Impact of antenna correlation on the performance of partial relay selection

Antenna correlation is generally viewed as an obstacle to realize the desired performance of a wireless system. In this article, we investigate the performance of partial relay selection in the presence of antenna correlation. We consider both channel state information (csi)-assisted and fixed gain amplify-and-forward (AF) relay schemes. The source and the destination are equipped with multiple antennas communicating via the best first hop signal-to-noise ratio (SNR) relay. We derived the closed form expression for outage probability, average symbol error rate (SER) for both schemes. Further, an exact expression is derived for the ergodic capacity in the csi-assisted relay case and an approximated expression is considered for the fixed gain case. Moreover, we provide simple asymptotic results and show that the diversity order of the system remains unchangedwith the effect of antenna correlation for both types of relay schemes. Introduction Two-hop amplify-and-forward (AF) relay networks have been investigated extensively in recent research [1-4]. The systemwith a source and a destination both equippedwith multiple antennas communicating via a single antenna relay has received significant interest in most of the previous literature [5-11]. Different transmission and receive techniques were used and use of maximal ratio transmission (MRT) and maximal ratio combining (MRC) were among the most significant ones [5-9]. The analyzes in these cases were carried out with different fading channel environments for performance evaluation. Antenna correlation is generally considered as a detrimental effect which degrades the system performance. To investigate this loss, several authors have studied the effect of antenna correlation in AF relay schemes. Authors in [7] have analyzed the channel state information (csi)assisted AF relay network under antenna correlation with distinct eigenvalue distribution of correlation matrices and the fixed gain scheme has been considered in [12]. Then the general case of arbitrary distributed correlation matrix structures has been investigated by the authors in [9]. However, these evaluations are limited to the single source, relay and destination scenario. *Correspondence: [email protected] Department of Communications Engineering, University of Oulu, Oulu, Finland It has been proven that the use of multiple relays with different selection methods can enhance the diversity and the performance [13-23]. There are several ways of selecting a relay for transmission. One method is referred to as the opportunistic relay selection [13,14] in which the relay with maximum instantaneous end-to-end signal-to-noise ratio (SNR) is considered. Synchronization is very important in this case. Another is the partial relay selection method, which can be carried out in two ways; by selecting either the first-hop relay [15,19,21] or the second hop relay [13,17,19] with the maximum instantaneous SNR. All these studies have been concentrated on the independent and identically distributed fading environments with some considering the effect of feedback delay. Contribution Although authors in the previous literature have studied the AF relay network under the effect of antenna correlation, all these works have been limited to single relay network. Hence, it motivated us to investigate the performance of partial relay selection with the effect of antenna correlation. We consider two types of AF relay schemes; csi-assisted and fixed gain relay. The exact closed form expressions for outage probability and average symbol error rate (SER) are derived for both schemes and an exact ergodic capacity expression is derived for the csi-assisted case and an approximation is found for the other case. Further, we study the system in high SNR and derive simple © 2012 Ferdinand et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Ferdinand et al. EURASIP Journal onWireless Communications and Networking 2012, 2012:261 Page 2 of 13 http://jwcn.eurasipjournals.com/content/2012/1/261 asymptotic expressions for outage probability and average SER for both cases. Our asymptotic analysis provide the depth in the system performance and it shows the variation of diversity gain. Finally, we give Monte Carlo simulations to verify our results. Systemmodel Consider an AF relay network where a source (S) communicates with a destination (D) via the best relay (R). Both S and D are equipped with ns and nd antennas, respectively, and relays are equipped with a single antenna. Direct path between source to destination is assumed to be unavailable due to heavy shadowing. The csi is assumed to be available at S. When the csi is available at the transmitter, the optimal transmission scheme is maximal ratio transmission (MRT) [24], hence, S uses MRT as the transmission scheme and destination uses MRC. We consider a system where all the relays are homogeneously located having the same average SNR and we further assume that S − Ri∀i channels are independent of each other. Source uses the csi to find the maximum SNR relay from L number of relays in the first hop as, ||hsm||F = max 1<i<L ||hsi||F (1) where || · ||F denotes the Frobenius norm and hsi is the ns × 1 channel vector between S− R and the elements of hsi are modeled as mutually correlated Ralyeigh fading entries. Let ns×ns correlationmatrix at source be s, then s = E[hsihsi], where E[ ·] and (·)† denote the expectation operator and the Hermitian transpose, respectively. The communication happens in two time slots as presented in numerous literature. During the first time slot, S transmits the signal x to the selected relay Rm and the received signal at Rm is given as, yr = √ Pshsmwsx+ vm (2) where Ps is the transmitted power and hsm is given as in (1) and ws is the MRT weight vector which is defined as ws = hsm/||hsm||F . Additive white Gaussian noise (AWGN) component with Vm variance at Rm is denoted as vm. Then Rm multiplies the received signal by gain G and transmits to the D and the received signal at D is given as, yd = hmdG √ Pr( √ Pshsmwsx+ vm)+ vd (3) where G is defined differently for the two relay schemes and is given in the next section. Pr is the transmitted power at Rm and 1× nd channel vector between Rm − D is hmd and its elements are mutually correlated such that the correlation matrix at D is d = E[hmdhmd]. vd is noise vector at D and it elements are AWGN with Vd variance. Now D performs MRC to obtain the signal as, yd = wdhmdG √ Pr( √ Pshsmwsx+ vm)+ wdvd (4) where wd = hmd/||hmd||F is the MRC weight vector. Now after some mathematical manipulations, we obtain the end-to-end SNR as, γe = Ps Vm ||hsm||F Pr Vd ||hmd||F Pr Vd ||hmd||F + 1 G2Vm (5) Notation: Let ρ1 = Ps Vm and ρ2 = Pr Vd and define γ1 = ||hsm||Fρ1 and γ2 = ||hmd||Fρ2. Let the distinct eigenvalues of the correlation matrix at source s be φ1,φ2, . . . ,φns and those of the correlation matrix at the destination d be σ1, σ2, . . . , σnd . Statistics of SNRs We can derive the probability density function (pdf) of γ2 as [25,26], pγ2(z) = nd ∑ u=1 σ nd−2 u ∏nd k=1,k =u (σu − σk) exp ( −z ρ2σu )