Impact of Antenna Correlation on a New Dual-Hop MIMO AF Relaying Model

A novel system model is proposed for the dual-hop multiple-input multiple-output amplify-andforward relay networks, and the impact of antenna correlation on the performance is studied. For a semiarbitrary correlated source-relay channel and an arbitrary correlated relay-destination channel, the complementary cumulative distribution function (CCDF) and the moment-generating function (MGF) approximations of the end-to-end signal-to-noise ratio (SNR) are derived. The outage probability, the average symbol error rate (SER), and the ergodic capacity approximations are also derived. Two special cases are treated explicitly: (1) dual-antenna relay and multiple-antenna destination and (2) uncorrelated antennas at the relay and correlated antennas at the destination. For the ﬁrst case, the CCDF, the MGF and the average SER of an upper bound of the end-to-end SNR are derived in closed-form. For the second case, the CCDF, the MGF, the average SER, and the moments of SNR are derived in closed-form; as well, the high SNR approximations for the outage probability and the average SER are derived, and the diversity gain and coding gain are developed. Extensive numerical results and Monte Carlo simulation results are presented to verify the analytical results and to quantify the detrimental impact of antenna correlations on the system performance.

Single-antenna AF relaying over two hops with source and destination using multiple antennas is analyzed in [10,11,13]. In these works, beamforming or MRT and MRC technologies are considered. The difference between [11] and [13] is that the former considers independent Rayleigh fading whereas the latter considers independent Nakagamim fading. In particular, [10] extends [13] to study the effect of antenna correlation at the source and destination. Moreover, in [12], the performance in independent Rayleigh fading is derived for a system, where the source uses TAS and the destination, MRC.
References [6,15] analyze the performance of dual-hop multibranch cooperative systems with decode-and-forward (DF) relays equipped with multiple receive antennas and a single transmit antenna. However, the source and the destination are single-antenna terminals. In [6], the performance metrics are derived by considering threshold-based MRC and threshold-based selection diversity combining at the relay. Moreover, [15] extends [6] by employing distributed beamforming to achieve improved capacity gains.
In [14,16,17], the performance of single-relay system, where the source, the relay, and the destination terminals are equipped with multiple transmit/receive antennas, is analyzed. All these works employ space time block codes. The analysis in [14] considers both DF and AF relaying strategies. In [16], the performance analysis employs random matrix theory. Reference [17] derives the exact outage probability in closed form.
For the sake of completeness, we briefly mention some prior research of the second category dealing with singleantenna two-hop AF relay networks. Their performance over Rayleigh fading is analyzed in [4,5]. The performance bounds for the multibranch case of such networks over Nakagami-m fading are derived in [18]. Reference [19] derives their performance over nonidentical Nakagami-m fading links. The performance bounds of such networks over generalized Gamma fading channels and nonidentical Weibull fading channels are derived in [20,22]. In [21], the exact expressions and lower bounds for mixed Rayleigh and Rician fading channels are derived.
Motivation. Although the dual-hop MIMO relay models in [10][11][12]15] provide significant performance gains over single-antenna relaying [4,5,[18][19][20], the following issues may arise in practical network deployments of such networks. In the emerging cellular dual-hop relay networks, employing multiple antennas at the mobile stations (MS) is strictly limited due to power and space constraints. However, there are no such constrains at the base stations (BS). On the other hand, the hardware cost and complexity associated with the relay should be low compared to a traditional BS. Although relaying can be performed by an MS as well, in this work, an infrastructure (fixed) relay [7] is considered. Such a relay can employ multiple antennas.
In this paper, we consider a practical scenario where a single-antenna MS communicates with a multiple-antenna BS via a fixed relay equipped with multiple antennas. This particular setup is shown in Figure 1. The relay uses selection diversity combining (SDC) for signal reception and uses one transmit antenna for forwarding the amplified signal. Although several other antenna setups are possible for the relay, we focus on this setup for several reasons. First, alternatives such as MRC are more costly; if the relay employs MRC reception, a separate receiver chain is required for each receive antenna, and this requirement increases the cost and complexity. Second, a single transmit antenna at the relay keeps the cost comparable to that of a single antenna relay, which requires only one transmit chain. Although this particular transmit/receive strategy adopted at the relay may not be optimal, it is designed to gain MIMO diversity benefits yet keep the costs/complexity at the relay as low as possible. Finally, from a theoretical point of view, there is no difficulty in analyzing other antenna configurations at the relay, but space limitations prevent us from doing so. The proposed system model may thus readily be used for emerging cellular networks with MIMO relaying, where the use of multiple antennas in a BS is reasonable; however the use of multiple antennas at MS may be strictly limited due to the terminal size and power constraints.
Our Contribution. Although MIMO techniques achieve diversity/SNR gains, these gains decrease when there is spatial correlation among the signals received by antenna elements. Therefore, the performance losses due to antenna correlations must be quantified. In this paper, in particularly, we consider the impact of spatial (antenna) correlation on our proposed dual-hop MIMO AF relay network (Figure 1). To the best of our knowledge, the proposed network setup has not been analyzed before, and the differences between our work and [6,10] are as follows. The setup in [10] studies a multiple-antenna source and single-antenna relay. Moreover, in [10], MRT and MRC are employed at the source and destination whereas our setup employs SDC and MRC at the relay and destination, respectively. Although reference [6] considers a relay identical to ours, the destination is a single-antenna terminal. The relaying strategy in [6] is DF whereas AF relaying is considered here. Further, [6] considers independent fading whereas our work treats consider correlated fading.
In our work, for a semiarbitrary correlated source-relay channel and arbitrary correlated relay-destination channel, we derive integral expressions for the complementary cumulative distribution function (CCDF) and the momentgenerating function (MGF) of the end-to-end signal-tonoise ratio (SNR). The numerically efficient Gauss-Laguerre quadrature rule [23] is employed to evaluate the integrals. The outage probability, the average symbol error rate (SER), and the ergodic capacity expressions are also derived. Closedform expressions are derived for the performance of two special cases of antenna correlation: (1) dual-antenna relay and multiple-antenna destination and (2) uncorrelated antennas at the relay and correlated antennas at the destination. We develop closed-form expressions for the CCDF and the MGF of an upper bound of the SNR for the first case. The average SER is evaluated by using the Gauss-Chebyshev quadrature rule [23]. Exact closed-form expressions are derived for the CCDF, the MGF, the average SER, and the moments of SNR of the second case. In particular, for the second case, the high SNR approximations for the outage probability and the average SER are derived and used to obtain valuable insights such as the diversity and the coding gains. Numerical and Monte Carlo simulation results are provided to analyze the system performance, obtain valuable insights, and validate our analysis. The insights provided by our analysis may well be used for designing of MIMO relay networks.
The rest of this paper is organized as follows. Section 2 presents the system and the channel model. In Section 3, the performance analysis is presented. Section 4 contains the numerical and simulation results. Section 5 concludes the paper, and the proofs are annexed.
Notations. K ν (z) is the Modified Bessel function of the second kind of order ν [23, 9.6]. 2

