On the Performance of Selection Cooperation with Imperfect Channel Estimation

# On the Performance of Selection Cooperation with Imperfect Channel Estimation

Mehdi Seyfi,  Sami (Hakam) Muhaidat*,  and Jie Liang,  The work of S. Muhaidat was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant RGPIN372049. The work of Jie Liang was supported in part by NSERC of Canada under grants RGPIN312262-05, EQPEQ330976-2006, and STPGP350416-07. The material in this paper was presented in part at the 2010 IEEE Vehicular Technology Conference, VTC’10, Taipei, Taiwan.M. Seyfi, S. Muhaidat and J. Liang are with the School of Engineering Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada. Phone: 778-782-7376. Fax: 778-782-4951. E-mail: msa119@sfu.ca, muhaidat@ieee.org, jiel@sfu.ca.Corresponding author
###### 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 amplify-and-forward protocol over frequency flat fading channels. In the selection scheme, only the “best” relay which maximizes the effective signal-to-noise ratio (SNR) at the receiver end is selected. We present lower and upper bounds on the effective SNR and derive closed-form 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.

{keywords}

Imperfect channel estimation, cooperative communication, relay selection, average symbol error rate, outage probability, average capacity.

## I Introduction

\PARstart

The 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 multi-input multi-output (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 closed-form 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 amplify-and-forward protocol. Building upon a similar set-up, Gedik and Uysal [21] have extended the work of [20] to a system with relays, assuming the amplify-and-forward 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 closed-form formula for the ASER performance with imperfect channel estimation. We also derive closed-form 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

 ysd = √Esdhsdx+nsd, (1) ysi = √Esihsix+nsi, (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 source-relay and the source-destination 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]

 xi=√Esihsix+nsi√Esi|^hsi|2+N0. (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]

 yid = √Eidhidxi+nid, (4) = √EsiEid√Esi|^hsi|2+N0hsihidx+~nid, = αihsihidx+~nid. (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

 ω2i=1+Eid|hid|2Esi|^hsi|2+N0. (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

 ^x = ^h∗sd√EsdN0ysd+α∗i^h∗si^h∗id^ω2iN0yid, (7)

where

 ^ωi= ⎷Esi|^hsi|2+Eid|^hid|2+N0Esi|^hsi|2+N0, (8)

since the destination only uses the estimate of the relay-destination 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

 ^h=h+e, (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]

 E{h|^h}=σ2hσ2^h^h, (10)

and variance of

 var{h|^h}=(1−σ2hσ2^h)σ2h. (11)

By (10) and (11) and assuming a least squares (LS) estimator, we can consider the channel as

 h=ρ^h+d (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)

 σ2D = (1−ρ)σ2h (13) = (ρ−ρ2)σ2^h.

Using (12), we can rewrite the first term of (7) as

 Dsd = ^h∗sd√EsdN0(√Esd(ρsd^hsd+dsd)x+nsd) (14) = ρsdEsd|^hsd|2N0x+√Esd^h∗sdN0(√Esddsdx+nsd).

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

 γeffsd = Ex{|E{Dsd}|2}var{Dsd} (15) = EsdN0ρ2sd|^hsd|21+EsdN0σ2D,sd=ρ2sd^γsd1+ϵsd,

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

 Did =αi^h∗si^h∗id^ω2iN0[αi(ρsi^hsi+dsi)(ρid^hid+did)x+~nid]. (16)

We can extract the message part, error part due to channel estimation error, and the noise part of the received signal as111Conditioned 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.

 M=α2i^ω2iN0ρsiρid|^hsi|2|^hid|2x, (17) D=α2i^ω2iN0[ρsi|hsi|2h∗iddid+ρid|hid|2h∗sidsi +h∗sih∗iddsidid]x, (18) N=αi^h∗si^h∗id^ω2iN0~nid, (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

 γeffi=Esiρ2si|^hsi|2Eidρ2id|^hid|2N20(EsiN0|^hsi|2(1+ρ2siEidN0σ2D,id)+EidN0|^hid|2(1+ρ2idEsiN0σ2D,si)+EsiN0σ2D,siEidN0σ2D,id+1). (20)

With a simple manipulation, we may write (20) as follows

 γeffi=ρ2siρ2id^γsi^γid^γsiλsi+^γidλid+ϵsiϵid+1, (21)

where , , , , , and . Following [22], can be ignored 222For 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]

 γeffi=ρ2siρ2id^γsi^γid^γsiλsi+^γidλid. (22)

## Iv selection Strategy

In the selection based amplify-and-forward 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

 γr = γeffsd+γeffi⋆, (23)

where

 γeffi⋆=arg maxi{γeffi}, (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 source-to-relay 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 relay-to-destination channel and will start a timer which is given by

 τi=1γeffi,

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 source-to-relay link and the relay-to-destination 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

 fY(γ)=1|m|fX(γm).
• Theorem 2: For two independent random variables and if , then

 fZ(γ)=fX(γ)+fY(γ)−fX(γ)FY(γ)−fY(γ)FX(γ).
• Theorem 3: Let , , where s are independent random variables, then

 FY(γ)=M∏i=1FXi(γ).

