting an-tennas transmit to the base station with N r receiving antennassuch that N r − KN t (cid:29) . The input and output relationshipcan be modeled by ¯ r = ¯ H ¯ F ¯ s (9)in which ¯ H = [ H , H , . . . , H K ] , ¯ F =diag { F , F . . . , F K } , and ¯ s = [ s T , s T , . . . , s TK ] T . Forthe multiuser case, we assume that the channel are correlatedfor themselves but uncorrelated between each other since theyare geographically separated, with covariance matrix I ⊗ Σ k for the channel matrix H k . Since the derivations of the SEPexpressions for PAM, PSK and QAM are similar, in thefollowing discussion, we simply take the PAM constellation asan example. We also notice that E [ ¯ F H ¯ H H ¯ H ¯ F ] = E [ ¯ F H ¯ Σ ¯ F ] ,where ¯ Σ = diag (cid:8) Σ , Σ , . . . , Σ K (cid:9) , and also (cid:0) ¯ F H ¯ Σ ¯ F (cid:1) − = diag (cid:8) [ F H Σ F ] − , . . . , [ F HK Σ K F K ] − (cid:9) . (10)For the considered multiuser system, the average SEP overall the users with precoding matrix ¯ F is given by:P PAM ( ¯ F ) = 2( M − M KN t π KN t (cid:88) k =1 (cid:90) π ... E ¯ H exp (cid:32) − η ( M − (cid:0) ¯ F H ¯ H H HF (cid:1) − ] kk sin θ (cid:33) dθ ( a ) = 2( M − M KN t π KN t (cid:88) k =1 (cid:90) π .... E ¯ γ exp (cid:32) − η ¯ γ ( M − (cid:0) ¯ F H ¯ Σ ¯ F (cid:1) − ] kk sin θ (cid:33) dθ = 2( M − M KN t π Γ( N r − KN t + 1) KN t (cid:88) k =1 (cid:90) π (cid:90) ∞ ... exp (cid:34) − (cid:32)
1+ 3 η ( M − (cid:0) ¯ F H ¯ Σ ¯ F (cid:1) − ] k sin θ (cid:33) ¯ γ (cid:35) ¯ γ N r − KN t dγdθ (11) where in ( a ) , likewise in (22), we have used the fact that ¯ γ = [ ( ¯ F H ¯ Σ ¯ F ) − ] kk [ ( ¯ F H ¯ H H ¯ H ¯ F ) − ] kk is subject to X N r − KN t +1) . Now, wehaveP PAM ( ¯ F ) P PAM ( ¯ F ) = 2( M − M KN t π KN t (cid:88) k =1 (cid:90) π ... (cid:32) η ( M − (cid:0) ¯ F H ¯ Σ ¯ F (cid:1) − ] kk sin θ (cid:33) − ( N r − KN t +1) dθ ( b ) = 1 KN t K (cid:88) (cid:96) =1 N t (cid:88) k =1 ¯ G PAM (cid:16) [ (cid:0) ¯ F H(cid:96) ¯ Σ (cid:96) ¯ F (cid:96) (cid:1) − ] kk (cid:17) , (12)we have used (10) in ( b ) and ¯ G PAM ( x ) = M − Mπ (cid:82) π (cid:16) η ( M − x sin θ (cid:17) − ( N r − KN t +1) dθ . Wecan see from (12) that the SEP over K user is simply thearithmetic mean of the SEPs of all the users. As each useris subject to their individual power constraint, the precoderdesign can be decomposed into K design problems, whichcan all be optimally solved by Theorem 1.IV. A SYMPTOTIC
SEP A
NALYSIS FOR O PTIMALLY P RECODED MASSIVE
MIMO S
YSTEMS
In this section, our main purpose is to investigate the asymp-totic behavior of SEP for the optimally precoded correlatedmassive MIMO systems equipped with the ZF receiver. Thearray size of both the transmitter and the receiver is increasedwhile maintaining a constant ratio between them.
A. Array Correlation Model with Toeplitz Covariance Matrix
In general, MIMO techniques can yield linear increasing inthe data rate against the minimum number of the transmitterand the receiver antennas in rich scattering environment,particularly when the array elements are uncorrelated [48].However, in a practical radio propagation process, correlationis almost inevitable, especially for a massive MIMO archi-tecture. In this paper, we assume that the transmitter arrayis arbitrarily correlated and that the correlation between eachelement of the receiver array is negligible. This case canbe considered as a MIMO system in the uplink where thetransmitter is a mobile terminal with correlated array andthe receiver is a base station, where the distance betweenadjacent antenna elements can be made as large as desiredto eliminate correlation. To facilitate our analysis, we alsoassume that the correlation matrix is a Hermitian Toeplitzmatrix [29]. This is a simplified model of measurement inpractical environment, but it can capture the main phenomenonof spatial correlation between antennas (see e.g., [26] for othermodels). This model enables us to completely take advantageof the structure provided by the optimal system as well as ofthe Szeg¨o’s theorem on large Hermitian Toeplitz matrices sothat we can attain a simple closed-form solution in terms of thecorrelation coefficients, from which some important insightfulinformation on the effect of correlation can be extracted.