System and Channel Model
We consider the dual-hop relay network in Figure 1. The single-antenna source (S) communicates with the destination (D) having N d ≥ 1 antennas via an AF relay (R). The relay has N r ≥ 1 receive antennas and uses only one antenna among them for forwarding. ( Although TAS can be used at the relay for the second time slot, it provides diversity gains only when N r > N d despite the additional CSI feedback requirement (see Remark 1 in Section 3). However, in practice, N r > N d is unlikely even since the relays should usually be more cost/complexity effective than BS.) Halfduplex transmission is assumed. Since we consider a MIMOenabled infrastructure relay, which is used primarily for extending the network coverage, the direct channel between S and D, which is far apart, is not considered assuming heavy shadowing and path loses. Cooperation occurs in two timeslots. In the first timeslot, the source transmits to the relay. In the second timeslot, the relay forwards an amplified version of the source signal to the destination. The relay and destination employ SDC and MRC receptions, respectively. Perfect channel state information is assumed to be available at the relay and destination.

Source-to-Relay Channel
Model. An arbitrary correlation model for the source-to-relay (S → R) channel appears to be analytically intractable for dual-hop AF relay networks. We instead consider the semiarbitrary correlation model. (The use of the semiarbitrary correlation model simplifies the N rfold numerical integration [26] in evaluating the CCDF of the SDC output SNR for the S → R channel into a single integral [27] and thereby enables the efficient computation of distribution functions of the end-to-end SNR of dual-hop AF relaying.) In other words, the S → R single-input multipleoutput (SIMO) channel is a semiarbitrary correlated [27] flat Rayleigh fading channel. The S → R channel vector is given by Ψ 1/2 R h sr , where Ψ R is the N r × N r covariance matrix at the relay, and h sr is an N r × 1 vector with independent Rayleigh fading entries. The (p, q)th element of Ψ R is given by [28, 8.1.5] Thus, Ψ p,q R can be parameterized by a 1 × N r vector ρ with the pth element ρ p .

