On the Performance of Selection Cooperation with Imperfect Channel Estimation
Abstract
In this paper, we investigate the performance of selection cooperation in the presence of imperfect channel estimation. In particular, we consider a cooperative scenario with multiple relays and amplifyandforward protocol over frequency flat fading channels. In the selection scheme, only the “best” relay which maximizes the effective signaltonoise ratio (SNR) at the receiver end is selected. We present lower and upper bounds on the effective SNR and derive closedform expressions for the average symbol error rate (ASER), outage probability and average capacity per bandwidth of the received signal in the presence of channel estimation errors. A simulation study is presented to corroborate the analytical results and to demonstrate the performance of relay selection with imperfect channel estimation.
Imperfect channel estimation, cooperative communication, relay selection, average symbol error rate, outage probability, average capacity.
I Introduction
\PARstartThe promise of high spectral efficiency and capability of providing great capacity improvements in a wireless fading environment, as reported by [1] and [2], has led to widespread interest in multiinput multioutput (MIMO) communications. However, due to size, cost, and/or hardware limitations, a wireless device may not be able to support multiple transmit antennas. Cooperative diversity was proposed as an alternative to MIMO systems. It has been demonstrated that cooperative diversity provides an effective way of improving spectral and power efficiency of the wireless networks [3, 4]. The main idea behind cooperative diversity is that in a wireless environment, the signal transmitted by the source () is overheard by other nodes, which can be defined as “relays” [5]. The source and its partners can then jointly process and transmit their information, thereby creating a “virtual antenna array”, although each of them is equipped with only one antenna. It is shown in [6, 7, 8, 9, 10, 11, 12, 13, 14] that cooperative diversity networks can achieve a diversity order equal to the number of paths between the source and the destination, however, the need of transmitting the symbols in a time division multiplexing (TDMA) fashion limits the improvement in capacity. Additionally due to the power allocation constraints, multiple relay deployment is not economic. Relay selection then was proposed to alleviate the loss in spectral efficiency caused by multiple relay schemes and also to moderate the power allocation constraints.
Recently, there have been considerable research efforts on the performance analysis of cooperative diversity including the derivation of closedform formulas for the average symbol error rate (ASER) [5, 6, 8, 10, 11, 12, 15], the outage probability [7, 9, 10, 13, 16], and the average capacity (ergodic capacity) [7, 10, 11, 17]. Most of the current works, however, assume the availability of perfect channel state information (CSI) at the destination () terminal. Channel estimation for relay communication has also been studied in literature as in[18, 19].
Related work and contributions: The impact of channel estimation error on the ASER performance of distributed space time block coded (DSTBC) systems has been investigated in [20], assuming the amplifyandforward protocol. Building upon a similar setup, Gedik and Uysal [21] have extended the work of [20] to a system with relays, assuming the amplifyandforward protocol. In [22], the symbol error analysis is investigated for the same scenario as in [21]. In this paper, we investigate the effect of channel estimation error on the ASER, outage probability and capacity of a cooperative diversity system with relay selection. We first introduce our system model in Sec. II. In Sec. III, we explain the Gaussian error model introduced by [23] and use it in the selection scenario to obtain the maximum ratio combiner (MRC) output SNR. In Sec. V, we study the ASER performance of our system in the case of imperfect channel estimation. We develop upper and lower bound analysis for the general relay case to formulate the relay selection scheme. We then derive a closedform formula for the ASER performance with imperfect channel estimation. We also derive closedform formula for the ASER performance of our system in the high SNR regime as introduced in [5, 24]. We also study the outage probability and average capacity per bandwidth in Sec. VI and Sec. VII, respectively. Diversity analysis is also provided in VIII. Simulation results are presented in Sec. IX, followed by conclusions in Sec. X.
Notation: , and , denote the expected value, the complex conjugate and the estimate of the variable , respectively. and stand for the lower bound and upper bound , respectively. indicates that the relay with SNR is selected. and denote the probability density function (PDF) and the cumulative density function (CDF), respectively.
Ii system model
In this section we consider a system in which a source node transmits information to a destination node with the help of the best relay node , which is selected under a defined criterion from M available relays. The transmissions are orthogonal, either through time or frequency division.