B. Asymptotic Behaviour of Large Toeplitz Matrices
To fully make use of the optimal structure provided by (8)for our analysis on the asymptotic behavior of the statisticalaverage SEP, let us review an important property on a sequenceof large Hermitian Toeplitz matrices { T K } ∞ K =1 . Without lossof generality, we let T K = t (0) t ( − · · · t ( − ( K − t (1) t (0) · · · t ( − ( K − ... ... . . . ... t ( K − t ( K − · · · t (0) . (13)where t ( k ) = t ∗ ( − k ) and t ( k ) are assumed to be absolutelysquare-summable, i.e., (cid:80) ∞ k =0 | t ( k ) | < ∞ . Thus, the follow-ing pair of discrete-time Fourier transforms exists, s T ( ω ) = ∞ (cid:88) k = −∞ t ( k ) e − jkω , t ( k ) = 12 π (cid:90) π s T ( ω ) e jkω dω. It is worth noting that the function s T ( w ) is real, since T is Hermitian and s T ( w ) is also known as the power spectraldensity (PSD) function. The above relationship is also knownas the Wiener-Khinchin theorem of discrete-time process. Lemma 2 (Szeg¨o’s theorem): [49] Let { T K } ∞ K =1 be asequence of Hermitian Toeplitz matrices with K eigenval-ues of T K given by µ K, ≤ µ K, ≤ · · · ≤ µ K,K , and (cid:80) ∞ k =0 | t ( k ) | being convergent. Then for any function F ( x ) that is continuous on [ L s T , U s T ] , where L s T = ess inf s T ( ω ) is the essential infimum [50] of s T ( ω ) and defined to be thelargest value of c for which s T ( ω ) ≥ c except on a set ofmeasure 0, and U s T = ess sup s T ( ω ) is the smallest number d for which s T ( ω ) ≤ d except for a set of measure 0, we have lim K →∞ K K (cid:88) (cid:96) =1 F ( µ K,(cid:96) ) = 12 π (cid:90) π F ( s T ( ω )) dω. (14)The above Lemma 2 plays a vital role in the asymptoticerror performance analysis of the considered massive MIMOsystem. C. Asymptotic SEP Analysis for Massive MIMO with ToeplitzCovariance Matrix
We are now ready to present the asymptotic SEP analysisfor the precoded massive MIMO system. From now on,we assume that the ratio of the number of the receiverantennas to that of the transmitter antennas is fixed, i.e., N r /N t = β > is constant. The average SEP-optimal codingdesign fits both conventional and massive MIMO. However, animportant question is how the SEP behaves when the numberof transmitting antennas goes to infinity while keeping β = N r N t and the transmitting power fixed. By strategically resorting tothe Sezg¨o’s theorem, we manage to show that the SEP quicklyconverges to a fixed value when the system SNR η is fixed.The main result of this paper can be formally stated as thefollowing theorem. Theorem 2:
Let us consider massive MIMO systems usingthe optimal precoder in (7), the ZF detector and the M -aryPAM, PSK or QAM constellations. If the entries of the channel covariance matrix Σ are absolutely square-summable and theresulting L s Σ > , then, lim N t →∞ P N t ( (cid:101) F ) = ¯P opt exists and • ¯P opt , PAM = M − M Q (cid:16)(cid:113) η ( β − M − (cid:17) ; • ¯P opt , PSK = π (cid:82) ( M − π/M exp (cid:16) − η ( β −
1) sin ( π/M )Λ sin θ (cid:17) dθ ; • ¯P opt , QAM = √ M − √ M Q (cid:16)(cid:113) η ( β − M − (cid:17) − √ M − M Q (cid:16)(cid:113) η ( β − M − (cid:17) .where Λ is defined by Λ = π (cid:82) π dω √ s Σ ( ω ) with s Σ ( ω ) = (cid:80) ∞ k = −∞ σ ( k ) e − jkω , where σ ( m − n ) = [ Σ ] mn denotes the ( m, n ) -th entry of Σ .The proof can be found in Appendix-D and we would liketo make the following two comments:1) From Theorem 1 we can see that the diversity gain forthe optimally precoded MIMO system for a fixed N t with the ZF receiver is N r − N t + 1 . However, when N t tends to infinity, Theorem 2 reveals that the limitingSEP of the optimally precoded MIMO system equippedwith the ZF detector decays exponentially.2) Despite the fact that the assumption of Theorem 2requires that the correlation matrix is Toeplitz so as tomake use of the Szeg¨o’s theorem, we can infer fromthe following proof that the assumption can be actuallyrelaxed to any invertible correlation matrix Σ with thecondition that lim N t →∞ N t (cid:80) N t n =1 λ − / k exists, where λ ≤ λ ≤ · · · ≤ λ N t are the eigenvalues of Σ .We now show that Theorem 2 can be used for the SEP eval-uation of precoded massive MIMO with correlated antennas.In particular, when the channel covariance matrix Σ is the non-symmetric Kac-Murdock-Szeg¨o (KMS) matrix that has beenused widely in the literature [27]–[34], [51], the ( m, n ) -thentry of which is denoted by σ ( m − n ) , i.e., σ ( m − n ) = [ Σ ] mn = (cid:26) ρ n − m m ≤ n, [ Σ ] ∗ nm m > n, (15)where < | ρ | < indicates the degree of correlation, we havethe following corollary. Corollary 1:
Consider massive MIMO systems with the op-timal precoder (7), ZF detector and the M -ary PAM, PSK andQAM constellation. If < | ρ | < , then, lim N t →∞ P N t ( (cid:101) F ) = (cid:101) P exists and (cid:101) P PAM = 2( M − M Q (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) π η (1 − | ρ | )( β − | ρ | )( M − E (cid:18) √ | ρ | | ρ | (cid:19) ; (cid:101) P PSK = 1 π (cid:90) ( M − πM exp − π η (1 − | ρ | )( β −
1) sin ( πM )4(1 + | ρ | ) E (cid:18) √ | ρ | | ρ | (cid:19) sin θ dθ ; (cid:101) P QAM = 4( √ M − √ M Q (cid:32)(cid:118)(cid:117)(cid:117)(cid:116) π η (1 − | ρ | )( β − | ρ | )( M − E (cid:16) √ | ρ | | ρ | (cid:17) (cid:33) − √ M − M Q (cid:32)(cid:118)(cid:117)(cid:117)(cid:116) π η (1 − | ρ | )( β − | ρ | )( M − E (cid:16) √ | ρ | | ρ | (cid:17) (cid:33) , where E ( k ) = (cid:82) π/ (cid:112) − k sin θ dθ denotes the completeelliptic integral of the second kind [52].The proof is provided in Appendix-E. D. Convex Region for KMS Matrices