Relay-to-Destination Channel Model.
The relay-todestination (R → D) SIMO channel is an arbitrary correlated [29] flat Rayleigh fading channel. In this case, the R → D channel vector is given by Ψ 1/2 D h rd , where Ψ D is the covariance matrix at the destination, and h rd is an N d ×1 vector with independent Rayleigh fading entries. Ψ D is constructed according to a practical channel model [30,31] with the (p, q)th element given by where l d is the relative antenna spacing between adjacent antennas (measured in number of wavelengths) of the linear array of antennas at the destination, θ d is the mean angle of arrival, and σ 2 d is the destination angular spread. The actual angle of arrival is given by . Such a correlation model may arise in practice in uniform linear antenna arrays, and this model appears to be adequate to describe a real-world scattering environment [30].

The End-to-End SNR.
The received signal vector at the relay can be written as where P 1 is the transmit power; x is the transmitted symbol satisfying E {|x| 2 } = 1; n r is the additive white Gaussian noise (AWGN) vector with mean zero and variance N o . The relay employs SDC to obtain the scalar signal as where |h I sr | = max 1≤i≤Nr |h i sr | and {h i sr } Nr i=1 are the elements of Ψ 1/2 R h sr and n r is the AWGN at the Ith antenna. In the second timeslot, the signal y r is amplified with the gain G and forwarded to the destination. The received signal at the destination can be written as where P 2 is the relay transmit power factor, n d is the and w H is the receive weight vector for MRC operation [32] given by (5) to obtain the following model for the received signal at the destination: By using (6), the resulting SNR can be obtained as The gains of practical channel state information-(CSI-) assisted relays can be defined as in [1,4]. Thus, by (7), the SNR of the dual-hop system with the practical CSI-assisted relay can be written as follows: where γ 1 = P 1 |h I sr | 2 /N o is the instantaneous SNR of the SDC output at the relay in semiarbitrary correlated Rayleigh fading, and γ 2 = P 2 Ψ 1/2 D h rd 2 F /N o is the instantaneous SNR of the MRC output at the destination over arbitrary correlated Rayleigh fading. We also define γ 1 = E {γ 1 } and γ 2 = E {γ 2 } as the average of γ 1 and γ 2 , respectively.
We can also consider ideal CSI-assisted relays that invert the S → R channel regardless of its fading state [4]. For this case, the relay gain is given by G = 1/P 1 |h I sr | 2 , and the SNR γ eq,ideal is given by [4] Moreover, the γ eq can be upper bounded as [5,18] γ eq ≤ γ eq,ideal ≤ γ ub eq = min γ 1 , γ 2 .

Performance Analysis
This section presents a comprehensive performance analysis of our proposed MIMO relay network model by taking into account the spatial correlation of antenna elements at the MIMO-enabled terminals. The CCDF and the MGF of the end-to-end SNR γ eq are derived and used to obtain accurate closed-form approximations the outage probability, the average symbol error rate (SER), and the ergodic capacity.

Statistical Characterization of the End-to-End SNR.
An accurate closed-form approximation for the CCDF of the end-to-end SNR in (8) can be derived as (Appendix A) where λ 1 > · · · > λ Nd are the eigenvalues of Ψ D , Δ(Ψ D ) is the determinant of the Vandermonde matrix of the eigenvalues of Ψ D , and Δ Nd, j (M) is the determinant of the matrix M with N d th row and jth column removed. The (i, j)th are the weights and the nodes of the Gauss-Laguerre quadrature rule, respectively. The nodes (y p , t q ) and weights (w p , w q ) can be efficiently computed by using the approach proposed in [33]. Moreover, L is the number of terms used for the Gauss-Laguerre quadrature rule.
The MGF of the end-to-end SNR is a useful statistic which can be used to analyze a wide range of performance metrics. The MGF of γ eq can accurately be approximated as follows (Appendix A): where

