New results for Shannon capacity over generalized multipath fading channels with MRC diversity
Faissal El Bouanani
1
Hussain Ben-Azza
0
Mostafa Belkasmi
1
0
ENSAM, Mekne`s,
Morocco
1
ENSIAS,
Mohammed V-Souissi University
, Rabat,
Morocco
In this article, we investigate the Shannon capacity for L-branch maximal combining ratio (MRC) over generalized multipath fading channel. We derive closed-form expressions of the maximal spectral efficiency over Rayleigh, Rician, Nakagami-m, and Weibull multipath fading channel under flat fading conditions. The results are expressed in terms of Meijer G-functions, which can be evaluated numerically using mathematical tools such as Mathematica and Maple. We show, in particular, that the more the number L increases, the larger the Shannon capacity is. We deduce that four branches are sufficient in several cases to mitigate the fading effect and the channel model will approaches the one of AWGN.
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Introduction
The channel capacity is an important parameter in the
design of any communication system. It provides an upper
bound of maximum transmission rate in a given channel.
In 1948, Shannon derived the AWGN channel capacity
[1,2]. Recently, in wireless mobile communication
system, the diversity techniques have been used to combat
multipath fading and multiuser interference [3].
In last years, several articles have been published
regarding the Shannon capacity of fading channels with
various important diversity schemes, such as maximum
ratio combining (MRC), postdetection equal gain
combining (EGC), and selective combining (SC), in terms of
generalized special functions. The capacity with MRC in
correlated Rayleigh fading in terms of Poisson distribution
and Exponential integral was obtained in [4,5]. In [6,7],
an expression of the capacity of single branch receivers
operating over Rician, Nakagami-m, and Weibull fading
channel was obtained in term of Meijer G-function. Some
statistics properties, such as the probability density
function (PDF) and the cumulative distribution function, of
*Correspondence:
1ENSIAS, Mohammed V-Souissi University, Rabat, Morocco
Full list of author information is available at the end of the article
the instantaneous signal-to-noise ratio (SNR) per symbol
at the output of MRC receiver in correlated
Nakagamim fading was derived in terms of Foxs H and Gamma
functions [8] in [9,10]. A statistical analysis for the
capacity over Nakagami-m fading with MRC/SC/switch and
stay combining (SSC) in terms of Meijer G-function was
presented in [11]. In [12], the capacity expressions of
correlated Nakagami-m fading with sual-branch MRC, EGC,
SC, and SSC were obtained in terms of Gamma
function. Expressions for the capacity of generalized fading
channel with MRC/EGC for MIMO/SISO systems based
on moment generating function (MGF) approach was
obtained in terms of Foxs H and Meijer functions [13,14].
Recently, a novel expression for the BER of modulations
and Shannon capacity over generalized-K and
Nakagamim fading channels in terms of Meijer G-function and its
generalization [15] was investigated in [16,17].
The equivalent channel model of a multipath fading
channel using a MRC Rake receiver and flat fading has
been approximated by a fading channel with fading
amplitude is a square root of a sum of a square amplitude of
each fading [18,19]. The equivalent channel model in
DSCDMA system with MRC-Rake receiver was investigated
in [18].
In this article, we present novel closed-form and
analytical expressions, in terms of Meijer G-function, for
the ergodic Shannon capacity for L-branch MRC-Rake
receiver over Rician, Rayleigh, Nakagami-m, and simple
approximation for Weibull multipath fading channel in
the non-frequency selective channels case. We generalize
for L-branch MRC the capacity expression given for
single path case (L = 1) in [6,7]. All the results are validated
by numerical Monte Carlo simulations. The study include
both DS-CDMA system and no-spreading system cases.
This article is structured as follows. In Section Channel
model, the equivalent channel models of both DS-CDMA
communication system and system without spreading
using a Rake receiver is introduced. In Section
Channel capacity, the closed-form expression of the channel
capacity for multipath fading channel (case of Rayleigh,
Rice, Nakagami-m, and Weibull) is derived. The main
results are summarized and some conclusions are given in
Section Conclusion. For the convenience of the reader, an
short appendix is added, regarding Meijer G-function.
Channel model
In this section, we present the equivalent channel model
of communication systems using coherent MRC receiver
in both system without spreading and Direct spread
spectrum system (DS-CDMA).
System with MRC diversity
Consider MRC diversity systems in flat fading
environment. Let
xi, yi, bi be the i th transmitted symbol, i th combined
received symbol, i th zero-mean, N0/2-variance
Gaussian noise added,
N0 is the noise power spectral density,
hil be the fading amplitude corresponding to the ith
symbol and the l th antenna, assumed (...truncated)