Let ζ ( Σ ) ≤ ζ ( Σ ) ≤ . . . ≤ ζ N t ( Σ ) be the eigen-values of the KMS matrix and then from [53], we have ζ k ( Σ ) = −| ρ | | ρ | +2 | ρ | cos θ k , for k = 1 , , . . . , N t , where cos θ ≥ cos θ ≥ . . . ≥ cos θ N t , in which θ k is the solution to | ρ | (cid:16) sin( N t + 1) θ k + 2 | ρ | sin N t θ k + | ρ | sin( N t − θ k (cid:17) = 0 .Since | cos θ k | ≤ and < | ρ | < , then for the KMS matrix, ζ ( Σ ) ≥ −| ρ | (1+ | ρ | ) = −| ρ | | ρ | > . Recall that the optimalitycondition for the precoder is ζ ( F H F ) ≥ ζ ( Σ ) T (16)where ζ ( F H F ) is the minimum eigenvalue of F H F . Finally,for KMS covariance matrix, the constraint (35) have thefollowing simple sufficient form ζ ( F H F ) ≥ | ρ | T (1 − | ρ | ) . E. Limiting Performance of the Individual SNR
To further appreciate the asymptotic SEP properties de-rived for the optimally precoded massive MIMO systemsin the previous subsection, we are also motivated to studythe asymptotic distribution of the SNR for each sub-channelwhen the array size is large. Notice that at the output of theZF receiver for each sub-channel, the average signal poweris E [ | s k | ] = 1 , and the power of the equalized noise is σ (cid:104) ( (cid:101) F H H H H (cid:101) F ) − (cid:105) kk . Therefore, the instantaneous SNRof each sub-channel as a function of the random channelrealization is τ k = 1 σ (cid:104) ( (cid:101) F H H H H (cid:101) F ) − (cid:105) kk for k = 1 , , . . . , N t , Now, we have the following remark on τ k when N t goes tobe unlimited. Remark 1:
For the asymptotic behaviour on individualSNR for each subchannel, we have lim N t →∞ τ k almost surely −−−−−−−→ η ( β − for k = 1 , , . . . , N t . Proof:
First, by Assumption 5, we have η = 1 /σ andhence τ k = η [ ( (cid:101) F H H H H (cid:101) F ) − ] kk = N t η (cid:101) γ k (cid:16)(cid:80) Ntm =1 λ − / m (cid:17) , for k =1 , , . . . , N t , where (cid:101) γ k = [ ( (cid:101) F H Σ (cid:101) F ) − ] kk [ ( (cid:101) F H H H H (cid:101) F ) − ] kk and (cid:101) F is the opti-mal precoder given in (7). Therefore, the mean and varianceof τ k can be determined as follows: E [ τ k ] = N t ( N r − N t + 1) η (cid:16)(cid:80) N t m =1 λ − / m (cid:17) , var[ τ k ] = N t ( N r − N t + 1) η (cid:16)(cid:80) N t m =1 λ − / m (cid:17) . When scaling up the array size, and with the help of (42), wehave lim N t →∞ E [ τ k ] = η ( β − , lim N t →∞ var[ τ k ] = lim N t →∞ β − σ Λ N t = 0 . Then, by the law of large numbers (LLN), we have lim N t →∞ τ k almost surely −−−−−−−→ η ( β − for k = 1 , , . . . , N t . Remark 1 suggests that when the size of the antennaarray goes to infinity, the instantaneous SNR of each sub-channel becomes stable, i.e., it converges to a fixed value.This verifies the results in Theorem 2. Here, note that the exactconvergence requires the array size goes to infinity and it doesnot necessarily work well when the array size is small. In whatfollows, we give an intuitive approximation to the distributionof the SNR for each receiver branch, which is very accuratewhen the array size is moderate large. From the convergenceof Szeg ¨ o’ Theorem in (42), we know that lim N t →∞ τ k = lim N t →∞ N t η (cid:101) γ k (cid:16)(cid:80) N t m =1 λ − / m (cid:17) ≈ η (cid:101) γ k N t Λ (17)for a large N t . Now letting ˜ τ k = η (cid:101) γ k N t Λ and as the Szeg¨o’stheorem converges very fast for the considered correlationmatrix, ˜ τ k ≈ τ k when N t is reasonably large. Since f ( (cid:101) γ k ) = N r − N t +1) e − (cid:101) γ k (cid:101) γ N r − N t k , ˜ τ k is subject to the Gamma dis-tribution with mean ( N r − N t +1) ηN t Λ ≈ η ( β − and variance ( N r − N t +1) η N t Λ ≈ η ( β − N t Λ when N t is large. Now, by the well-known central limit theorem, we have ˜ τ k ˙ ∼ N (cid:18) η ( β − , η ( β − N t Λ (cid:19) (18)where ˙ ∼ means approximately with the same distributionwhen the array size is large. Hence, lim N t →∞ τ k = lim N t →∞ ˜ τ k ∼ N (cid:18) η ( β − , lim N t →∞ η ( β − N t Λ (cid:19) almost surely −−−−−−−→ η ( β − . It is worth pointing out that the approximation in (18) is prettyaccurate when the array size is relatively small, say, N t = 10 ,as can be seen in Figs. 8 and 9. F. Uniform Power Allocation Strategy
As a comparison, we are also interested with the systemperformance when no channel information is available at thetransmitter. In this scenario, the transmitter cannot performoptimization on the input covariance matrix or carry out powerallocation across transmitter antennas. Since in this case therewould be no bias in terms of the mean or covariance ofthe channel matrix H , the best precoding strategy would beto allocate equal power to each transmitter antenna and tomake the covariance matrix omni-directional. As a result,we consider asymptotic SEP for uniformly precoded massiveMIMO channels, i.e., ˆ F = √ N t I . We then have the followingtheorem: Theorem 3:
Consider massive MIMO systems using theuniform precoder, ZF detector and the M -ary PAM, PSK andsquare QAM constellations. If the channel covariance matrix Σ is the KMS matrix in (15), then, lim N t →∞ P( ˆ F ) = (cid:98) P U exists and (cid:98) P U , PAM = 2( M − M Q (cid:32)(cid:115) η ( β − − | ρ | )( M − | ρ | ) (cid:33) ; (cid:98) P U , PSK = 1 π (cid:90) ( M − πM exp (cid:32) − η ( β − − | ρ | ) sin ( πM )(1 + | ρ | ) sin θ (cid:33) dθ ; (cid:98) P U , QAM = 4( √ M − √ M Q (cid:32)(cid:115) η ( β − − | ρ | )( M − | ρ | ) (cid:33) − √ M − M Q (cid:32)(cid:115) η ( β − − | ρ | )( M − | ρ | ) (cid:33) . The proof is provided in Appendix-F.
