Performance of Coded Systems with Generalized Selection Diversity in Nakagami Fading

EURASIP Journal on Wireless Communications and Networking, Dec 2008

We investigate the performance of coded diversity systems employing generalized selection combining (GSC) over Nakagami fading channels. In particular, we derive a numerical evaluation method for the cutoff rate of the GSC systems. In addition, we derive a new union bound on the bit-error probability based on the code's transfer function. The proposed bound is general to any coding scheme with a known weight distribution such as convolutional and trellis codes. Results show that the new bound is tight to simulation results for wide ranges of diversity order, Nakagami fading parameter, and signal-to-noise ratio (SNR).

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Performance of Coded Systems with Generalized Selection Diversity in Nakagami Fading

EURASIP Journal on Wireless Communications and Networking Hindawi Publishing Corporation Performance of Coded Systems with Generalized Selection Diversity in Nakagami Fading Salam A. Zummo 0 0 Electrical Engineering Department, King Fahd University of Petroleum and Minerals (KFUPM) , Dhahran 31261 , Saudi Arabia We investigate the performance of coded diversity systems employing generalized selection combining (GSC) over Nakagami fading channels. In particular, we derive a numerical evaluation method for the cutoff rate of the GSC systems. In addition, we derive a new union bound on the bit-error probability based on the code's transfer function. The proposed bound is general to any coding scheme with a known weight distribution such as convolutional and trellis codes. Results show that the new bound is tight to simulation results for wide ranges of diversity order, Nakagami fading parameter, and signal-to-noise ratio (SNR). 1. INTRODUCTION Diversity is an effective method to mitigate multipath fading in wireless communication systems. Diversity improves the performance of communication systems by providing a receiver with M independently faded copies of the transmitted signal such that the probability that all these copies are in a deep fade is low. The diversity gain is obtained by combining the received copies at the receiver. The most general diversity combining scheme is the generalized selection combining (GSC), which provides a tradeoff between the high complexity of maximal-ratio combining (MRC) and the poor performance of selection combining (SC). In GSC, the largest Mc branches out of M diversity branches are combined using MRC. The resulting signal-to-noise ratio (SNR) at the output of the combiner is the sum of the SNRs of the largest Mc branches. A general statistical model for multipath fading is the Nakagami distribution [1]. The error probability and the cutoff rate of GSC over Rayleigh fading channels was analyzed in [2, 3], respectively. In [4], the performance of some special cases of GSC systems over Nakagami fading channels was analyzed. A more general framework to the analysis of GSC systems over Nakagami fading channels was presented in [5] and more recently in [6]. In [7], the cutoff rate and a union bound on the bit-error probability of coded SC systems over Nakagami fading channels were derived. The derivation is based on the transfer function of the code. To the best of our knowledge, no analytical results on the performance of coded GSC systems over Nakagami fading channels exit yet. In [8], a new approach to analyzing the performance of GSC over Nakagami fading channels was presented. The approach is based on converting the multidimensional integral that appears in the error probability of GSC into a single integral that can be evaluated efficiently. In this paper, we generalize this approach to derive the cutoff rate and a union bound on the bit-error probability of coded GSC over Nakagami fading channels. The bound is based on the transfer function of the code and is simple to evaluate using the Gauss-Leguerre integration (GLI) rule [9]. Results show that the proposed union bound is tight to simulation results for a wide range of Nakagami parameter, SNR values, and diversity orders. The paper is organized as follows. The coded GSC system is described in Section 2. In Section 3, the cutoff rate of coded GSC systems is derived. In Section 4, the proposed union bound on the bit-error probability is derived, and results are discussed therein. Conclusions are discussed in Section 5. 2. SYSTEM MODEL The transmitter in a coded system is generally composed of an encoder, interleaver, and a modulator. The encoder might ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 10 ) ( 11 ) where fa2 (x) and Fa2 (x) are, respectively, the probability density function (pdf) and cumulative distribution function (CDF) of the SNR of each diversity branch, and φa2 (d, x) is the marginal MGF [8] defined as φa2 (d, x) = e−dt fa2 (t)dt. ∞ x For Nakagami fading channels, the pdf and CDF are given, respectively, by mm fa2 (x) = Γ(m) xm−1e−mx, Fa2 (x) = γ(m, mx), x ≥ 0, m ≥ 0.5, x ≥ 0, m ≥ 0.5, where γ(a, y) = (1/Γ(a)) 0y e−tta−1dt is the incomplete Gamma function and Γ(·) is the Gamma function. The marginal MGF for Nakagami fading [8] is given by 1 1 φa2 (d, x) = Γ(m) (1 + d/m)m 1 − γ m, mx(1 + d/m) . Substituting ( 6 )–( 8 ) into ( 4 ), we obtain C si, sj = Mc M mm 1 Mc Γ(m)Mc (1 + d/m)m(Mc−1) be convolutional, turbo, trellis-coded modulation (TCM), or any other coding scheme. The encoder encodes a block of K information bits into a codeword of L symbols. The code rate is defined as Rc = K/L. For the lth symbol in the codeword, the matched filter output of the ith diversity branch is given by yl,i = Esal,i sl + zl,i, ( 1 ) where Es is the received signal energy per diversity branch M and al = {al,i}i=1 are the fading amplitudes affecting the M diversity branches, modeled as independent (...truncated)


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Salam A Zummo. Performance of Coded Systems with Generalized Selection Diversity in Nakagami Fading, EURASIP Journal on Wireless Communications and Networking, 2008, pp. 670503, Volume 2008, Issue 1, DOI: 10.1155/2008/670503