In the first data sharing time slot, the source node communicates with the destination as well as the relay nodes. In this phase, the signals received by the destination and each relay are
(1)  
(2) 
where , , and denote the transmitted signal with unit energy and the signals received at the destination and the relay node, respectively. and are the channel coefficients of the sourcerelay and the sourcedestination channels, which include the effect of fading. and represent the average signal energies received at destination and the relay terminals, respectively, taking into account the path loss and shadowing effects. and are additive white Gaussian noises (AWGN) in the corresponding channels with the same variance , i.e., .
In the next time slot each relay node normalizes its received signal and retransmits it to the destination. For the relay, the normalization factor is and the signal transmitted from the this relay is [7]
(3) 
Since the relays practically do not have a perfect knowledge about the channel , , the power normalization procedure in each relay is performed based on the estimate of the corresponding channel. We assume in this paper that . Based on (3), the signal received by the destination from the relay node is [7]
(4)  
(5) 
where is the channel gain from this node to the destination. is the average signal energy received at the destination via the relay in its corresponding time slot which accounts for the shadowing and path loss effects and . denotes the AWGN of the relay destination channel. is the equivalent noise term in . It can easily be shown that, conditioned on the channel realizations , where
(6) 
Supposing only the relay participates in cooperation we look for the condition under which we can select the best relay, , by searching s for .
Since each relay only amplifies the signal from the source, the destination is the only place where an estimate of the information symbol is computed. In order to achieve maximum likelihood performance, the signals from both diversity branches (direct branch and the branch via the relay) are combined using a maximum ratio combiner (MRC). Decoding of is delayed until the relayed signal containing the information symbol is received at the destination. Since the noise power is not the same on the two sub channels, both diversity branches must be weighted by their respective complex fading gain over total noise power on that particular branch before the combiner. Thus we obtain the estimated information symbol as
(7) 
where
(8) 
since the destination only uses the estimate of the relaydestination channel.
Iii Modeling of channel estimation error
Let the true channel gain and its estimate be and , respectively. It is shown in [23, 25] that it is possible to consider the channel estimate as
(9) 
where is a zero mean complex Gaussian noise. As and , are jointly Gaussian, the conditional probability density function (PDF), , would also be Gaussian with the mean [26]
(10) 
and variance of
(11) 
By (10) and (11) and assuming a least squares (LS) estimator, we can consider the channel as
(12) 
where is the correlation coefficient between and and . In this paper, and are the channel estimation error variances and their corresponding factors of the , , and links, respectively.
According to (11)
(13)  
Using (12), we can rewrite the first term of (7) as
(14)  
Here, we have inserted (12) in (1) then the resultant expression is substituted into (7). Following [22], the received direct path signal after the MRC can be decomposed into the message part given as and the noise part which is given as . Finally, due to the independence of the , and , the effective SNR due to the direct path is given by
(15)  
where is the estimated received SNR due to the direct path and .
Following the same approach, we can write the second term of (7) as
(16) 
We can extract the message part, error part due to channel estimation error, and the noise part of the received signal as^{1}^{1}1Conditioned on and the distribution of (III) is approximated as Gaussian. The validity of this approximation has been confirmed by simulation. It should be further noted that the same conclusion has been reported by [22] and the references therein.