10 15 20 25 30 35 40 45
SNR (dB) -4 -3 -2 -1 A v e r age sy m bo l e rr o r p r obab ili t y ( SEP ) N t =50, N r =100, =0.1*exp(0.5j)
16 QAM, theoretical16 QAM, simulated16 PSK, theoretical16 PSK, simulated16 PAM, theoretical16 PAM, simulated64 QAM, theoretical16 QAM, simulated64 PSK, theoretical64 PSK, simulated64 PAM, theoretical64 PAM, simulated
Fig. 1. Average SEP performance against SNR η , where the correlationcoefficient ρ = 0 . ∗ exp(0 . j ) , N t = 50 , N r = 100 .
10 15 20 25 30 35SNR (dB)10 -5 -4 -3 -2 -1 A v e r age sy m bo l e rr o r p r obab ili t y ( SEP ) N t =50, N r =100, 16 QAM =0.1*exp(0.5j), theoretical =0.1*exp(0.5j), simulated =0.5*exp(0.5j), theoretical =0.5*exp(0.5j), simulated =0.9*exp(0.5j), theoretical =0.9*exp(0.5j), simulated Fig. 2. Average SEP performance against SNR η with 16-QAM constellation. V. N
UMERICAL S IMULATIONS
In this section, we verify our theoretical results throughcomputer simulations. In order to validate the theoretical SEP
10 15 20 25 30 3510 − − − − − SNR, dB A v e r age sy m bo l e rr o r p r obab ili t y ( SEP ) − N t =5, optimal precoderN t =5, uniform power loadingN t =50, optimal precoderN t =50, uniform power loading ρ =0.1*exp(0.5j) ρ =0.5*exp(0.5j) ρ =0.9*exp(0.5j) Fig. 3. Average SEP performance against SNR, with 16-QAM, β = 2 anddifferent ρ . ρ | d B Precoding gain
Fig. 4. The precoding gain compared with uniform power allocation versus | ρ | . expression, Monte Carlo simulations are carried out. Let usfirst consider a uniform linear array with N t = 50 transmittingantennas and N r = 100 receiving antennas, where the receiverknows the CSI perfectly and the transmitter knows only thecorrelation matrix Σ . In this simulation, the correlation matrix Σ is taken as the Kac-Murdock-Szeg¨o matrix. The theoreticaland the simulated SEP results for the optimal precoder areplotted in Fig. 1 and Fig. 2 with different constellations (PAM,PSK and square QAM) and channel correlation coefficientsagainst the SNR η . It can be observed that the simulatedresult matches with the theoretical expression very well, whichverifies the correctness of our analysis. Therefore, in thefollowing, we will use the theoretical result to examine someasymptotic properties.We first compare the error performance of the optimalprecoder with a uniform power allocation scheme. Considerthe case where N t = 5 and , β = 2 , and using a 16-QAM constellation. The average SEPs are given for differentcorrelation coefficient ρ in Fig. 3. Again, we can find thatas | ρ | increases, the SEP is becoming significantly worse. Theoptimal precoder always leads to better error performance evenwhen N t = 5 . The gap between the optimal precoder and the (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) Number of transmitter antennas, N t A v e r age sy m bo l e rr o r p r obab ili t y ( SEP ) (cid:239) QAM, SNR=20 dB 0 50010 (cid:239) Exact SEPLimiting SEP (cid:108) =0.5*exp(0.5j) (cid:108) =0.1*exp(0.5j) (cid:108) =0.75*exp(0.5j)
Fig. 5. Average SEP performance against the number of transmitter antennas,with 16-QAM, β = 2 , and SNR = 20dB. -5 -4 -3 -2 A v e r age sy m bo l e rr o r p r obab ili t y ( SEP ) -1
16 120
SNR, dB
Number of transmitter antennas, N t Exact SEPLimiting SEP
Fig. 6. Average SEP performance against the number of transmitter antennas N t and SNR, with 16-QAM, β = 2 , and ρ = 0 . ∗ exp(0 . j ) . uniform power allocation strategy becomes larger when | ρ | increases. The reason is that when | ρ | is very small, Σ is veryclose to a diagonal matrix with equal diagonal entries. Then,the optimal precoder will degrade into the uniform powerallocation case. However, for general Σ , the performancegap is non-negligible. To show this phenomenon clearly, theratio of the SNR of individual sub-channel between optimalprecoder and uniform power allocation transmitter are givenin Fig. 4. The SNR ratio is a monotonic increasing function of | ρ | , which verifies the results in Fig. 3. Therefore, precodingat the transmitter side can yields better performance over theuniform power allocation strategy. Also, increasing the numberof transmitter antennas N t while keeping β = N r N t fixed willdecrease the transmitted power of each data stream, but willincrease the diversity order, where the limiting performancecan be characterized by using the Sezg¨o’s theorem.On the other hand, to demonstrate the convergence rate interms of the number of the transmitter antennas, the exacttheoretical SEPs and their corresponding limits are depictedversus the number of transmitting antennas in Fig. 5. Withoutloss of generality, 16-QAM constellation is adopted and the Fig. 7. Average SEP performance against the number of transmitter antennas N t and β , with 16-QAM, ρ = 0 . ∗ exp(0 . j ) , and