New results for Shannon capacity over generalized multipath fading channels with MRC diversity
 Faissal El Bouanani^{1}Email author,
 Hussain BenAzza^{2} and
 Mostafa Belkasmi^{1}
https://doi.org/10.1186/168714992012336
© El Bouanani et al.; licensee Springer. 2012
Received: 17 December 2011
Accepted: 12 September 2012
Published: 7 November 2012
Abstract
In this article, we investigate the Shannon capacity for Lbranch maximal combining ratio (MRC) over generalized multipath fading channel. We derive closedform expressions of the maximal spectral efficiency over Rayleigh, Rician, Nakagamim, and Weibull multipath fading channel under flat fading conditions. The results are expressed in terms of Meijer Gfunctions, 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.
Keywords
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, Nakagamim, and Weibull fading channel was obtained in term of Meijer Gfunction. Some statistics properties, such as the probability density function (PDF) and the cumulative distribution function, of the instantaneous signaltonoise ratio (SNR) per symbol at the output of MRC receiver in correlated Nakagamim fading was derived in terms of Fox’s H and Gamma functions[8] in[9, 10]. A statistical analysis for the capacity over Nakagamim fading with MRC/SC/switch and stay combining (SSC) in terms of Meijer Gfunction was presented in[11]. In[12], the capacity expressions of correlated Nakagamim fading with sualbranch 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 Fox’s H and Meijer functions[13, 14]. Recently, a novel expression for the BER of modulations and Shannon capacity over generalizedK and Nakagamim fading channels in terms of Meijer Gfunction 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 MRCRake receiver was investigated in[18].
In this article, we present novel closedform and analytical expressions, in terms of Meijer Gfunction, for the ergodic Shannon capacity for Lbranch MRCRake receiver over Rician, Rayleigh, Nakagamim, and simple approximation for Weibull multipath fading channel in the nonfrequency selective channels case. We generalize for Lbranch 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 DSCDMA system and nospreading system cases.
This article is structured as follows. In Section ‘Channel model’, the equivalent channel models of both DSCDMA communication system and system without spreading using a Rake receiver is introduced. In Section ‘Channel capacity’, the closedform expression of the channel capacity for multipath fading channel (case of Rayleigh, Rice, Nakagamim, 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 Gfunction.
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 (DSCDMA).
System with MRC diversity
Consider MRC diversity systems in flat fading environment. Let

x_{ i }, y_{ i }, b_{ i } be the i th transmitted symbol, i th combined received symbol, i th zeromean, N_{0}/2variance Gaussian noise added,

N_{0} is the noise power spectral density,

h_{ il } be the fading amplitude corresponding to the i th symbol and the l th antenna, assumed being i.i.d.,

L be the number of diversity channels,

E_{ s } be the average symbol energy.
DSCDMA system

$\left(\right)close="">{b}_{i}^{\left(k\right)}$ represents the zeromean, unitvariance Gaussian noise added to the i th transmitted code symbol of the k th user,

$\left(\right)close="">\left{h}_{\mathit{\text{il}}}^{\left(k\right)}\right$ is the fading amplitude associated to the k th user, the l th path and the i th symbol (the $\left(\right)close="">{h}_{\mathit{\text{il}}}^{\left(k\right)}$ are complex numbers). The amplitudes are assumed independent identically distributed (i.i.d.)

