Impact of Antenna Correlation on a New Dual-Hop MIMO AF Relaying Model
© Gayan Amarasuriya et al. 2010
Received: 24 November 2009
Accepted: 25 April 2010
Published: 30 May 2010
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.
Cooperative relay networks have been the focus of a flurry of research activities and standard deployment [1–5]. The use of multiple antennas at the source, relay, and/or destination of relay networks offers significant performance gains [6–15]. Such cooperative multiple-input multiple-output (MIMO) relaying opens up the possibility of deploying diversity transmission techniques such as beamforming, maximal ratio transmission (MRT), maximal ratio combining (MRC), and transmit antenna selection (TAS) strategies [10–12, 15]. In this paper, a suboptimal yet a simple and efficient system model, which achieves a better trade-off among the hardware cost, complexity, and the performance, is proposed and analyzed for dual-hop MIMO amplify-and-forward (AF) relay networks.
Prior Related Research
The prior work can be divided into two broad categories. The first category deals with multiple-antenna terminals (source, relay, and destination) [6, 10–17]. The second category considers single-antenna terminals only [4, 5, 18–21].
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  and  is that the former considers independent Rayleigh fading whereas the latter considers independent Nakagami- fading. In particular,  extends  to study the effect of antenna correlation at the source and destination. Moreover, in , 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 , the performance metrics are derived by considering threshold-based MRC and threshold-based selection diversity combining at the relay. Moreover,  extends  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  considers both DF and AF relaying strategies. In , the performance analysis employs random matrix theory. Reference  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 single-antenna 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- fading are derived in . Reference  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 , the exact expressions and lower bounds for mixed Rayleigh and Rician fading channels are derived.
Although the dual-hop MIMO relay models in [10–12, 15] provide significant performance gains over single-antenna relaying [4, 5, 18–20], 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  is considered. Such a relay can employ multiple antennas.
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  studies a multiple-antenna source and single-antenna relay. Moreover, in , 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  considers a relay identical to ours, the destination is a single-antenna terminal. The relaying strategy in  is DF whereas AF relaying is considered here. Further,  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 moment-generating function (MGF) of the end-to-end signal-to-noise ratio (SNR). The numerically efficient Gauss-Laguerre quadrature rule  is employed to evaluate the integrals. The outage probability, the average symbol error rate (SER), and the ergodic capacity expressions are also derived. Closed-form expressions are derived for the performance of two special cases of antenna correlation: ( ) dual-antenna relay and multiple-antenna destination and ( ) 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 . 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.
is the Modified Bessel function of the second kind of order [23, ]. is the Gauss Hypergeometric function [23, ]. denotes the Gaussian Q-function[23, ]. is the th-order Marcum -function [24, equation ( )]. For the sake of brevity, we write to denote . is the th-order Modified Bessel function of the first kind [23, ]. is the Exponential integral function [25, ]. denotes the expected value of over . and are the Frobenius norm and conjugate transpose of .
2. System and Channel Model
We consider the dual-hop relay network in Figure 1. The single-antenna source ( ) communicates with the destination ( ) having antennas via an AF relay ( ). The relay has 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 despite the additional CSI feedback requirement (see Remark 1 in Section 3). However, in practice, is unlikely even since the relays should usually be more cost/complexity effective than BS.) Half-duplex transmission is assumed. Since we consider a MIMO-enabled infrastructure relay, which is used primarily for extending the network coverage, the direct channel between and , 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.
2.1. Source-to-Relay Channel Model
2.2. Relay-to-Destination Channel Model
where is the relative antenna spacing between adjacent antennas (measured in number of wavelengths) of the linear array of antennas at the destination, is the mean angle of arrival, and is the destination angular spread. The actual angle of arrival is given by with . 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 .
2.3. The End-to-End SNR
where is the instantaneous SNR of the SDC output at the relay in semiarbitrary correlated Rayleigh fading, and is the instantaneous SNR of the MRC output at the destination over arbitrary correlated Rayleigh fading. We also define and as the average of and , respectively.
3. 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 are derived and used to obtain accurate closed-form approximations the outage probability, the average symbol error rate (SER), and the ergodic capacity.