Outage Probability.
The outage probability is the probability that the instantaneous SNR γ eq falls below a predefined SNR threshold γ th . Thus, the outage probability P out can readily be obtained by using (11) as follows:

Average Symbol Error Rate.
The average SER is an important performance metric of wireless networks. The conditional error probability (CEP) of the coherent binary frequency-shift keying (C-BFSK) and M-ary pulse amplitude modulation (PAM) is given as P e | γ = aQ( bγ) [32], where a and b are constants dependent on the modulation scheme. For example, the cases (a = 1, b = 2) and (a = 1, b = 1) yield the exact bit error rate of the coherent binary phaseshift keying (BPSK) and C-BFSK, respectively. Further, the SER of M-ary PAM is given by (a = 2(M − 1)/M) and (b = 6 log 2 M/M 2 − 1) [32]. An accurate closed form average SER approximation can be derived by integrating P e | γ over the PDF of the SNR γ eq as follows (Appendix B): where α k,r = 2ρ k t r /(1 − ρ k ) and ζ k,p,q = 4z q (2z q + (1 + EURASIP Journal on Wireless Communications and Networking 5

Ergodic Capacity.
The channel capacity is defined as the maximum data rate at which information can be transmitted across a noisy channel with arbitrary reliability.
The ergodic capacity C γeq is defined as the expected value of the instantaneous maximum mutual information I between S and D. For a dual-hop cooperative relay system operating over two timeslots per burst, I is given by I = (1/2)log 2 (1 + γ eq ) [1]. Then, the ergodic capacity can be expressed as By substituting (A.4) into C γeq and applying the Gauss-Laguerre quadrature rule, an approximation for C γeq in closed form is obtained as follows: where

Special Cases.
In this section, two special cases of our proposed system model are analyzed.

Two
Receive Antennas at the Relay (N r = 2). Consider the case where only two receive antennas are at the relay. This case may arise in practise due to the space limitations at the relay or due to the cost factor. In this section, the lower bounds for the outage probability and the average SER are derived in closed form by using the upper bound of the SNR γ ub eq in (10). The CCDF of γ ub eq can be written as [34] When N r = 2, the CCDF of SDC in the equally correlated Rayleigh fading channels given in (A.2) reduces to the wellknown result [35,36] The CCDF of γ 2 can readily be obtained by using (A.3) and Then, the closed-form expression for the CCDF of γ ub eq can be obtained by substituting (17) and (18) into (16) as follows: where The MGF of γ ub eq can be derived in closed-form by substituting (19) into (A.5) as follows: where σ = 2ρ/γ 1 (1 − ρ 2 ) and ε = (1 + ρ 2 )/γ 1 (1 − ρ 2 ). With the aid of (19), a lower bound of the outage probability can readily be computed as follows: P lb out = 1 − F γ ub eq (γ th ). Next, we present an accurate and computationally efficient closed-form approximation for computing a lower bound for the average SER. The CEP of the coherent binary frequency-shift keying (C-BFSK) and M-ary PAM can be expressed in an alternative form [37] as P e | γ = aQ( bγ) = (a/π) (b/2) ∞ 0 (exp(−γ(s 2 + b/2))/(s 2 + b/2))ds. The average error rate can be obtained in the following form by averaging the alternative CEP expression over the PDF of γ ub eq and manipulating it with the variable transformation s 2 + b/2 = b/(x + 1) [37]: We use an accurate and computationally efficient method in [37] which uses the Gauss-Chebyshev quadrature rule [23] to obtain a very compact closed-form expression for (21): where N p is a positive integer, θ n = (2n−1)π/4N p , and R Np is the remainder term. R Np becomes negligible as N p increases 6 EURASIP Journal on Wireless Communications and Networking [37]. Thus, the lower bound of average SER can be obtained by substituting (20) into (22). Moreover, for the CEP of the noncoherent binary frequency-shift keying (NC-BFSK) and differential BPSK (D-BPSK) is given in the form [32] P e | γ = ae −bγ , with (a = 0.5, b = 1) and (a = 0.5, b = 0.5) for D-BPSK and NC-BFSK, respectively. Now, the average SER of NC-BFSK and D-BPSK can be derived by substituting (19) into P lb e = a − ab ∞ 0 F γ ub eq (x)e −bx dx as follows: where = b+(γ 1 +2γ 2 λ j )/γ 1 γ 2 λ j and ρ = (b + ε + μ) 2 − σ 2 .

