Analysis of Correlated MRC with Transmit Antenna Selection under eta-mu Fading

0018-9545 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2685683, IEEE Transactions on Vehicular Technology 1 Analysis of Correlated MRC with Transmit Antenna Selection under η-µFading Concepci´ on T´ ellez-Labao, Juan P. Pe˜ na-Mart´ ın and Juan M. Romero-Jerez, Senior Member, IEEE Abstract—A wireless link considering correlated Maximal Ratio Combining (MRC) and Transmit Antenna Selection (TAS) under the general η-µfading is analyzed. Exact and asymptotic expressions for the cumulative distributing function (CDF) and probability density function (PDF) are derived. These new results are used to evaluate the outage capacity and the asymptotic average bit error rate (average BER) for different binary modulations. The obtained theoretical results are validated through Monte Carlo simulations. Index Terms—Bit error rate, asymptotic analysis, generalized fading. I. INTRODUCTION AMONG the different possible Multiple Input - Multiple Output (MIMO) schemes, which employ multiple transmit and receive antennas, Transmit Antenna Selection (TAS) with Maximal Ratio Combining (MRC) at the receiver provides full diversity gain and reduces the implementation complexity by selecting the transmit antenna that maximizes the output SNR [1]. Thus, TAS/MRC is currently the focus of much attention by the research community, as it has been proposed as a part of present and future wireless communications systems, including multi-hop, spectrum-sharing and securecommunications systems (see, for example, [2]–[5] and the references therein). Multi-antenna signal reception in a portable or handheld terminal will typically yield to signal correlation due to insufficient antenna separation, resulting in a performance degradation that needs to be quantified for a proper system design. In spite of the vast amount of literature dealing with TAS/MRC systems, few works consider receive antenna correlation, and results seem to be available only for correlated Rayleigh and Nakagami-mfading, e.g. [6]–[9]. The η-µfading is a general fading model that can be used to better represent the small-scale variations of the radio signal in a non-line-of-sight condition [10], and includes the Hoyt (Nakagami-q), the Nakagami-mand the Rayleigh fading models as particular cases. On the other hand, the generality of the model yields to a considerable increased mathematical complexity, which precludes the derivation of simple expressions for the performance analysis of wireless Copyright (c) 2017 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported by the European Regional Development Fund and the Spanish Ministry of Economy and Competitiveness under Grant TEC201342711-R. The authors are with the Department of Electronic Technology, E.T.S.I. Telecomunicaci´ on, Universidad de M´ alaga, 29071 M´ alaga, Spain (e-mail: [email protected], [email protected], [email protected]). communication systems. Thus, to the author’s knowledge, there is no previous result for TAS/MRC in correlated η-µ fading, and results seem to be available only for the case of uncorrelated antennas [11]. A general framework for the asymptotic evaluation of the symbol error rate (SER) suitable for arbitrary fading was presented in [2], however, uncorrelated receive antennas was assumed. A comprehensive analysis of TAS/MRC under η-µconsidering arbitrary receive antenna correlation is presented in this work. The CDF and PDF of the instantaneous output signal-to-noise ratio (SNR) are derived both in exact form, in terms of a confluent Lauricella function, and in asymptotic form, which tends to the exact values as the average SNR per receive antenna increases. These chief statistics are new in the literature, to the authors’ knowledge, and are used to find exact and asymptotic expressions of the outage capacity, as well as asymptotic BER expressions for several binary modulations. The