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

 Q(x)=1√2π∫∞xexp(−t22) dt,

and depends on the modulation scheme.333For 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.,

 ¯Pe=∫∞0Q(√kγr) fγr(γr) dγr. (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]

 12min(X,Y)≤XYX+Y≤min(X,Y). (26)

Applying (26) to in (22) yields

 γi,lb≤γeffi≤γi,ub, (27)

where

 γi,ub = ρ2siρ2idλsiλidmin(^γsiλsi,^γidλid), (28) γi,lb = 12ρ2siρ2idλsiλidmin(^γsiλsi,^γidλid), (29)

Finally, , can be bounded as

 γlb≤γr≤γub, (30)

where and are the lower bound and the upper bound SNR values defined as

 γub = ρ2sd^γsd1+ϵsd+γi⋆,ub, (31) γlb = ρ2sd^γsd1+ϵsd+γi⋆,lb, (32)

respectively, where

 γi⋆,ub \lx@stackrelΔ= maxi{γi,ub}, (33) γi⋆,lb \lx@stackrelΔ= maxi{γi,lb}. (34)

Using (30), (31) and (32), (25) can be bounded as

 ¯Plb≤¯Pe≤¯Pub, (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 444Here we have to note that s are not i.i.d random variables, since s are not analogous for different relay paths.

 fγi,ub(γ)=1βiexp(−γβi), (36)

where . Using Theorem 2 and noting that , we obtain

 fγi⋆,ub(γ) = ∂Fγi⋆,ub(γ)∂γ (37) = ∂∏Mi=1Fγi,ub(γ)∂γ (38) = M∑l=1fγl,ub(γ)M∏i=1i≠lFγi,ub(γ). (39)

The closed-form formulation for would then be (see Appendix I)

 fγub(γ)=M∑p=1(Mp)∑m=1(−1)(p+1)[Ψpb]m¯μ[Ψpb]m−1[exp(−γ¯μ)−exp(−γ[Ψpb]m)], (40)

where and is a binary permutation matrix with dimension of which is defined in the Appendix I and

 b\lx@stackrelΔ=[1β1 1β2 … 1βM]T, (41)
 Q(γ)=12−12Erf(γ√2). (42)

Having noted that

 ¯Plb=∫∞0[12−12Erf(√k2γub)]fγub(γub)dγub (43)

by substituting (40) into (43) and setting , we can write as

 ¸¯Plb=12−12M∑p=1(Mp)∑m=1(−1)(p+1)[Ψpb]m¯μ[Ψpb]m−1[¯μ√k/2¯μ−1+k/2−[Ψpb]−1m√k/2[Ψpb]m+k/2]. (44)

Here, we have used the fact [31]

 ∫∞0γ Erf(aγ)exp(−b2γ2)dγ=a2b2√b2+a2    b≠0. (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.,

 f^γub=a^γnub+o(^γub), (46)

where , if the derivatives of up to order are null and

 o(^γub)=∞∑t=n+11t!∂tf^γub∂^γtub(0)^γtub.

It can be easily shown that by replacing (46) into (43), the asymptotic behavior of ASER is given by [5]

 ¯Plb→∏n+1m=1(2m−1)2(n+1)k(n+1)1n!∂nfγub∂γnub(0). (47)

where we have used the fact that .

Similarly, the asymptotic upper bound ASER can be found as

 ¯Pub→∏n+1m=1(2m−1)2(n+1)k(n+1)1n!∂nfγlb∂γnlb(0). (48)

Theorem 4: For all , the values of and are zero and for

 ¯Plb→∏M+1m=1(2m−1)2(M+1)k(M+1)1μM∏i=11βi, (49) ¯Pub→∏M+1m=1(2m−1)(M+1)k(M+1)1μM∏i=11βi, (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 source-destination mutual information falls below the transmission rate . Defining

 Isd\lx@stackrelΔ=12log2(1+γeffsd+γeffi⋆)≤R, (51)

the outage probability is given by

 Pout = (52) =

Substituting the introduced upper bound for leads to adopt an upper bound on the outage probability, i.e.,

 Pout≤Pubout = (53) = Fγub(22R−1).

We can simply find by integrating (40) with respect to the result of which is given by

 ¸Fγub(γ) = M∑p=1(