SNR= 12dB. ρ =0.1*exp(0.5j)Equivalent SNR of each sub − channel P r obab ili t y den s i t y f un c t i on ( P D F ) t =400N t =300N t =200 N t =10N t =100 Fig. 8. The distribution of the equivalent SNRs of each sub-channel with β = 2 , and ρ = 0 . ∗ exp(0 . j ) , SNR = 20dB. theoretical SEPs are in solid lines while their limits are denotedby dash lines. Three different correlation matrices are gener-ated according to ρ . It can be noticed that as the magnitude ofthe correlation coefficient ρ decreases, the correlation betweenthe adjacent antennas reduces, and as a consequence, thecorresponding SEP reduces substantially. In Fig. 6, the limitingSEPs are also plotted against N t and different SNRs. It isexpected that the SEPs drop as SNR increases. In both theabove figures, it can be seen clearly that for a given SNR,as the array size is scaled up, the theoretical SEPs and thecorresponding asymptotic results gradually meet together. Theapproximation is accurate for moderate and large number ofantennas. Note that the mean of SNR for each sub-channel is adecreasing function of | ρ | and an increasing function of systemSNR η . Hence, either decreasing | ρ | or increasing η willeventually increase the mean of SNR for each sub-channel,and thus, result in a lower convergence rate for theoretic SEPapproach to its limit expression against N t . This phenomenonis also observed in [29] in the scenario of the approximationof channel capacity. In Fig. 7, the average SEP curves areplotted against N t and β = N r /N t . As can be observed that,increasing the number of β while letting N t be a constant t =100Equivalent SNR of each sub − channel P r obab ili t y den s i t y f un c t i on ( P D F ) ρ =0.9*exp(0.5j) ρ =0.7*exp(0.5j) ρ =0.5*exp(0.5j) ρ =0.1*exp(0.5j) Fig. 9. The distribution of the equivalent SNRs of each sub-channel with β = 2 , and N t = 100 , SNR= 20dB. will reduce the SEP significantly as predicted in Theorem 2.Actually increasing β will improve the array gain, which willresult in higher end-to-end channel gain.In addition, we also show the convergence characteristicof the approximated distribution of individual SNR in eachreceiver branch for the optimally precoded massive MIMOsystems. Both the approximated PDF and the simulated PDFare given in Fig. 8, from which it can be seen clearly thatthe Gaussian approximation is very accurate even when weonly have a very small number of antennas, say, 10 trans-mitter antennas. Moreover, the case with different correlationcoefficients but of the same number of transmitter antennasis studied in Fig. 9. Clearly, as the correlation coefficient | ρ | increases, the mean value of the average SNR decreases asexpected and the intervals spanned by the equivalent SNR arealso narrowed down.VI. C ONCLUSIONS
In this paper, we have derived an explicit convex regionin terms of the modulated signals, system SNR and channelstatistics for the optimal precoder minimizing the average SEPof the ZF detector. A simple expression with a very fastconvergence rate for the SEP limit of the massive MIMOsystems with the PAM, PSK and square QAM constellationsand the ZF receiver is attained. The intuitive understandingof this convergent process has also been provided in terms ofthe approximation to the distribution of the individual SNRfor each sub-channel. The main technical approach proposedin this paper to deriving our results is to fully take advantageof the characteristic of the MIMO channels, the structure ofthe transmitter as well as of the ZF receiver, the Szeg¨o’stheorem [49] on large Hermitian Toeplitz matrices, and thewell-known limit: lim x →∞ (1 + 1 /x ) x = e . Numerical resultsshowed that when second-order channel statistics are availableto the transmitter, the optimally precoded massive MIMO sys-tem outperforms its uniformly precoded counterpart, especiallywhen the antenna correlation is strong. A PPENDIX
A. Proof of Lemma 1
We analyze the SEP expression for M -ary PAM, PSK, andQAM constellations as follows:
1) PAM signals:
The SEP for M -ary PAM signal s k is P PAM ( H , F , s k ) = M − M Q (cid:16)(cid:113) η ( M − F H H H HF ) − ] kk (cid:17) .Therefore, the arithmetic average of all SEPs in one block isP PAM ( H , F )= 2( M − M N t N t (cid:88) k =1 Q (cid:32)(cid:115) η ( M − F H H H HF ) − ] kk (cid:33) . (19)For our purpose, we now prefer to use another expression forGaussian Q -function [54], i.e., Q ( x ) = 1 π (cid:90) π exp (cid:18) − x θ (cid:19) dθ, x ≥ . (20)Substituting (20) into (19) yieldsP PAM ( H , F ) = 2( M − M N t π N t (cid:88) k =1 (cid:90) π ... exp (cid:32) − η ( M − F H H H HF ) − ] kk sin θ (cid:33) dθ. (21)It is known [42, Th. 3.2.12] that γ = [ ( F H ΣF ) − ] kk [( F H H H HF ) − ] kk issubject to X N r − N t +1) , i.e., its