Abstract
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 amplifyandforward (AF) relay schemes. The source and the destination are equipped with multiple antennas communicating via the best first hop signaltonoise 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 csiassisted 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.
Introduction
Twohop amplifyandforward (AF) relay networks have been investigated extensively in recent research [14]. 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 [511]. 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 [59]. 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.
It has been proven that the use of multiple relays with different selection methods can enhance the diversity and the performance [1323]. 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 endtoend signaltonoise 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 firsthop 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; csiassisted 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 csiassisted 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.
System model
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 n_{s}and n_{d}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, Suses 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−R_{i}∀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,
where ·_{F} denotes the Frobenius norm and h_{si} is the n_{s}×1 channel vector between S−R and the elements of h_{si} are modeled as mutually correlated Ralyeigh fading entries. Let n_{s}×n_{s}correlation matrix at source be Φ_{s}, then , 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 R_{m} and the received signal at R_{m}is given as,
where P_{s} is the transmitted power and h_{sm} is given as in (1) and w_{s} is the MRT weight vector which is defined as . Additive white Gaussian noise (AWGN) component with V_{m}variance at R_{m} is denoted as v_{m}. Then R_{m} multiplies the received signal by gain G and transmits to the Dand the received signal at Dis given as,
where G is defined differently for the two relay schemes and is given in the next section. P_{r} is the transmitted power at R_{m} and 1×n_{d} channel vector between R_{m}−Dis h_{md}and its elements are mutually correlated such that the correlation matrix at D is . v_{d} is noise vector at D and it elements are AWGN with V_{d}variance. Now Dperforms MRC to obtain the signal as,
where is the MRC weight vector. Now after some mathematical manipulations, we obtain the endtoend SNR as,
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
We can derive the probability density function (pdf) of γ_{2} as [25,26],
and the cumulative density function (cdf) of γ_{2}can be derived using with the help of ([27], Equation 2.321.2) as,
Now, we derive the cdf of γ_{1} as,
Proof
See Appendix 1. □
Channel state information assisted relay
Here the relay uses the csi to amplify the received signal. It has been proven that the csiassisted relay outperforms the fixed gain relay in general. However, it has a higher complexity in implementation when compared to the fixed gain one. In this section, we derive exact expressions for outage probability, average SER and ergodic capacity. First, we select the gain G for csiassisted relay as,
Then the endtoend SNR given in (5) can be rewritten as,
where c=1 for exact endtoend 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: csiassisted relay
The outage probability is the probability that γ_{e}drops below a predefined threshold Θand it is mathematically represented as,
We can derive the exact closed form expression for outage probability as,
where K_{1}(·) is the first order modified Bessel function of first kind.
Proof
Appendix 1. □
Outage probability: approximation
We simplify (12) by substituting c=0 to obtain the tight upper bound as given in (11) as,
Average SER
In this section, we derive the closed form expressions for average SER (P_{ser}) which is valid for several modulation schemes. According to technical literature P_{ser} is defined as,
where E[·] is the expectation operator and Q(.) is the Gaussian Q function. The modulation schemes is defined by ab, mainly BPSK (a=1, b=1), Mary PAM () and MPSK () [28,29]. Carrying out integration by parts of (14), we obtain,
Now, we substitute in (13) to (15) and perform the integration with the help of ([27], Equation 6.621.3), to obtain the closed form expression for the average SER of γ_{e}as,
where
where F(μν;a;b) is the Gauss hypergeometric function defined in ([27], Equation 9.10–9.13) and Γ(·) represents the Gamma function.
Ergodic capacity
The exact closed form expression for ergodic capacity has a significant importance since it has not been derived even for a single user relay network with antenna correlation. The ergodic capacity (C_{erg}) can be mathematically expressed as,
Closed form expression for C_{erg}can be derived as,
where
where is the exponential integral. χ_{2}is given as,
and
where
Proof
Appendix 2 □
High SNR analysis
Here we derive the high SNR expressions for the outage probability and the average SER. Let and ρ_{2}=μρ_{1}.
High SNR outage probability
We rewrite the (7) by expanding the using Maclaurin series as follows,
It is observed at high SNR the lower order terms () sum to zero, hence, by collecting the higher order terms, we can express (28) in high SNR as,
We can further simplify (29) as,
Similarly, we can derive the in high SNR as
Simplification yields,
Following the same procedure as in ([30], Equations (A.09) and (A.10)), we can obtain the high SNR expression for as,
where,
It is observed from (33) that the diversity gain G_{d}= minLn_{s}n_{d}. This shows that the performance of the system is dominated by one of the single links unless Ln_{s}=n_{d}, hence, in order to fully utilize the resources, we like to keep Ln_{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
Average SER in high SNR can be obtained using [31]
where a and b define the modulation scheme and ψis given as in (34)–(36). t=minLn_{s}n_{d}−1 and diversity gain G_{d}=t + 1.
Fixed gain relay
In this section, we investigate the end to end performance of a dualhop fixed gain network with multiple relays in the presence of antenna correlation at both ends. We derive closedform expressions for outage probability, average SER, generalized moments of the endtoend SNR and ergodic capacity. The asymptotic behavior of the system under high SNR is also considered. After performing some algebraic manipulations of (5), the endtoend SNR for fixed gain can be expressed as,
According to literature, there are two common techniques for selecting a fixed gain G in (38). If the gain is selected as,
then the constant Cgiven by
If the gain is selected as,
the constant Cgiven by
Outage probability
If outage probability is denoted by P_{out}, then
The closed form expression for outage probability can be derived as,
where K_{1}(.) is 1st order modified Bessel function of second type and C is the fixed gain.
Proof
Appendix 3 □
From Equation (8), cdf of γ_{1}can be written as,
Where,
and
Case 1
Using ([9], Equation 46) the expected value for random variable γ_{1} is calculated as,