(17)  
(18)  
(19) 
respectively. To obtain the effective SNR expression, we need to find the ratio of the signal power to the overall noise power. Alternatively, the received SNR can be calculated as [27]. Since , , and are zero mean independent processes, we can write the effective SNR of the selected path as
(20) 
With a simple manipulation, we may write (20) as follows
(21) 
where , , , , , and . Following [22], can be ignored ^{2}^{2}2For a fixed SNR, i.e., dB, we have noticed that the mean square error MSE between in (21) and (22) is percent of for dB.; hence, (22) can be approximated as [6, 22]
(22) 
Iv selection Strategy
In the selection based amplifyandforward scheme studied in [5, 6, 7, 8, 9, 28, 29] the best relay selected by the destination terminal is the one that leads to the maximum total received SNR, , which is given by
(23) 
where
(24) 
for . , which is the total received SNR at the destination, is the addition of the SNR received from the direct path, , and the SNR received via the selected relay path . The selection steps are mainly as follows. The source terminal sends a ready to send (RTS) packet to all the relays and the destination. As the relays receive this packet they estimate their corresponding sourcetorelay channel . They divide their signal by the factor and wait for the clear to send (CTS) packet from the receiver side. As each relay receives the CTS overhead, it estimates its corresponding relaytodestination channel and will start a timer which is given by
where is given by (21). Each timer counts down to zero right after receiving the CTS packet. The first timer that reaches zero sends a flag to all the relays and the destination, introducing itself as the best relay. Then all the relays would remain silent, and the selected relay forwards its signal to the destination, with an overhead package containing information about its sourcetorelay link and the relaytodestination channel . It is also possible for the relays to use a separate bit feedback link to send the CSI to the destination. Further details on the selection procedures can be found in [16, 14].
V ASER analysis
In this section, we derive an ASER expression for the relay selection scheme. We first introduce the following theorems [30]

Theorem 1: If and are two random variables with , then

Theorem 2: For two independent random variables and if , then

Theorem 3: Let , , where s are independent random variables, then
It is shown in [5, 6, 8, 10, 8, 11, 12, 15] that the symbol error rate (SER) in a cooperative scenario is given by , where
and depends on the modulation scheme.^{3}^{3}3For instance for BPSK modulation . The analysis can be generalized to QPSK or QPAM modulation as well[5].
To find the average symbol error rate, we need to integrate over the probability density function (PDF) of , i.e.,
(25) 
Finding the exact expression for can be quite cumbersome, therefore, in the following and to simplify analysis, we develop lower and upper bounds on . It is straightforward to show that for two arbitrary independent random variables, the following inequality holds [6]
(26) 
Applying (26) to in (22) yields
(27) 
where
(28)  
(29) 
Finally, , can be bounded as
(30) 
where and are the lower bound and the upper bound SNR values defined as
(31)  
(32) 
respectively, where
(33)  
(34) 
Using (30), (31) and (32), (25) can be bounded as
(35) 
where and , are the lower and the upper bounds for ASER’s. In this paper, we limit our discussion to the lower bound analysis. The upper bound analysis can be obtained similarly. Since the ASER is a decreasing function of SNR, a lower bound for would simply be .
Since our final goal is to obtain the PDF of we need to find the PDF of , for , first. Note that the PDF of can be expressed in terms of the average SNR as . Similar expressions can be found for the PDF of and as well. Thus, using Theorem 1, the PDF of can be written as ^{4}^{4}4Here we have to note that s are not i.i.d random variables, since s are not analogous for different relay paths.
(36) 
where . Using Theorem 2 and noting that , we obtain
(37)  
(38)  
(39) 
The closedform formulation for would then be (see Appendix I)
(40) 
where and is a binary permutation matrix with dimension of which is defined in the Appendix I and
(41) 
(42) 
Having noted that
(43) 
by substituting (40) into (43) and setting , we can write as
(44) 
Here, we have used the fact [31]
(45) 
Note that the upper bound analysis can be derived similarly.
To gain further insight into the performance of the selection scheme under consideration, we resort to the high SNR analysis. Defining , where is the average SNR, and following [24, 5], we can approximate the asymptotic behavior of in the high SNR regime with its Mclaurin series expansion, i.e.,
(46) 
where , if the derivatives of up to order are null and
It can be easily shown that by replacing (46) into (43), the asymptotic behavior of ASER is given by [5]
(47) 
where we have used the fact that .
Similarly, the asymptotic upper bound ASER can be found as
(48) 
Theorem 4: For all , the values of and are zero and for
(49)  
(50) 
where .
Proof of Theorem 4: (see Appendix II)
Vi Outage probability Analysis
The outage probability , which is a valid measure of performance in slowly fading channels, is defined as the probability that the sourcedestination mutual information falls below the transmission rate . Defining
(51) 
the outage probability is given by
(52)  
Substituting the introduced upper bound for leads to adopt an upper bound on the outage probability, i.e.,
(53)  
We can simply find by integrating (40) with respect to the result of which is given by