IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 10, OCTOBER 2011 1047 On the Error Exponent of Amplify and Forward Relay Networks Bappi Barua, Student Member, IEEE, Mehran Abolhasan, Member, IEEE, Farzad Safaei, Member, IEEE, and Daniel R. Franklin, Member, IEEE Abstract—In this letter we derive the exact random coding error exponent of a dual hop amplify and forward (AF) relay network with channel state information (CSI) assisted ideal relay gain. Numerical results have been presented, which provide in- sight about the performance tradeoff between the error exponent and the data rate of the network. Finally we present the capacity analysis of this relay network. Index Terms—Random coding error exponent, amplify and forward, ergodic capacity. I. I NTRODUCTION C Ooperative relay communication has been proven to provide better reliability against the multipath fading pro- cess. Depending on the CSI and allowable complexity, relays retransmit the received signals utilizing different approaches. Amplify and forward is the commonly used relaying protocol due to its simplicity and ease of deployment. Significant research results on cooperative relay networks are already available [1]–[3]. The performance of a simple dual hop network has been studied in [3] for Rayleigh fading channels. In particular, performance analysis utilizing random coding error exponent (RCEE) has received considerable attention in recent years. The error exponent defined in [4], [5] is the exponent along with codeword length that imposes a tight upper bound on the probability of error. Error exponent mea- surement provides information about the design requirement of a codeword to achieve any target rate R below the capacity C of the channel. Particularly, we can derive different capacity terms such as, the ergodic capacity, cut-off rate and the critical rate of a network utilizing the error exponent expression [4]. Using Gallager’s exponent over Nakagami- fading channels performance analysis of a dual hop relay network has been conducted in [6] assuming ideal inverted channel gain in AF relays. More recently [7] has analyzed the system performance using error exponent method in two way relay communication channels. In [6] the authors have derived the RCEE using the hypo- thetical relay gain which is, in fact, a simplified assumption of CSI assisted power constraint relay gain factor 2 = ∣ℎ1∣ 2 +1 proposed by [2] when 1 is set to zero. and are the source and the relay power respectively, and Manuscript received March 28, 2011. The associate editor coordinating the review of this letter and approving it for publication was D. Michalopoulos. This research was supported by the Australian Research Council (ARC) discovery research grant No. DP0879507. B. Barua, M. Abolhasan, and D. R. Franklin are with the Faculty of Engi- neering and IT, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia (e-mail: bappi.barua@student.uts.edu.au, {mehran.abolhasan, daniel.franklin}@uts.edu.au). F. Safaei is with the Faculty of Informatics, University of Wollongong, NSW 2522, Australia (e-mail: farzad@uow.edu.au). Digital Object Identifier 10.1109/LCOMM.2011.081211.110668 ℎ 1 is the channel gain of Source-Relay hop. 1 is the one- sided noise spectral density at the relay node. Avoiding the noise figure in the denominator of the relaying gain factor has allowed the authors to produce the probability density function (PDF) of the receiver SNR in more mathematically tractable form. However this assumption may not be viable in a power constraint AF relay system when the channel gain of the source-relay hop is very low. In this letter we derive the exact random coding error exponent with CSI assisted AF relay without avoiding the denominator noise figure. II. SYSTEM MODEL Consider a single source-destination pair communicating via a single antenna relay without any direct link. A half duplex AF protocol has been considered over independent Rayleigh fading channels. We assume the receiver and the relay have full CSI while the transmitter has no CSI. Using the CSI assisted relay gain the end-to-end signal-to-noise ratio (SNR) at the receiver using maximal ratio combining (MRC) is given by, = 1 2 1 + 2 +1 (1) Due to the Rayleigh fading assumption, the first and second hop SNR 1 and 2 are exponentially distributed with parame- ter 1 and 2 respectively 1 . The instantaneous and the average SNR of 1st and 2nd hop are denoted as ≜ ∣ℎ∣ 2 and ≜ Ω respectively, where, ∈{ , }, ℎ and Ω are the instantaneous and the average channel gain of ∈{1, 2}th hop and is the variance of zero mean circularly symmetric complex Gaussian noise at the relay or at the receiver node. The PDF of the end-to-end SNR can be invoked from [8] as, ( )=2 −(1+2) [ 1 2 (2 + 1) 0 ( 2 √ 1 2 ( + 1) ) +( 1 + 2 ) √ 1 2 ( + 1) 1 ( 2 √ 1 2 ( + 1) )] (2) where, ( ) is the th order modified Bessel’s function of second kind. III. ERROR EXPONENT:DUAL HOP NETWORK The random coding error exponent is defined as a function of input distribution function (), a factor ∈ [0, 1] and rate ≤ (for details please read ch. 5 of [4]), which is jointly optimized over () and at a desired rate . However the Gaussian input distribution has often been used in 1 where, = 1 ; ∈{1, 2} is the inverse of the average SNR of the corresponding (1st or 2nd) hop. 1089-7798/11$25.00 c ⃝ 2011 IEEE