Uncorrelated
Antennas at the Relay (ρ = 0). When the antenna spacing at the relay is sufficiently large, the S → R channels experience independent Rayleigh fading. In this case, the CCDF of γ eq can be obtained by substituting ρ = 0 in (A.4) and by using the identity of the Marcum-Q function, Q(0, x) = e −x 2 /2 , as follows (Appendix D): Let us consider an ideal CSI-assisted relay. For this case, the SNR γ eq,ideal is given in (9), and the CCDF of γ eq,ideal can be easily obtained with the aid of (24) as follows: The PDF of γ eq and γ eq,ideal can easily be derived by differentiating (24) and (25) with respect to x and by using x(∂K ν (x)/∂x) + νK ν (x) + xK ν−1 (x) = 0 [38, 8.486.12]. However, for the sake of brevity, the PDF results are omitted.
A closed-form expression for the MGF of γ eq,ideal can be derived by substituting (A.4) into (A.5) as shown in Appendix E: where ψ = (γ 1 + γ 2 λ j l)/γ 1 γ 2 λ j and ϕ = 2 l/γ 1 γ 2 λ j . By substituting (24) into (B.1), a closed-form expression for the average SER can be derived as follows (Appendix F): A generalized closed-form expression for the moments of the SNR can be obtained by substituting (25) into γ n eq,ideal = ∞ 0 nx n−1 F γeq,ideal (x)dx and evaluating the resulting integral as follows (Appendix G): The SNR moments can be used to study the higherorder metrics, such as the skewness and the kurtosis that characterize the distribution of γ eq,ideal . The skewness (S), which is a measure of the symmetry of the distribution, can be obtained as S = γ 3 eq,ideal /(γ 2 eq,ideal ) 3/2 . The kurtosis (K), which quantifies the degree of peakedness of the distribution, is given by K = γ 4 eq,ideal /(γ 2 eq,ideal ) 2 . On the other hand, the amount of fading (AoF) is a performance metric which quantifies the severity of the fading that the signal experienced from the source to the destination. The AoF is given by AoF = γ 2 eq,ideal /(γ eq,ideal ) 2 − 1.

High SNR Analysis.
This section presents the high SNR analyses for the proposed system model when the antennas at the relay are uncorrelated.

The Outage Probability at High SNR.
The behavior of the cumulative distribution function (CDF) of γ eq for large γ is equivalent to the behavior of F γeq (y) around y = 0 [39]. By substituting γ 1 = γ 2 = γ and x = γy into (25) and by expressing the exponential function and the Bessel function in terms of their Taylor series expansions around y = 0 [38, equations (1.211) and (8.446)], one obtains the expression with the lowest powers of y. Now, by collecting the first-order terms, the high SNR approximation (i.e., when γ = γ 1 = γ 2 → ∞) for the CDF of the end-to-end SNR (24) can be derived as follows: where , At high SNR, the outage probability can easily be obtained by substituting (29) into P out = F ∞ γeq (γ th ).

The Symbol Error Rate at High SNR.
The average SER at high SNR can be derived by substituting (29) into P e = (a/2) (b/2π) In the high SNR regime, the average SER can be represented by P ∞ e ≈ [G c γ] −Gd , where G d and G c are referred to as the diversity gain and coding gain, respectively [39]. By using (31), G d and G c are given by respectively.

Remark 1.
The diversity order of the proposed system is given by G d = min(N r , N d ) (32). If a single-antenna relay is used, then G d = min(1, N d ) = 1 even though the destination is equipped with multiple antennas. Thus, our analysis shows that in order to retain MIMO diversity benefits for dual-hop relay networks with single-antenna sources, the relay should be equipped with multiple antennas.
Remark 2. In our proposed system model, the relay uses only one transmit antenna for forwarding the amplified signal to the destination. Since the relay is already equipped with multiple antennas, transmit antenna selection (TAS) can also be employed at the relay. If the transmit antenna, which maximizes the end-to-end SNR, is used at the relay, then the diversity order of the system can be written by following (32) as Thus, the system with TAS guarantees a diversity order of N r whereas the system without TAS provides a diversity order of min(N r , N d ). Hence TAS improves the diversity benefits only when N r > N d . However, when N r ≤ N d , the system with TAS achieves coding gains despite no diversity advantages.