density function is f ( γ ) = 1Γ( N r − N t + 1) e − γ γ N r − N t for γ > , (22)where Γ( x ) denotes the gamma function. Now, taking theexpectation in (21) over random channel H yieldsP PAM ( F ) ( a ) = 2( M − M N t π N t (cid:88) k =1 (cid:90) π ... E H exp (cid:32) − η ( M − F H H H HF ) − ] kk sin θ (cid:33) dθ ( b ) = 2( M − M N t π N t (cid:88) k =1 (cid:90) π ... E γ exp (cid:32) − ηγ ( M − F H ΣF ) − ] kk sin θ (cid:33) dθ ( c ) = 2( M − M N t π Γ( N r − N t + 1) N t (cid:88) k =1 (cid:90) π (cid:90) ∞ . . . exp (cid:34) − (cid:32) η ( M − F H ΣF ) − ] k sin θ (cid:33) γ (cid:35) γ N r − N t dγdθ ( d ) = 2( M − M N t π N t (cid:88) k =1 (cid:90) π ... (cid:32) η ( M − F H ΣF ) − ] kk sin θ (cid:33) − ( N r − N t +1) dθ = 1 N t N t (cid:88) k =1 G PAM (cid:16) [ (cid:0) F H ΣF (cid:1) − ] kk (cid:17) , (23) where ( a ) is obtained by inserting (20) into (19)and then averaging over the channel statistics, (b)is true for the definition of γ above, (c) holds dueto (22), and ( d ) can be attained by finishing theintegral over the distribution of γ given above. Inaddition, function G PAM ( x ) is defined as G PAM ( x ) = M − Mπ (cid:82) π/ (cid:16) η ( M − x sin θ (cid:17) − ( N r − N t +1) dθ . Thesecond-order derivative of G PAM ( x ) is given by d G PAM ( x ) dx = 2( M − M π (cid:90) π (cid:18) η ( M − x sin θ (cid:19) − ( N r − N t +3) × η ( N r − N t + 1)( M − x sin θ (cid:18) η ( N r − N t )( M −
1) sin θ − t (cid:19) dθ. Since η ( N r − N t ) / (cid:0) ( M −
1) sin θ (cid:1) − t ≥ η ( N r − N t ) / (cid:0) ( M − (cid:1) − t , then if the followingcondition is satisfied, i.e., < x ≤ η ( N r − N t )2( M −
1) = T PAM , we have d G PAM ( x ) /dx ≥ . This implies that G PAM ( x ) isconvex in this interval.
2) PSK signals:
The SEP for M -ary PSK signal s k isP PSK ( H , F , s k ) = 1 π (cid:90) ( M − πM ... exp (cid:32) − η sin ( π/M )[( F H H H HF ) − ] kk sin θ (cid:33) dθ. (24)Therefore, the arithmetic mean of all SEPs isP PSK ( H , F ) = 1 N t π N t (cid:88) k =1 (cid:90) ( M − πM exp (cid:32) − η sin ( π/M )[( F H H H HF ) − ] kk sin θ (cid:33) dθ. (25)Similarly, taking the expectation of (25) over random channel H producesP PSK ( F ) = 1 N t π N t (cid:88) k =1 (cid:90) ( M − πM ... (cid:32) η sin ( π/M )[( F H ΣF ) − ] kk sin θ (cid:33) − ( N r − N t +1) dθ = 1 N t N t (cid:88) k =1 G PSK (cid:16) [ (cid:0) F H ΣF (cid:1) − ] kk (cid:17) , where function G PSK ( x ) is defined as G PSK ( x ) = 1 π (cid:90) ( M − πM (cid:18) η sin ( π/M ) x sin θ (cid:19) − ( N r − N t +1) dθ. Now, the second-order derivative of G PSK ( x ) is d G PSK ( x ) dx = 1 π (cid:90) ( M − πM (cid:18) η sin ( π/M ) x sin θ (cid:19) − ( N r − N t +3) × η ( N r − N t + 1) sin ( πM ) x sin θ (cid:32) η ( N r − N t ) sin ( πM )sin θ − t (cid:33) dθ. Since η ( N r − N t ) sin ( π/M ) / (sin θ ) − t ≥ η ( N r − N t ) sin ( π/M ) − t ≥ , we have that if < x ≤ η ( N r − N t ) sin ( π/M )2 = T PSK , (26)then, in this interval, d G PSK ( x ) /dx ≥ . This shows that G PSK ( x ) is convex in this range.
3) QAM signals:
The SEP for M -ary QAM signal s k isP QAM ( H , F , s k )= 4 (cid:16) − / √ M (cid:17) Q (cid:18)(cid:115) η ( M − F H H H HF ) − ] kk (cid:19) − (cid:16) − / √ M (cid:17) Q (cid:18)(cid:115) η ( M − F H H H HF ) − ] kk (cid:19) . (27)The first term in (27) can be replaced by (20). Similarly, Q ( · ) function also has a very nice formula [54], Q ( x ) = 1 π (cid:90) π/ exp (cid:18) − x θ (cid:19) dθ. (28)Substituting (20) and (28) into (27) and then, taking theexpectation over the random channel matrix, we can obtainP QAM ( F ) = 1 N t N t (cid:88) k =1 G QAM (cid:16) [ (cid:0) F H ΣF (cid:1) − ] kk (cid:17) , (29)where function G QAM ( x ) is defined as G QAM ( x )= 4( √ M − √ M π (cid:90) π π (cid:18) η M − x sin θ (cid:19) − ( N r − N t +1) dθ + 4( √ M − M π (cid:90) π/ (cid:18) η M − x sin θ (cid:19) − ( N r − N t +1) dθ. (30)For QAM signals, the second-order derivative of G QAM ( x ) is d G QAM ( x ) dx = (cid:16) π − √ M π (cid:17) × (cid:90) π π (cid:16) η M − x sin θ (cid:17) − ( N r − N t +3) × η ( N r − N t + 1)2( M − x sin θ (cid:18) η ( N r − N t )2( M −
1) sin θ − t (cid:19) dθ + 4 (1 − / √ M )( √ M π (cid:90) π (cid:18) η M − x sin θ (cid:19) − ( N r − N t +3) × η ( N r − N t + 1)2( M − x sin θ (cid:18) η ( N r − N t )2( M −
1) sin θ − t (cid:19) dθ. Since η ( N r − N t ) / (2( M −
1) sin θ ) − t ≥ η ( N r − N t ) / (2( M − − t ≥ , if the following condition meets, < x ≤ η ( N r − N t )4( M −
1) = T QAM , (31)then, d G QAM ( x ) /dx ≥ and as a result, G QAM ( x ) is aconvex function in this interval.Combing all the above results, we complete the proof ofLemma 1. B. Proof of Proposition 1
To develop an explicit constraint from (6), we need tointroduce the following two lemmas.
