On the Performance of Optimum Noncoherent Amplify-and-Forward Reception for Cooperative Diversity - eScholarship
Ramesh Annavajjala, Pamela C. Cosman, Larry B. Milstein · 2005
On the Performance of Optimum Noncoherent Amplify-and-Forward Reception for Cooperative Diversity Ramesh Annavajjala, Pamela C. Cosman and Laurence B. Milstein Department of Electrical and Computer Engineering University of California, San Diego, La Jolla, CA 92093, U.S.A Abstract— In this paper, we present receiver structures for maximum-likelihood (ML) noncoherent amplify-and-forward (AF) communication links when multiple relay nodes are em- ployed. We consider both on-off keying (OOK) and binary fre- quency shift keying (BFSK) modulation schemes on Rayleigh fad- ing relay channels with no channel state information. Even for the simple case of having only one relay node, the optimum non- coherent receiver is quite involved, and the ML metric computa- tion requires certain integral evaluations. To lower bound the av- erage bit error rate (BER), we assume that the link between the relay and the destination node is unfaded, a reasonable assump- tion when there is a strong line-of-sight path between the relay and the destination, and obtain simple closed-form expressions for the average BER with an arbitrary number of relays. Upper bounds on the average BER are also presented by numerically evaluating the Bhattacharyya distance between the likelihood functions. Fur- ther, simple suboptimum receiver structures are proposed, for both OOK and BFSK, along with an analytical performance evaluation, and an asymptotic diversity order analysis. Keywords: User cooperation, cooperative diversity, relay channels, nonco- herent communication. I. I NTRODUCTION Cooperative diversity is attractive for mobile terminals having single-antenna transceivers. A distributed antenna array can be formed by collaboration among M nodes, with a potential to achieve the full diversity order of M . Sendonaris et al. in [1] showed that, with transmitter channel state information (CSI), the sum-capacity of an ergodic fading channel can be improved with user cooperation, whereas in [2] Laneman showed that with CSI only at the receiver, the sum-capacity cannot be increased over no-cooperation. The performance of coherent binary PSK (BPSK) signaling with an amplify-and-forward (AF) commu- nication protocol and receiver CSI was studied in [3]. An im- proved analysis of error probability, using the moment generat- ing function approach, was presented in [4]. Recently, the per- formance of multi-branch, multi-hop, relay channels was con- sidered in [5] and [6]. The analysis of [3]-[6] showed that with M relay nodes and perfect CSI at the receiver, coherent multi- branch AF reception over independent channels achieves the full diversity order of M + 1. In order to acquire the CSI, the relay channel has to be trained (typically by pilot signaling), which results in a throughput penalty. If the variation of the channel over time is high rela- tive to the signaling duration, then the estimates become out- dated. In such a scenario, one is inclined to employ nonco- herent detection techniques which do not require knowledge of the instantaneous channel realization. In this context, Chen and Laneman in [7] and [8] studied the performance of noncoherent binary FSK (BFSK) signaling with a decode-and-forward (DF) protocol. They showed that, with M relays, the diversity order achievable with a noncoherent DF protocol is at most (M/2)+1 when M is even, and (M + 1)/2 when M is odd. That is, with the DF protocol, noncoherent signaling loses approximately half of the available diversity order. In this contribution, we consider noncoherent communication over Rayleigh fading relay channels with an AF protocol. While neither the relays nor the destination have knowledge of the in- stantaneous CSI, we assume that the statistical averages of the channel gains are known to them. The amplification gain of the relay is chosen to satisfy an average power constraint. We consider both on-off keying (OOK) and BFSK modulation, and derive the maximum likelihood (ML) noncoherent AF (NCAF) receiver structures at the destination. Unfortunately, even for the case of single relay node, no closed-form expression for the ML NCAF receiver is available, and the ML metric computation re- quires numerical evaluation of certain integrals. To gain some understanding of the receiver performance, we assume that the relay-to-destination link is unfaded 1 . This is reasonable when there is a strong line-of-sight path from the relay to the desti- nation. With this, we are able to derive simple closed-form ex- pressions for the average bit error rate (BER), with an arbitrary number of relay nodes, that serve as lower bounds on the opti- mal performance. We derive the upper bounds on the average BER by employing the Bhattacharyya bound [10]. We propose simple suboptimum receivers, for both OOK and BFSK, along with their performance evaluations. We also show that, using asymptotic diversity order analysis [11], with M relay nodes plus a link between the source and the destination, the OOK achieves a diversity order of at least (M +1)/2, but never M +1, whereas BFSK achieves the full diversity of M + 1. However, one of our more surprising results is that for the OOK system, without a relay, the asymptotic diversity analysis predicts a di- versity order of less than unity. Since we could not find a good physical interpretation of this result in the context of diversity, it suggests that asymptotic diversity analysis should be used with 1 of 9 1 A similar assumption is made in [9] for diversity analysis.