Substituting to (42), the closed form expression for fixed gain can be derived as,
where α_{4}and β_{4} are as mentioned in (46) and (47), respectively.
Case 2
Substituting to (40)
where Ei(·) is the exponential integral.
Average SER
Then substituting from (44) to (15) and through the mathematical simplification with the help of ([27], Equation 6.614.5), the closed form expression for SER is obtained.
where K_{1}(.) and K_{0}(.) are the 1st and 2nd order modified Bessel functions of second type, respectively.
Generalized moments of SNR
In this section, we derive closed form expression for generalized moments of γ_{e} which is essential to the obtain ergodic capacity and the performance evaluation of the system using the average output SNR and the degree of fading. Substituting (44) into ([9], Equation 46) and after some mathematical manipulations with the help of ([27], Equation 6.643.3), we obtain
where W_{k,μ}(.) is the whittaker function defined in ([27], Equation 9.220.4).
Ergodic capacity
We derive a closed form expression for the ergodic capacity in multi relay network which is significant in determining the system performance especially in a correlated environment. Based on the literature ergodic capacity can be expressed as follows,
applying the expectation operator as in (54), an approximated result can be obtained as given below ([18], Equation 6)
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
In this section, we analyze the fixed gain relay system in high SNR. Let C=Dρ_{1}ρ_{2}=μρ_{1}, and z=Θ/ρ_{1} then we can rewrite (44) as,
then (56) can be rewritten as,
where
and
Now, we expand the exponential function using Maclaurin series and Bessel function using ([27], Equation 8.446) in (57) to obtain,
Now it is observed that sum to zero and after limiting to high order terms with some simplifications we have,
where N=min(Ln_{s}n_{d}) and
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 csiassisted relay scheme.
High SNR analysis: average SER
We can write the asymptotic average SER as [31],
where
whereΨ as in (58) and a and b define the modulation scheme. t=minLn_{s}n_{d}−1 and diversity gain G_{d}=t + 1.
Numerical analysis
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 ρ_{1}=ρ_{2}(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.
Figure 1 shows the outage probability variation with the average SNR of the first hop. Curves are plotted for different antenna configurations and correlation parameters. It is observed from the figure that the increase of the number of antennas, improves the outage probability. High SNR curves are also plotted where we can clearly see how diversity gain is varying. One can notice that the left three curves have a diversity gain of four and the two right most curves have a diversity gain of two. It is noticed that the increase of correlation decreases the performance. The outage probability variation for fixed gain relay is depicted in Figure 2. Here we see an improvement in the performance when the number of antennas and number of relays increase. However, as in the csiassisted case we can observe that this improvement depends on the diversity gain. It is further observed that the increase of correlation decreases the performance. Moreover, one can notice that the csiassisted relay outperforms the fixed gain one by approximately 3 dB. Moreover Monte Carlo simulation results exactly match with the analytical ones.
Average SER figures are depicted in Figures 3 and 4 for csiassisted and fixed gain relay schemes, respectively. In both figures it is observed that the increase of number of antennas and the number of relays improve the average SER. Conversely, the increase of correlation parameters decreases it. Without loss of generality we have considered the BPSK, QPSK, and QAM schemes to demonstrate the average SER variation. High SNR curves are plotted and they are compatible with the exact ones in medium to high SNR and they show the diversity gain variation. Further, we can notice that the csiassisted relay performs better than the fixed gain relay. Monte carlo simulations coincide with the analytical ones and it shows the accuracy of our results.
Figures 5 and 6 show the ergodic capacity variation for csiassisted and fixed gain relay cases. Without loss of generality, we fixed the number of antennas to be n_{s}=n_{d}=2 and ρ_{1}=ρ_{2}=2 dB. We have plotted the ergodic capacity variation against the number of relays to demonstrate the fact that the ergodic capacity can be improved with the increase of correlation for a higher number of relays. From the figures it is noticed that the increase of correlation at the destination ρ_{d}, decreases the ergodic capacity , however, the increase of correlation parameter at the source increases it. The reason for this behavior can be explained as follows; the relay to destination is a point to point link, hence, the increase of correlation decreases the ergodic capacity, however, the source to relay link is a pointtomultipoint link, hence the increase of correlation parameter reduces the channel hardening effect [26,32] which results in a higher capacity. Moreover, Monte Carlo simulations exactly coincide with the analytical ones for csiassisted one and are closely compatible with the approximated fixed gain ergodic capacity.
Conclusion
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; csiassisted 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.
Appendix 1
Let γ_{i}=ρ_{1}h_{si}_{F}and we rewrite (1) as,
We find the pdf of γ_{i}as [25,26],
and cdf of γ_{i}as,
We assume that the relays are distributed homogeneously such that they have equal average SNR, further, we assume that the S−R_{i}∀i channels are independent. Then we can derive the cdf of γ_{1}as,
Using multinominal theorem and after some simplifications, we derive as in (8),
Appendix 2
Outage probability
Pdf of γ_{2} can be obtained from (6) and the cdf of γ_{1}is derived in (8). Following the same procedure as mentioned in ([4], Appendix A), can be expressed as,
where . By substituting (8) and (6) to (70) and by mathematical simplifications we obtain the as,
Performing the integration with the help of ([27], 3.471.9), we can obtain the closed form solution as in (12).
Ergodic capacity
We can rewrite (20) as,
After some mathematical manipulations [33],
Now, we can rewrite (73)
where γ_{3}=γ_{1} + γ_{2}. Now can be derived using pdf of γ_{2}as,
Performing the integration with the help of ([27], Equation 4.337.5) we obtain the closed form expression for χ_{2} as in (23). Similarly, we can derive as in (22). Performing convolution operation we obtain the pdf of γ_{3}=γ_{1} + γ_{2}as,
We carry out the integration to obtain as,
where
and
Now by using the same procedure as in the derivation of χ_{2}, we can obtain the closed form expression for . We use χ_{1}χ_{2}, and χ_{3} to get the closed form expression for the ergodic capacity as in (21).
Appendix 3
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This research was supported by the Finnish Funding Agency for Technology and Innovation (Tekes), Renesas Mobile, Nokia Siemens Networks, Elektrobit.
References