Lemma 3 (Rayleigh-Ritz): [55] Let A ∈ C K × K be Her-mitian and let ζ ( A ) ≤ ζ ( A ) ≤ . . . ≤ ζ K ( A ) be theeigenvalues of A . Then ζ ( A ) = min (cid:107) x (cid:107)(cid:54) =0 x H Axx H x , ζ K ( A ) = max (cid:107) x (cid:107)(cid:54) =0 x H Axx H x . Since ( F H ΣF ) − is Hermitian and [( F H ΣF ) − ] kk = e Hk ( F H ΣF ) − e k , where e k = [0 , · · · , , , , · · · , H has 1only in the k -th entry, by Lemma 3 we have ζ (cid:0) ( F H ΣF ) − (cid:1) ≤ [( F H ΣF ) − ] kk ≤ ζ N t (cid:0) ( F H ΣF ) − (cid:1) , ∀ k, where ζ (cid:0) ( F H ΣF ) − (cid:1) and ζ N t (cid:0) ( F H ΣF ) − (cid:1) are the mini-mum and maximum eigenvalues of ( F H ΣF ) − , respectivelyand the equality is attainable when F diagonalizes Σ . There-fore, if F satisfies ζ N t (cid:0) ( F H ΣF ) − (cid:1) ≤ T, (32)then, such F also satisfies (6). To further simplify the con-straint, we need another lemma. Lemma 4 (Ostrowski): [55] Let A ∈ C K × K be Hermitianand S ∈ C K × K be nonsingular. If we let the eigenvalues of A and SS H be given by ζ ( A ) ≤ ζ ( A ) ≤ . . . ≤ ζ K ( A ) and ζ ( SS H ) ≤ ζ ( SS H ) ≤ . . . ≤ ζ K ( SS H ) , respectively, then,for each i = 1 , , . . . , K , there exists a positive real number κ i such that ζ ( SS H ) ≤ κ i ≤ ζ K ( SS H ) and ζ i ( SAS H ) = κ i ζ i ( A ) .Let the eigenvalue decomposition (EVD) of Σ and thesingular value decomposition (SVD) of F be Σ = WΛW H , (33a) F = UDV , (33b)where W , U , and V are all unitary matrices. We alsoassume Λ = diag( λ , λ , · · · , λ N t ) , where < λ ≤ λ ≤· · · ≤ λ N t , and D = diag( √ d , √ d , · · · , (cid:112) d N t ) , where < d ≤ d ≤ · · · ≤ d N t , since F is assumed to be offull-rank (nonsingular). Then, by Lemma 4 with K = N t , S = F − and A = Σ − , we have ζ N t (cid:0) ( F H ΣF ) − (cid:1) = ζ N t (cid:0) F − Σ − F − H (cid:1) ≤ ζ N t ( Σ − ) ζ N t (cid:0) ( F H F ) − (cid:1) = 1 λ d , (34) where the equality is also attainable when U = W . As d = ζ ( F H F ) , then if ζ ( F H F ) = d ≥ λ T , (35)we can conclude that F satisfies constraint (6). This completesthe proof of Proposition 1. C. Proof of Theorem 1
First, by Lemma 1, the average SEP with precoding matrix F is given byP ( F ) = 1 N t N t (cid:88) k =1 G (cid:16) [ (cid:0) F H ΣF (cid:1) − ] kk (cid:17) , (36)where G ( x ) is convex for < x < T . Now, followingthe same way as [35], [40], [47] and applying the Jensen’sinequality [56] to function G ( x ) under the constraint (35)result in N t N t (cid:88) k =1 ,d ≥ λ T G (cid:0) [( F H ΣF ) − ] kk (cid:1) ≥ G (cid:32) N t N t (cid:88) k =1 [( F H ΣF ) − ] kk (cid:33) , (37)where the equality in (37) holds if and onlyif [( F H ΣF ) − ] = [( F H ΣF ) − ] = · · · =[( F H ΣF ) − ] N t N t . Recall that, in (33), we let Σ = WΛW H ,and F = UDV , then by a well known trace-inequality [57],we have tr (cid:0) ( F H ΣF ) − (cid:1) = tr (cid:0) F − Σ − F − H (cid:1) = tr (cid:0) ( FF H ) − Σ − (cid:1) = tr (cid:0) UD − U H WΛ − W H (cid:1) ≥ N t (cid:88) k =1 d − N t +1 − k λ − k , (38)where the equality in (38) holds if U = WP , in which P isan anti-diagonal permutation matrix given by P = · · · · · · ... . . . ... ... · · · . Then, using the Cauchy-Schwarz inequality, we can at-tain (cid:80) N t k =1 (cid:112) d N t +1 − k · √ d Nt +1 − k λ k ≤ (cid:113)(cid:80) N t (cid:96) =1 d (cid:96) · (cid:113)(cid:80) N t k =1 1 d Nt +1 − k λ k . Combining this with the power constraint tr( F H F ) = 1 gives us N t (cid:88) k =1 d − N t +1 − k λ − k ≥ (cid:32) N t (cid:88) k =1 λ − / k (cid:33) . (39)The equality in (39) holds if and only if d N t +1 − k = λ − / k (cid:80) N t (cid:96) =1 λ − / (cid:96) , k = 1 , , . . . , N t . (40) Since G ( x ) monotonically increases, combining (40) with (37)leads to N t N t (cid:88) k =1 ,d ≥ λ T G (cid:0) [( F H ΣF ) − ] kk (cid:1) ≥ G (cid:18)(cid:16) N t (cid:88) k =1 λ − / k (cid:17) /N t (cid:19) , where the equality holds if U = WP , the square of thesingularvalues of F meets (40) and V is chosen as thenormalized DFT matrix. Therefore, the optimal solution is (cid:101) F = 1 (cid:112) tr( Λ − / ) WΛ − / (cid:101) V , (41)where (cid:101) V is an arbitrary unitary matrix. By specifically chose (cid:101) V be a normalized unitary maxtrix, we complete the proofof Theorem 1. D. Proof of Theorem 2