remainder of this work is organized as follows: The system model is presented in Section II. In Section III, exact and asymptotic expressions of the CDF and PDF of the instantaneous SNR are derived. In Section IV, expressions for the outage capacity and the asymptotic BER for several binary modulations are found. Some application results are shown in Section V. Finally, main conclusions are drawn in Section VI. II. SYSTEM MODEL We consider a MIMO system with Lrreceive antennas and Lttransmit antennas. Spatial diversity is exploited at the transmit end by means of TAS, while at the receive end MRC is performed. We assume a slowly varying (with respect to the symbol duration) and non-selective (flat) fading. The transmit antennas are assumed to be uncorrelated, while the receive antennas are assumed to be spatially correlated with arbitrary correlation between any pair of antennas. An errorfree feedback channel is assumed to allow the transmitter to select the transmit antenna which maximizes the output SNR. The channel between any transmit and receive antenna is assumed to undergo η-µfading. This is a general fading model that considers the received signal to be the composition of 2µsignal clusters, when 2µis an integer, which is assumed here. Format 1 of this fading model is considered, in which the ηparameter represents the power ratio between the inphase and quadrature components in each signal cluster [10]. The instantaneous SNR at receive antenna iconditioned on antenna jbeing selected for transmission will be γi|j= 2µi,j X k=1 X2 i,k|j+Y2 i,k|j(1) 0018-9545 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2685683, IEEE Transactions on Vehicular Technology 2 where Xi,k|jand Yi,k|jare mutually independent zero-mean Gaussian random variables with variances σ2 Xi,k|j= ¯γi|jηi,j/(2µi,j (1 + ηi,j)) and σ2 Yi,k|j= ¯γi|j/(2µi,j (1 + ηi,j)), where ¯γi|jdenotes the average SNR at receive antenna iconditioned on antenna jbeing selected for transmission, while ηi,j and µi,j are the fading parameters in the link between transmit antenna jand receive antenna i. The SNR at the output of the MRC combiner, conditioned on antenna jbeing selected for transmission is given by γj= Lr X i=1 γi|j.(2) It is proved in [12] that, when 2µi.j are integers, the MGF of γjfor correlated MRC reception under η-µfading is given by Mγj(s) = Vj Y v=1 1−2λ(Xj) vs−ξ(Xj) v21−2λ(Yj) vs−ξ(Yj) v2, (3) where {λ(Xj) v}Vj v=1 and {λ(Yj) v}Vj v=1 are the distinct eigenvalues, and {ξ(Xj) v}Vj v=1 and {ξ(Yj) v}Vj v=1 their associated algebraic multiplicities, of the covariance matrices KXj= Cov XT j,Xjand KYj=Cov YT j,Yj, where Xj= hX1|j, . . . , XLr|jiwith Xi|j=Xi,1|j, . . . , Xi,2µi,j |j, and Yj=hY1|j, . . . , YLr|jiwith Yi|j=Yi,1|j, . . . , Yi,2µi,j |j. Note that KXjand KYjare square matrices of dimension PLr i=1 2µi,j and their elements are determined considering Cov Xi,k|j, Xp,q|j=σXi,k|jσXp,q|j√ρi,p and Cov Yi,k|j, Yp,q|j=σYi,k|jσYp,q|j√ρi,p if k=q, where ρi,p is the correlation coefficient between the SNR at receive antennas iand p. while Cov Xi,k|j, Xp,q|j= Cov Yi,k|j, Yp,q|j= 0 if k6=q, with i, p = 1, . . . , Lr, k= 1, . . . , 2µi,j and q= 1,...,2µp,j. For the sake of compactness, in the following, and unless explicitly stated otherwise, we will consider equal channel parameters for any pair of transmit and receive antennas, and the sub-indices will be dropped from the notation when unnecessary. In this case, the MGF given in (3) can be written in terms of the eigenvalues of the signal correlation matrix of the received antennas [13], which permits to avoid constructing matrices KXand KY. Actually, we demonstrate in the next Proposition that, in this case, the sum of correlated η-µrandom variables is statistically equivalent to the sum of independent η-µrandom variables where the individuals SNRs, assumed to be the same at every receive antenna, are weighted by the eigenvalues of the signal correlation