JN Laneman, DNC Tse, GW Wornell (eds), Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans. Inf. Theory 50, 3062–3080 (2004). Publisher Full Text

MO Hasna, M Alouini (eds), Endtoend performance of transmission systems with relays over Rayleighfading channels. IEEE Trans. Wirel. Commun 2, 1126–1131 (2003). Publisher Full Text

MO Hasna, M Alouini (eds), A performance study of dualhop transmissions with fixed gain relays. IEEE Trans. Wirel. Commun 3, 1963–1968 (2004). Publisher Full Text

D Senarathne, C Tellambura (eds), Unified Exact Performance Analysis of TwoHop AmplifyandForward Relaying in Nakagami Fading. IEEE Trans. Veh. Technol 59, 1529–1534 (2010)

RHY Louie, Y Li, B Vucetic (eds), in Perform. Anal. Beamforming in Two Hop Amplify and Forward Relay Networks (IEEE ICC , Beijing, China, 2008)

DB da Costa, S Aïssa (eds), in Beamforming in DualHop Fixed Gain Relaying Syst (IEEE, Dresden, Germany, 2009)

RHY Louie, Y Li, HA Suraweera, B Vucetic (eds), Performance analysis of beamforming in two hop amplify and forward relay networks with antenna correlation. IEEE Trans. Wirel. Commun 8, 3131–3142 (2009)

DB da Costa, S Aïssa (eds), Cooperative dualhop relaying systems with beamforming over Nakagamim fading channels. IEEE Trans. Wirel. Commun 8, 3950–3954 (2009)