By using Lemma 2 with K = N t , T N t = Σ and F ( x ) =1 / √ x , we have Λ = lim N t →∞ (cid:80) N t k =1 λ − / k N t = 12 π (cid:90) π (cid:112) s Σ ( ω ) dω (42)where λ , λ . . . , λ N t are the eigenvalues of Σ . Fornotational simplicity, let ¯ λ N t = (cid:0) (cid:80) N t k =1 λ − / k (cid:1) /N t . Now,using the optimal precoder given in Theorem 1, theresulting minimum average SEP is lim N t →∞ P min ( (cid:101) F ) =lim N t →∞ G (cid:16)(cid:0) (cid:80) N t k =1 λ − / k (cid:1) /N t (cid:17) . Correspondingly, forPAM signal, we obtain ¯P opt , PAM ( a ) = 2( M − M π × (cid:90) π lim N t →∞ (cid:16) η ( M − N t ¯ λ N t sin θ (cid:17) − ( β − N t − dθ ( b ) = 2( M − M π (cid:90) π exp (cid:16) − η ( β − M − sin θ (cid:17) dθ = 2( M − M Q (cid:18)(cid:115) η ( β − M − (cid:19) , (43)where equality ( a ) follows from the fact that (cid:18) η ( M − N t ¯ λ Nt sin θ (cid:19) − ( β − N t − < for all N t ≥ and θ ∈ [0 , π/ and thus, by the Lebesgue’s DominatedConvergence Theorem [58], we can change the order oflimitation and integration. The equality ( b ) is due to thewell-known limit of the Euler’s number. Following the similarargument, for PSK signal, we have ¯P opt , PSK = 1 π (cid:90) ( M − πM lim N t →∞ (cid:32) η sin ( π/M ) N t ¯ λ N t sin θ (cid:33) − ( β − N t − dθ = 1 π (cid:90) ( M − πM exp (cid:18) − η ( β −
1) sin ( π/M )Λ sin θ (cid:19) dθ (44) and for QAM signal, we can attain ¯P opt , QAM = 4( √ M − √ M π (cid:90) π ... lim N t →∞ (cid:32) η M − N t ¯ λ N t sin θ (cid:33) − ( β − N t − dθ − √ M − M π (cid:90) π ... lim N t →∞ (cid:32) η M − N t ¯ λ N t sin θ (cid:33) − ( β − N t − dθ = 4( √ M − √ M Q (cid:18)(cid:115) η ( β − M − (cid:19) − √ M − M Q (cid:18)(cid:115) η ( β − M − (cid:19) . (45)This completes the proof of Theorem 2. E. Proof of Corollary 1
We know that, s Σ ( ω ) = (cid:80) ∞ k = −∞ σ ( k ) e − jkω = −| ρ | | ρ | − Re [ ρe jω ] . for < | ρ | < , we have s Σ ( ω ) ≥ −| ρ | | ρ | and as a result, L s Σ ≥ −| ρ | | ρ | > . Now by Lemma 2 with F ( x ) = 1 / √ x , we attain Λ KMS = 1 π (cid:112) − | ρ | (cid:90) π (cid:112) | ρ | − | ρ | cos ωdω = 2 π (cid:115) | ρ | − | ρ | E (cid:32) (cid:112) | ρ | | ρ | (cid:33) . Combining this with Theorem 2 completes the proof ofCorollary 1.
F. Proof of Theorem 3
Note that KMS matrix Σ has a simple tridiagonal in-verse [51], given by Σ − = 11 − | ρ | − ρ · · · − ρ ∗ | ρ | − ρ · · · . . . . . . . . . · · · − ρ ∗ | ρ | − ρ · · · − ρ ∗ . Combining this with (29) and the uniform precoder yieldsP
QAM ( ¯ F ) = N t G QAM (cid:16) N t −| ρ | (cid:17) + N t − N t G QAM (cid:16) N t (1+ | ρ | )1 −| ρ | (cid:17) . Since lim N t →∞ (cid:16) η (1 −| ρ | )2( M − N t Φ sin θ (cid:17) − ( N r − N t +1) =exp (cid:18) − η ( β − −| ρ | )2( M − θ (cid:19) , where Φ = 1 or | ρ | , we have ¯P U , QAM = lim N t →∞ G QAM (cid:16) N t (1 + | ρ | )1 − | ρ | (cid:17) = 4( √ M − √ M Q (cid:32)(cid:115) η ( β − − | ρ | )( M − | ρ | ) (cid:33) − √ M − M Q (cid:32)(cid:115) η ( β − − | ρ | )( M − | ρ | ) (cid:33) . The expressions of ¯P U , PAM and ¯P U , PSK can be obtainedin a similar fasion and hence are omitted for brevity. Thiscompletes the proof of Theorem 3.R
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