matrix. Proposition 1. Let us assume a correlated MRC receiver under η-µfading with equal parameters at every branch, then, the MGF of the output SNR, given that antenna jis selected for transmission, can be written as Mγj(s) = V Y v=1 1−η¯γ µ(1 + η)λvs−µζv ×1−¯γ µ(1 + η)λvs−µζv (4) where {λv}V v=1 are the distinct eigenvalues, and {ζv}V v=1 their associated algebraic multiplicities, of the signal correlation matrix Mof the receive antenna array, which is assumed to be non-singular. Note that Mis a square matrix of dimension Lrand PV v=1 ζv=Lr. Proof: When the fading parameters at the receive antennas are equal, matrices KXand KYare constructed considering that Cov Xi,k|j, Xp,q|j=ηCov Yi,k|j, Yp,q|j,(5) Cov Yi,k|j, Yp,q|j=   ¯γ 2µ(1+η),for i=p, k =q, ¯γ 2µ(1+η)√ρi,p,for i6=p, k =q, 0,for the rest of cases, (6) where i, p = 1, . . . , Lr;k, q = 1,...,2µ; and where ρi,p is the correlation coefficient between the received SNR at antennas i and pof the receive array. Thus, the correlation matrices KX and KYcan be expressed in compact form as KX=ηKY,(7) KY=γ 2µ(1+η)M⊗I2µ,(8) where Indenotes the n×nidentity matrix, ⊗denotes the kronecker product and Mis the signal correlation matrix which elements are given by M(i, p) = √ρi,p, i, p = 1, . . . , Lr.(9) From (7) and (8) and the fact that the eigenvalues of the kronecker product of two matrices are given by the product of the eigenvalues of the involved matrices, it is clear that the eigenvalues of KXand KY(λ(X) vand λ(Y) v) are related to the eigenvalues of Mby λ(X) v=¯γη 2µ(1 + η)λv, λ(Y) v=¯γ 2µ(1 + η)λv,(10) which implies that KX,KYand Mhas the same number of different eigenvalues, satisfying ξ(X) v=ξ(Y) v= 2µζv, as I2µ has a unity eigenvalue with multiplicity 2µ, which together with (3) yields to (4). Note that for 2µ=m,η= 1, the result in (4) collapses to [9, eq. (2)] for Nakagami-mfading. As TAS is performed at the transmitter, the unconditional output SNR will be given by γ= maxj{γj},j= 1, . . . , Lt. In the next section we derive both exact and asymptotic expressions for the PDF and CDF of γ. III. OUTPUT SNR STATISTICS We now show that it is possible to find exact expressions for both the PDF and the CDF of the output SNR of the proposed system model. Lemma 1. The PDF and CDF of the output SNR of a correlated MRC receiver with transmit antenna selection under η-µ fading with equal parameters can be expressed in terms of the confluent Lauricella function Φ2(·), defined in [14, p. 34, (8)], as given, respectively, in (11) and (12), where we have defined Z={ζ1, . . . , ζV}and ∆=µ(1 + η){λ−1 1, . . . , λ−1 V}. 0018-9545 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2685683, IEEE Transactions on Vehicular Technology 3 fγ(x) = 2µLrLt [(2µLr)!]Lt[det (M)]2µLt µ2(1 + η)2 η!µLrLt1 ¯γx ¯γ2µLrLt−1 ×Φ(2V) 2µZ, µZ; 2µLr;−∆ η x ¯γ,−∆x ¯γΦ(2V) 2µZ, µZ; 2µLr+ 1; −∆ η x ¯γ,−∆x ¯γLt−1 , (11) Fγ(x) = 1 [(2µLr)!]Lt[det (M)]2µLt µ2(1 + η)2x2 η¯γ2!µLrLtΦ(2V) 2µZ, µZ; 2µLr+ 1; −∆ η x ¯γ,−∆x ¯γLt .(12) Proof: From (4), the MGF of γjcan be rewritten as Mγj(s) = 1 [det (M)]2µ µ2(1 + η)2 η¯γ2!µLr1 (−s)2µLr × V Y v=1 1−µ(1 + η) η¯γλvs−µζv1−µ(1 + η) ¯γλvs−µζv . (13) The PDF fγj(x)and CDF Fγj(x)of the output SNR conditioned on antenna jbeing selected for transmission are given by the inverse Laplace transformations fγj(x) = L−1[Mγj(−s); x], and Fγj(x) = L−1[Mγj(−s)/s;x], which can be found using [14, p. 290, (55)]. Considering now that the PDF and CDF of the output SNR are given, respectively, by fγ(x) = PLt j=1 fγj(x)QLt k=1,k6=jFγk(x)and Fγ(x) = QLt j=1 Fγj(x), the final results are obtained. Remark. For the case of non-necessarily equal channel parameters between the transmit and receive antenna pairs, the MGF of γjgiven in (3) can be rewritten as Mγj(x) = 1 (−2s)PLr i=1 2µi,j qdet KXjdet KYj × Vj Y v=1 1−1 2λ(Xj) vs−ξ(Xj) v21−1 2λ(Yj) vs−ξ(Yj) v2 , (14) and following the same approach as in Lemma 1, the PDF and CDF of the output SNR can be obtained for the general case. However, the resulting expressions will be less compact and not explicitly given in terms of the channel parameters and the average SNR per receive antenna, but in terms of matrices KXj,KYjand their eigenvalues.  