IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 5, MAY 2007 399 Exact Expression and Tight Bound on Pairwise Error Probability for Performance Analysis of Turbo Codes over Nakagami-m Fading Channels Syed Amjad Ali, Member, IEEE, Nirmal Singh Kambo, and Erhan AliRiza ˙ Ince, Member, IEEE Abstract— This letter presents derivation for an exact and efficient expression on pairwise error probability over fully interleaved Nakagami-m fading channels under ideal channel state information at the decoder. As an outcome, this derivation also leads to a tight upper bound on pairwise error probability which is close to the exact expression. Pairwise error probability plots for different values of Nakagami parameter m along with an already existing numerically computable expression are provided. As an application of pairwise error probability, average union upper bounds for turbo codes having (1, 7/5, 7/5) and (1, 5/7, 5/7) generator polynomials employing transfer function approach are presented to illustrate the usefulness of the new efficient results. Index Terms— Average union upper bound, ideal channel state information, Nakagami-m fading channel, pairwise error probability, turbo codes. I. I NTRODUCTION T HE impressive performance of turbo codes over both the additive white Gaussian noise (AWGN) and fading channels is thoroughly discussed in [1]–[4]. Bit error proba- bility bounds are mostly used for the performance analysis of turbo codes for large values of signal to noise ratio (RE b /N 0 ) to avoid extensive simulation time. On the contrary, both simulation and tight upper bounds beyond the channel cutoff rate can be used to study the code performance. Unfortunately, tight upper bounds exist only for the AWGN and Rician fading channel case with the expressions provided by Sason [5] being the prominent ones. Recently, authors in [6], [7] provided probability distri- bution based derivation for pairwise error probability over Rayleigh and Rician fading channels and showed that this approach led to computationally efficient results. This letter follows a similar approach to derive an exact and efficient expression for pairwise error probability over fully interleaved Nakagami-m fading channels for BPSK modulation. The newly derived expression is also approximated to yield a tight upper bound which stays close to the exact expression. Using these new results, union upper bounds are obtained which are computationally more efficient than an existing exact pairwise error probability expression which is based on numerical integration. A rate 1/3 turbo code with a memory of Manuscript received January 16, 2007. The associate editor coordinating the review of this letter and approving it for publication was Dr. Giorgio Taricco. S. A. Ali is with the Department of Computer Technology and Information Systems, Bilkent University, Ankara, Turkey (e-mail: syedali@bilkent.edu.tr). N. S. Kambo is Ex. Prof. IIT, Delhi, India and EMU, Famagusta, North Cyprus (e-mail: nskambo@yahoo.com). E. A. ˙ Ince is with the Department of Electrical and Electronic Engineering, Eastern Mediterranean University, Famagusta, North Cyprus, via Mersin 10, Turkey (e-mail: erhan.ince@emu.edu.tr). Digital Object Identifier 10.1109/LCOMM.2007.070082. two and an input block size of K bits and an output encoded stream of N = 3(K + 2) bits is used to study the pairwise error probability expressions. With both encoders terminating in the zero state, union upper bounds for (1, 7/5, 7/5) and (1, 5/7, 5/7) code structures are obtained to demonstrate the application of the proposed results. The remaining sections of this letter are organized in the following order. Section II discusses details regarding trans- fer function based average union upper bounds. Section III presents an existing expression for pairwise error probability together with the newly derived results. In Section IV results are discussed. Lastly, Section V summarizes the findings of the work. II. TRANSFER FUNCTION BASED UNION BOUNDS The rate 1/3 turbo code is a parallel concatenated error correction coding scheme. The encoder output is transmitted as the multiplexed version of the message sequence with Hamming weight i (the systematic bit) together with the two parity sequences with Hamming weights of d 1 and d 2 . The parity sequences are obtained by encoding the message sequence and the interleaved version of the message. Since the codeword is obtained by concatenating the three sequences, the Hamming weight d of the codeword becomes the sum of the individual weights of these three sequences (i.e., d = i + d 1 + d 2 ). Following [2] and [4], union upper bound on the ensemble average value of bit error probability can be written as P b ≤ K  i=1  d1  d2 i K  K i  p(d 1 |i)p(d 2 |i)P 2 (d) (1) where p(d|i) is the conditional probability of producing a codeword fragment of weight d for a randomly selected input sequence of weight i, P 2 (d) represents the pairwise error probability and the Hamming weight d of the code starts from d min = (i + d 1 + d 2 ) min which is based on the selected code [3]. In order to evaluate the expression in (1) the distribution of the weight of parity sequences needs to be determined. The expression for their distribution is attained as shown by [2] and equals p(d|i)= t(l, i, d) ( K i ) (2) where t(l, i, d) is obtained from the code’s transfer function and represents the total number of paths of length l, input weight i, and output weight d, emerging from and terminating in the zero state. 1089-7798/07$25.00 c  2007 IEEE