Numerical Results
This section presents the numerical and the simulation results for the proposed dual-hop AF MIMO relaying with antenna correlation. Monte Carlo simulation results are provided to verify the accuracy of the analytical derivations presented in Section 3. In computing the Gauss-Laguerre approximation for the outage probability (13), average bit error rate (BER) (14), and ergodic capacity (15), we choose L to be 25 for 1 ≤ min(N r , N d ) ≤ 3 and L to be 30 for min(N r , N d ) > 3.

Comparison of Semiarbitrary and Arbitrary Correlation
Models for the S → R Channel. In Figure 2, the average BER of BPSK for the proposed dual-hop MIMO relay network is plotted by taking into account two correlation models; (i) semiarbitrary correlation (ii) arbitrary correlation for the S → R channels. For both these cases, the R → D 8 EURASIP Journal on Wireless Communications and Networking channel is arbitrary correlated. The average BER curves for the former case are plotted by using our closed-form BER approximation (14), whereas the Monte Carlo simulation results are used for the latter case. Figure 2 reveals clearly that the semiarbitrary correlation model agrees well with the arbitrary correlation model for different antenna setups. Thus, our analytical results may well be used to obtain design insights for dual-hop MIMO AF relay networks over arbitrary correlated fading. Figure 2 also verifies that the maximum diversity order is given by the minimum number of antenna at the relay and destination; G d = min (N r , N d ).
This fact can also be proved by applying the min-max cut from the S → R and from R → D [40]. The BER curve corresponding to N r = 1, N d = 1 represents the dualhop relaying with single-antenna terminals. The proposed dual-hop relay network with multiple-antenna relay and destination provides a substantial BER performance gains compared to single-antenna relaying. Moreover, the exact agreement between the analytical curves and the Monte-Carlo simulation points verifies the accuracy of our analysis.

Impact of Spatial Correlation on the Average BER.
In Figure 3, the impact of antenna correlation at both the relay and destination on the average BER of the BPSK is depicted. Three different correlation effects are obtained by changing l d , ρ, and σ 2 . Since l d is the relative antenna spacing between the adjacent antennas in the linear array measured in the number of wavelengths, and σ 2 d is the angular spread at the destination, smaller values of l d and σ 2 d result in higher antenna correlation at the destination. Moreover, the amount of antenna correlation at the relay can also be changed by changing ρ, where ρ is a 1 × N r vector containing the elements ρ p and ρ q of S → R semiarbitrary correlation matrix (1). Thus, three correlation scenarios are obtained as (a) high correlation, (b) medium correlation, and (c) low correlation. Figure 3 reveals clearly that the higher correlation effect at both the relay and destination degrades the BER performance significantly.

Impact of Spatial Correlation on the Outage Probability.
In Figure 4, the outage probability is plotted against the correlation coefficient of the S → R channels for the low, medium, and high spatial correlation scenarios in the R → D channels. Here, the S → R channels undergo equally correlated fading. The outage probability performance significantly degrades as the correlation of antennas at the relay increases. Moreover, this figure also shows that the correlation of the antennas at the destination has a considerable impact on the outage probability. Figure 5 shows the impact of spatial correlation of antenna elements at relay and the destination on the ergodic capacity for semiarbitrary correlated S → R and arbitrary correlated R → D channels. Lower antenna spacing (i.e., higher spatial correlation) results in substantial losses in capacity. Figure 5 also reveals that the smaller angular spreads (i.e., higher spatial correlation) degrade the capacity benefits. A  higher number of antennas at both the relay and destination increase the capacity benefits of dual-hop systems. Figure 6 plots the average BER of BPSK against the antenna spacing for three different angular spreads (σ 2 d = π/6, σ 2 d = π/12, and σ 2 d = π/24). Here, the S → R channel is equally correlated whereas the R → D channel is arbitrary correlated. Two different antenna configurations at the relay and destination are considered. Lower antenna spacings and angular spreads at the destination degrade the BER performance considerably. The BER curves seem to be approaching a fixed value once the antenna spacing is increased beyond 2.5 λ, regardless of the angular spread.