The derived exact expressions can be accurately calculated numerically, however, they do not offer insights on the impact of system parameters on performance. Fortunately, they permit the derivation of asymptotic expressions in the high SNR regime. Corollary 1. The asymptotic behavior, as γ→ ∞, of the PDF and CDF of the output SNR of a correlated MRC receiver with transmit antenna selection under η-µfading with equal parameters can be obtained as fγj(x) = 2µLrLt [(2µLr)!]Lt[det (M)]2µLt × µ2(1 + η)2 η¯γ2!µLrLt x2µLrLt−1+o¯γ−2µLrLt, (15) Fγj(x) = 1 [(2µLr)!]Lt[det (M)]2µLt × µ2(1 + η)2 η!µLrLtx ¯γ2µLrLt +o¯γ−2µLrLt, (16) where the function o(g(x)) is any function such that limx→0o(g(x))/g(x) = 0. Proof: These expressions can be obtained from (11) and (12) and the fact that, from the definition of Φ2(·), given in [14, p. 34, (8)], it is clear that. Φ(n) 2(b1, . . . , b1;c;d1x, . . . , dnx) = 1 + o(x), with 0<<1. Note that the obtained PDF has the form fγ(x). =α γx γt +o¯γ−(t+1),(17) where parameters tand αare given by t= 2µLrLt−1,(18) α=2µLrLt [(2µLr)!]Lt[det (M)]2µLt µ2(1 + η)2 η!µLrLt .(19) Note that parameters αand tcan be used to readily obtain closed-form expressions of different performance metrics, such as the average error rate for different binary and M-ary modulations in the high SNR regime [15]. IV. PERFORMANCE ANALYSIS OF WIRELESS COMMUNICATIONS SYSTEMS As we have been able to obtain exact and asymptotic expressions of the correlated MRC receiver with TAS under η-µfading, we can calculate different performance metrics of wireless communications systems employing this MIMO scheme under the considered fading model. As an example of application, we compute the outage capacity probability and the asymptotic BER for several binary modulations. 0018-9545 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2685683, IEEE Transactions on Vehicular Technology 4 A. Outage capacity probability The instantaneous channel capacity per unit bandwidth is well known to be given by C= log2(1 + γ).(20) We define the outage capacity probability as the probability that the instantaneous channel capacity Cfalls below a predefined threshold RS(given in terms of rate per unit bandwidth), i.e., Pout =P(C < RS) = P(log2(1 + γ)< RS).(21) Therefore Pout =Pγ < 2RS−1,(22) yielding Pout =Fγ2RS−1,(23) which together with (12) permits to calculate the exact outage capacity probability. This result offers little insight on the effect of parameters on performance. Fortunately, we can obtain a simple expression in the high SNR regime as, from (16), we can write Pout . =1 [(2µLr)!]Lt[det (M)]2µLt × µ2(1 + η)2 η!µLrLt2RS−1 ¯γ2µLrLt ,¯γ→ ∞, (24) where the symbol . =denotes asymptotic equality as γ→ ∞. B. Asymptotic average BER The average error rate for several binary modulations can be expressed in compact form using [16, eq. (8.100)] as Pe=Z∞ 0 Γ(b, ax) 2Γ(b)fγ(x)dx, (25) where Γ(m, x) = R∞ xzm−1e−zdz denotes the incomplete Gamma function, Γ(m) = Γ(m, 0) denotes the Gamma function and where (a, b) = (1,0.5) for binary phase shift keying (BPSK), (a, b) = (0.5,0.5) for binary frequency shift keying (BFSK), (a, b) = (1,1) for differential binary phase shift keying (DBPSK) and (a, b) = (0.5,1) for non-coherent binary frequency shift keying (NCBFSK). Alternatively, integrating (25) by parts, the average error rate can be computed from the CDF as Pe=ab 2Γ(b)Z∞ 0 xb−1e−axFγ(x)dx. (26) From the derived exact expression of the CDF given in (12) it is not possible to find a closed-form exact expression for the average BER. However, (26) is well suited for numerical evaluation using the Gauss-Laguerre quadrature method [17, p. 890 (25.4.45)]. On the other hand, the introduction of (16) into (26) leads to a closed-form asymptotic expression as Pe. =Γ (b+ 2µLrLt) 2Γ (b) [(2µLr)!]Lt[det (M)]2µLt × µ2(1 + η)2 η!