Impact of antenna correlation on the performance of partial relay selection
© Ferdinand et al.; licensee Springer. 2012
Received: 13 April 2012
Accepted: 12 July 2012
Published: 16 August 2012
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 unchanged with the effect of antenna correlation for both types of relay schemes.
Two-hop amplify-and-forward (AF) relay networks have been investigated extensively in recent research [1–4]. The system with a source and a destination both equipped with 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  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 . Then the general case of arbitrary distributed correlation matrix structures has been investigated by the authors in . However, these evaluations are limited to the single source, relay and destination scenario.
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.
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 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.
Notation: Let and and define and . Let the distinct eigenvalues of the correlation matrix at source Φ s be and those of the correlation matrix at the destination Φ d be .
Statistics of SNRs
See Appendix 1. □
Channel state information assisted relay
where c=1 for exact end-to-end SNR. The approximation holds for the medium to high SNR and it provides a tight upper bound, we use the exact SNR to derive the outage probability and ergodic capacity, and use the approximation for average SER.
Outage probability: csi-assisted relay
where K1(·) is the first order modified Bessel function of first kind.
Appendix 1. □
Outage probability: approximation
Appendix 2 □
High SNR analysis
Here we derive the high SNR expressions for the outage probability and the average SER. Let and ρ2=μ ρ1.
High SNR outage probability
It is observed from (33) that the diversity gain G d = minL n s n d . This shows that the performance of the system is dominated by one of the single links unless L n s =n d , hence, in order to fully utilize the resources, we like to keep L n s =n d . It is observed here that the diversity gain of the system is not affected by the antenna correlation. However, we have to note that this condition is only true for the case where the correlation matrices have full rank.
High SNR average SER
where a and b define the modulation scheme and ψ is given as in (34)–(36). t=minL n s n d −1 and diversity gain G d =t + 1.
Fixed gain relay
where K1(.) is 1st order modified Bessel function of second type and C is the fixed gain.
Appendix 3 □
where α4and β4 are as mentioned in (46) and (47), respectively.
where Ei(·) is the exponential integral.
where K1(.) and K0(.) are the 1st and 2nd order modified Bessel functions of second type, respectively.
Generalized moments of SNR
By substituting (53) in to (55) for h=1, h=2 we can obtain approximated closed form expression for ergodic capacity.
High SNR analysis: outage probability
where ψ(·) is Euler Psi function. It is observed from the fixed gain asymptotic outage expression that the diversity gain of the system is similar to the csi-assisted relay scheme.
High SNR analysis: average SER
whereΨ as in (58) and a and b define the modulation scheme. t=minL n s n d −1 and diversity gain G d =t + 1.
Here we carry out the numerical analysis and verify our results using Monte Carlo simulations. We use the exponential correlation matrix structure where (i,j)th element of the matrix Φ s is and that of the correlation matrix Φ d is . Without loss of generality we consider ρ1=ρ2(μ=1) in all the cases shown in the figures. Fixed gain type 1 in (49) is used. Exponential correlation matrices defined above have full rank. Hence, we obtain the desired diversity.
We have investigated the performance of a partial relay selection network with the effect of antenna correlation at the source and the destination. Two relay schemes; csi-assisted and fixed gain relay schemes have been considered and exact closed form expressions for outage probability, average SER and ergodic capacity have been derived. Our results can be used to quantify the effect of antenna correlation in partial relay selection. Further, we have provided an asymptotic analysis which can be used to obtain an insight of the system performance. In addition, we have showed that for a higher number of relays, the ergodic capacity can be improved with higher correlation at the source.
Using multinominal theorem and after some simplifications, we derive as in (8),
Performing the integration with the help of (, 3.471.9), we can obtain the closed form solution as in (12).
Now by using the same procedure as in the derivation of χ2, we can obtain the closed form expression for . We use χ1χ2, and χ3 to get the closed form expression for the ergodic capacity as in (21).
This research was supported by the Finnish Funding Agency for Technology and Innovation (Tekes), Renesas Mobile, Nokia Siemens Networks, Elektrobit.
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