NS Ferdinand, N Rajatheva (eds), Unified performance analysis of twohop amplifyandforward relay systems with antenna correlation. IEEE Trans. Wirel. Commun 10, 3002–3011 (2011)

G Amarasuriya, C Tellambura, M Ardakani (eds), in Impact of antenna correlation on a new dualhop MIMO AF relaying model, vol. 2010, 1–14 (Article ID 956721, 2010)

TQ Duong, GC Alexandropoulos, TA Tsiftsis, HJ Zepernick (eds), Orthogonal spacetime block codes with CSIassisted amplifyandforward relaying in correlated Nakagamim fading channels. IEEE Trans. Veh. Technol 60, 882–889 (2011)

HA Suraweera, HK Garg, A Nallanathan (eds), in Beamforming in DualHop Fixed Gain Relay Systems with Antenna Correlation (IEEE ICC , Cape Town, South Africa, 2010)

A Bletsas, H Shin, MZ Win (eds), Cooperative communications with outageoptimal opportunistic relaying. IEEE Trans. Wirel. Commun 6, 3450–3460 (2007)

B Barua, H Ngo, H Shin (eds), On the SEP of cooperative diversity with opportunistic relaying. IEEE Commun. Lett 12, 727–729 (2008)

I Krikidis, J Thompson, S McLaughlin, N Goertz (eds), Amplifyandforward with partial relay selection. IEEE Commun. Lett 12, 235–237 (2008)

DB da Costa, S Aïssa (eds), Endtoend performance of dualhop semiblind relaying systems with partial relay selection. IEEE Trans. Wirel. Commun 8, 4306–4315 (2009)

DB da Costa, S Aïssa (eds), Performance analysis of relay selection techniques with clustered fixedgain relays. IEEE Signal Process. Lett 17, 201–204 (2010)

DB da Costa, S Aïssa (eds), Capacity analysis of cooperative systems with relay selection in Nakagamim Fading. IEEE Commun. Lett 13, 637–639 (2009)

DB da Costa, S Aïssa (eds), Amplifyandforward relaying in channelnoiseassisted cooperative networks with relay selection. IEEE Commun. Lett 14, 608–610 (2010)

M Torabi, D Haccoun (eds), Capacity analysis of opportunistic relaying in cooperative systems with outdated channel information. IEEE Commun. Lett 14, 1137–1139 (2010)

HA Suraweera, M Soysa, C Tellambura, HK Garg (eds), Performance analysis of partial relay selection with feedback delay. IEEE Signal Process. Lett 17, 531–534 (2010)

JL Vicario, A Bel, JA LopezSalcedo, G Seco (eds), Opportunistic relay selection with outdated CSI: outage probability and diversity analysis. IEEE Trans. Wirel. Commun 8, 2872–2876 (2009)

DS Michalopoulos, HA Suraweera, GK Karagiannidis, R Schober (eds), in AmplifyandForward Relay Sel. with Outdated Channel State Inf (IEEE Globecom, Miami, Florida, 2010)

TKY Lo (ed.), Maximum ratio transmission. IEEE Trans. Commun 47, 1456–1461 (1999)

L Musavian, M Dohler, MR Nakhai, AH Aghvami (eds), Closedform capacity expressions of orthogonalized correlated MIMO channels. IEEE Commun. Lett 8, 365–367 (2004). Publisher Full Text

NS Ferdinand, N Rajatheva, M Latvaaho (eds), in Effect of Antenna Correlation on the Performance of MIMO MultiUser Dual Hop Relay Network (IEEE Globecom, Houston, Texas, 2011)

IS Gradshteyn, IM Ryzhik, in Table of Integrals, Ser. Prod (Elsevier, New York, 2007)

MK Simon, MS Alouini, in Digital Commun. over Fading Channels: A Unified Approach to Perform. Anal (John Wiley and Sons, New York, 2010)

Z Fang, L Li, Z Wang (eds), Asymptotic performance analysis of multihop relayed transmissions over Nakagamim fading channels. IEICE Trans. Commun E91B, 4081–4084 (2008). Publisher Full Text

Z Wang, GB Giannakis (eds), A simple and general parameterization quantifying performance in fading channels. IEEE Trans. Commun 51, 1389–1398 (2003). Publisher Full Text

D Park, SY Park (eds), Performance analysis of multiuser diversity under transmit antenna correlation. IEEE Trans. Commun 56, 666–674 (2008)

L Fan, X Lei, W Li (eds), Exact closedform expression for ergodic capacity of amplifyandforward relaying in channelnoiseassisted cooperative networks with relay selection. IEEE Commun. Lett 15, 332–333 (2011)