Numerical Results for the Special Cases.
This section presents the numerical results for the special cases: (1) dualantenna relay and (2) uncorrelated antennas at the relay. Figures 7 and 8, we analyze the tightness of the lower bounds developed for the average BER and the outage probability for the special case of dualantenna relay. Both the lower bounds are tighter to the exact curves at moderate-to-high SNR regime. Thus, these bounds may serve as benchmarks for performance analysis of practical systems with practical CSI-assisted relays. In computing the lower bound for average BER by using the MGF approach, we choose N p to be 10 for the Gauss-Chebyshev approximation in (22). The exact agreement between the BER curves evaluated using (22) and Monte Carlo simulation results validates the computational accuracy of our analysis. Figures 7 and 8 also reveal that the average BER and the outage probability performance gap between the curves corresponding to high correlation and medium correlation are larger than those of curves corresponding to low correlation and medium correlation. The reason for this performance gap difference is because when l d > 0.38 [32], the effect of antenna correlation at the destination becomes negligible, and the performance degradation is resulted solely by the antenna correlation at the relay. However, when ρ = 0.9, l d = 0.1, the performance is degraded by antenna correlation at the relay and destination. Figure 9, the outage probability of dual-hop MIMO relaying with uncorrelated antennas at the relay and correlated antennas at the destination is plotted against the destination angular spread. It shows that the outage probability degrades considerably due to the spatial correlation of antennas at the destination. Higher angular spreads degrade the outage probability performance significantly. Lowering the antenna spacing at the destination adversely impacts the outage probability. Figure 10 shows the average BER performance of dualhop MIMO relaying with uncorrelated antennas at the relay and correlated antennas at the destination. Figure 10 also shows the high SNR analysis of the average BER for different antenna configurations at the relay and destination. The analytical curves obtained from (27)    Antenna spacing (wavelengths)

Uncorrelated Antennas at the Relay. In
Ergodic capacity (bits/s/Hz)  Antenna spacing (wavelengths) Average bit error rate       Figure 11) is suboptimal, it achieves most of the MIMO benefits while using only one transmit/receive chain at the relay.

Conclusion
A suboptimal yet simple and realistic dual-hop AF MIMO relay network model was developed. The impact of spatial correlation on the performance of the proposed system model over Rayleigh fading was investigated by using a semiarbitrary and arbitrary correlation models for the S → R and R → D channels, respectively. Accurate closed-form expressions for the CCDF, the MGF, the outage probability, the average symbol error rate, and the ergodic capacity were derived. The performance of two special cases was studied: (1) dual-antenna relay and multiple-antenna destination, and (2) uncorrelated antennas at the relay and correlated antennas at the destination. The high SNR approximations for the outage probability and the average were derived to obtain valuable system design insights such as the diversity order and coding gain. Numerical and Monte Carlo simulation results were presented to investigate the detrimental effect of the antenna correlation on the system performance and to validate our analyses. Our results show that in order to retain MIMO benefits for dual-hop relay networks that consist of single-antenna sources and multiple-antenna destinations, MIMO-enabled relays should be used. Our results may be useful in analyzing practical system scenarios that involve a single-antenna portable device communicating with a multiple-antenna base station via an infrastructurebased fixed relay equipped with multiple antennas.

Appendices
This section provides sketches of the proofs of some of the results presented in Section 3.

A. Statistical Characterization of the End-to-End SNR
The closed-form approximation for the CCDF of the endto-end SNR in (8) can be obtained by using the following integral expression [13]: where F γ1 (x) and f γ2 (x) are the CCDF and probability density function (PDF) of γ 1 and γ 2 , respectively. The CCDF of γ 1 can be written as in [27,36] The PDF of γ 2 can be written as in [10] f γ2 (x) = , we obtain an integral expression for the CCDF of the SNR as follows: where Υ 1 (t) = 2ρ k t/(1 − ρ k ) and Υ 2 (t, y) = 2x(x + γ 2 λ j y + 1)/γ 1 γ 2 λ j (1 − ρ k )y.
No closed-form solution for the double-integral (A.4) appears to be available. However, it is in the form of ∞ 0 f (x, y)e −x e −y dx dy, and, thus, it can be efficiently and accurately approximated by using the Gauss-Laguerre quadrature rule [23] in closed form as in (11).

D. CCDF of SNR with Uncorrelated Antennas at the Relay
When the S → R channel undergoes independent Rayleigh fading, one can obtain the CCDF of γ eq by letting