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)