3.1. Statistical Characterization of the End-to-End SNR
where are the eigenvalues of , is the determinant of the Vandermonde matrix of the eigenvalues of , and is the determinant of the matrix with th row and th column removed. The th element of is given by where denotes the eigenvalues of . Further, and are the weights and the nodes of the Gauss-Laguerre quadrature rule, respectively. The nodes and weights can be efficiently computed by using the approach proposed in . Moreover, is the number of terms used for the Gauss-Laguerre quadrature rule.
3.2. Outage Probability
3.3. Average Symbol Error Rate
3.4. Ergodic Capacity
3.5. Special Cases
In this section, two special cases of our proposed system model are analyzed.
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 in (10).
where is a positive integer, and is the remainder term. becomes negligible as increases . Thus, the lower bound of average SER can be obtained by substituting (20) into (22).
The PDF of and can easily be derived by differentiating (24) and (25) with respect to and by using [38, ]. However, for the sake of brevity, the PDF results are omitted.
The SNR moments can be used to study the higher-order metrics, such as the skewness and the kurtosis that characterize the distribution of . The skewness ( ), which is a measure of the symmetry of the distribution, can be obtained as . The kurtosis ( ), which quantifies the degree of peakedness of the distribution, is given by . 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 .
3.6. High SNR Analysis
This section presents the high SNR analyses for the proposed system model when the antennas at the relay are uncorrelated.
3.6.1. The Outage Probability at High SNR
3.6.2. The Symbol Error Rate at High SNR
The diversity order of the proposed system is given by (32). If a single-antenna relay is used, then 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.
Thus, the system with TAS guarantees a diversity order of whereas the system without TAS provides a diversity order of . Hence TAS improves the diversity benefits only when . However, when , the system with TAS achieves coding gains despite no diversity advantages.
4. 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 to be 25 for and to be 30 for .
4.2. Impact of Spatial Correlation on the Average BER
4.3. Impact of Spatial Correlation on the Outage Probability
4.4. Impact of Spatial Correlation on the Ergodic Capacity
4.5. Sensitivity to the Antenna Spacing and Angular Spread
4.6. Numerical Results for the Special Cases
4.6.1. Dual-Antenna Relay
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 , 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 , , the performance is degraded by antenna correlation at the relay and destination.
4.6.2. Uncorrelated Antennas at the Relay
4.7. Comparison of BER of BPSK for Several Dual-Hop MIMO AF Relay System Models
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 and 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: ( ) dual-antenna relay and multiple-antenna destination, and ( ) 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 infrastructure-based fixed relay equipped with multiple antennas.
A. Statistical Characterization of the End-to-End SNR
where and . No closed-form solution for the double-integral (A.4) appears to be available. However, it is in the form of , and, thus, it can be efficiently and accurately approximated by using the Gauss-Laguerre quadrature rule  in closed form as in (11).
B. Average Symbol Error Rate
where and . Since the triple integral (B.2) does not appear amenable to a closed-form solution, we again use the Gauss-Laguerre quadrature rule to obtain an accurate average SER approximation as in (14).
C. MGF of the Upper Bounded SNR with Two Receive Antennas at the Relay
The integral can be solved by substituting and by using [41, equations ( ) and ( )] as where , , and . The integral can be solved by using [42, ] as where . Substitution of and into (C.1) yields the desired result given in (20).
D. CCDF of SNR with Uncorrelated Antennas at the Relay
By evaluating the integral in (D.1) by using [25, ], one can obtain the result given in (25).
E. MGF of the SNR with Uncorrelated Antennas at an Ideal CSI-Assisted Relay
where The integral can be evaluated by using [25, ] to yield the desired result given in (26).
F. Average SER with Uncorrelated Antennas at an Ideal CSI-Assisted Relay
where We solve by using [25, ], and substituting into (F.1), we obtain the desired result given in (27).
G. Moments of SNR with Uncorrelated Antennas at an Ideal CSI-Assisted Relay
where . The desired result in (28) can be obtained by solving by using [25, ].
This section provides sketches of the proofs of some of the results presented in Section 3.
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