µLrLt1 (a¯γ)2µLrLt,¯γ→ ∞. (27) 5 10 15 20 25 30 35 10−18 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 Average SNR per branch (dB) Outage Capacity Probability η= 0.5,µ= 2 η= 0.2,µ= 1 η= 0.1,µ= 0.5 Asymptotic behavior Monte Carlo Simulation ρ= 0.2 ρ= 0.9 Figure 1. Outage capacity probability vs. average SNR per branch for a capacity threshold RS= 5 bps/Hz when Lr= 2 and Lt= 2. V. NUMERICAL RESULTS In this section, we show some representative results of the analyzed system. For simplicity of discussion, we have considered a constant correlation ρbetween each pair of receive antennas. The Φ2(·)function is fast and efficiently computed as a numerical Inverse Laplace Transform using [18]. Fig. 1 shows the outage capacity probability vs. the average SNR per branch for a given capacity threshold RS= 5 bps/Hz when Lr= 2 and Lt= 2 considering correlated receive antennas. The figure plots the exact results, the asymptotic results and also some points obtained by Monte Carlo simulation that show an excellent agreement with the theoretical results. We can appreciate how, for a given set of values for the channel parameters ηand µ, the higher the correlation the greater the outage probability, as expected due to the loss of channel variability among the receive antennas. We can also see in Fig. 1 the influence of the channel parameters: the severity of fading decreases when µincreases and, consequently, the outage probability becomes lower. On the other hand, the η parameter has a much smaller influence and it can be observed that the higher the value of η, for η∈(0,1], the lower the fading severity, thus yielding a better performance. Fig. 2 represents the outage capacity probability vs. correlation coefficient for a capacity threshold RS= 5 bps/Hz and γ= 15 dB. The influence of the sizes of the transmit and receive arrays with different channel parameter values can be analyzed. It is shown that, as the number of antennas increases, the outage capacity probability reduces. Comparing the cases, Lr= 2, Lt= 3 and Lr= 3, Lt= 2, it is interesting to note that a better performance (lower outage capacity probability) is obtained when more antennas are deployed at the receiver, except for the case of high correlation, as the MRC receiver is less effective when the receive antennas experience similar channels. Finally, Fig. 3 shows the asymptotic average BER for different modulation schemes, correlation and channel parameter values. As expected, the average asymptotic BER increases 0018-9545 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2685683, IEEE Transactions on Vehicular Technology 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−5 10−4 10−3 10−2 10−1 100 ρ(Correlation Coefficient) Outage Capacity Probability η= 0.2,µ= 1 η= 0.1,µ= 0.5 Monte Carlo Simulation Lr= 3, Lt= 3 Lr= 3, Lt= 2 Lr= 2, Lt= 3 Lr= 2, Lt= 2 Figure 2. Outage capacity probability vs. correlation coefficient for a capacity threshold RS= 5 bps/Hz and an average SNR γ= 15 dB. 10 15 20 25 30 35 10−15 10−10 10−5 100 Average SNR per branch(dB) Asymptotic Average BER η= 0.5,µ= 2 η= 0.2,µ= 1 η= 0.1,µ= 0.5 ρ= 0.2 BPSK ρ= 0.9 NCBFSK ρ= 0.9 BPSK Figure 3. Asymptotic average BER vs. average SNR per branch for Lr= 2 and Lt= 2.. when the channel fading is more severe (lower ηand/or µ). On the other hand, it is shown how correlation degrades performance, i.e., the lower ρthe lower the BER in the high SNR regime. Also, it is shown that the NCBFSK modulation shows a worse behavior than BPSK, as the former is a noncoherent modulation. VI. CONCLUSIONS In this work, expressions for the PDF and CDF of the output SNR are derived for the first time for TAS/MRC systems under η-µfading and arbitrary correlation at the receive end, both in exact and asymptotic form. 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