L is the Rake receiver branch number,

g is the spreading factor,

I _{0} is the total noise power spectral density (including multiple access interference, thermal noise, and multipath fading),
where ${A}_{\mathit{\text{iL}}}^{\left(k\right)}=\sqrt{\frac{\sum _{l=1}^{L}\phantom{\rule{0.3em}{0ex}}{\left{h}_{\mathit{\text{il}}}^{\left(k\right)}\right}^{2}}{\sum _{l=1}^{L}\phantom{\rule{0.3em}{0ex}}\mathbb{E}\left[{\left{h}_{\mathit{\text{il}}}^{\left(k\right)}\right}^{2}\right]}}.$
Remark 1
From (4) and (1), we can see that the equivalent fading amplitude in both DSCDMA and general case of system without spreading using MRC technique is the same. Consequently, the spectral efficiency is the same in the two cases.
For simplicity of notation, and regardless of the user and bit index, we use I_{0} as the total noise power, the A_{ L } as the equivalent fading amplitude, and h_{ l } as the l th path fading amplitude (that is, we drop the index i and index k).
Channel capacity
where $\stackrel{\u0304}{\gamma}=\frac{{E}_{s}}{\mathrm{TW}{I}_{0}}$ is the average received SNR, T is the symbol period, and f(r) denotes the probability density function (PDF) of the fading amplitude random variable R.
In the following, we give the exact Shannon capacity (5) for the channel models described in the previous section in the case where all $\left(\right)close="">\left{h}_{\mathit{\text{il}}}^{\left(k\right)}\right$ are Rayleigh, Rician, Nakagamim or Weibull distributed random variables (RVs).
Rayleigh multipath fading channel
where X_{j 1} (inphase component) and X_{j 2} (quadrature component) are independent normally distributed RVs with mean 0 and the same variance σ^{2}.
where $\Gamma \left(L\right)={\int}_{0}^{+\infty}{t}^{L1}{e}^{t}\mathit{dt}$ denotes the gamma function[8].
For singlepath Rayleigh fading channel (L = 1),the expression (13) is exactly that found by Sagias et al.[6] using the single branch receiver (SBR). This is due to the fact that the single finger Rake receiver and SBR have identical function.
Rician multipath fading channel
where I_{L−1}(.) is a modified Bessel function of the first kind of order L⏱−⏱1[8].
This formula generalizes the capacity expression founded by Sagias et al.[6] in the case of one path. It generalizes also our first result corresponding to the normalized Rayleigh multipath fading channel. Indeed, for Rayleigh fading, all LOS amplitudes s_{ j } equal zero $(\lambda =\varphi =0,\frac{{\beta}^{2}}{2}=L),$ and then we find the above expression (13).
Average energies of the multipath fading (LOS and non LOS)
L  3  2  4  

$\left(\right)close="">{s}_{i}^{\left(2\right)}$  $\frac{1}{4},\frac{1}{4}$  $\frac{1}{4},\frac{1}{8},\frac{1}{8}$  $\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{16}$  
2σ^{2}  $\frac{1}{4}$  $\frac{1}{6}$  $\frac{1}{8}$  
λ  2  $\sqrt{6}$  $2\sqrt{2}$  
β  $2\sqrt{2}$  $2\sqrt{3}$  4 
Since the received power in the case of Rician fading is greater than that in the case of Rayleigh, so the required Rake receiver finger number to reach the AWGN capacity must be less or equal to 4.
Nakagamim multipath fading channel
In the case of this fading, we show the expression of the channel capacity for two cases:

The received average energies are equal $\left({\sigma}_{i}=\sigma ,\phantom{\rule{1em}{0ex}}\forall i\u2a7dL\right),$

The received average energies are not necessarily equal. This case generalizes the first one.
Case 1: received average energies are equal
Let (N_{ l })_{1≤l≤L} be RVs Nakagamim distributed of the same average energy 2σ^{2}. Since the distribution of the of two independent gamma RVs of parameters (α_{1}β) and (α_{2}β) is a gamma distribution of parameters (α_{1} + α_{2}β)[25], the RV $\sum _{l=1}^{L}{\left{N}_{l}\right}^{2}$ is a gamma distribution of parameters $(\mathit{\text{mL}},\frac{2{\sigma}^{2}}{m})$ and mean 2L σ^{2}.
It’s clear that for m ≥ 1 (Figures2a and6a,b), L = 4 appears to be sufficient to approach the AWGN channel.
Practically, for 3path channel and m=3,the fading channel behaves like an AWGN channel.
Case 2: received average energies are not equal
This expression generalizes (25). Indeed, in the case of the same average energies (σ_{ i } = σ), $\varphi =\frac{\mathit{\text{mL}}}{\stackrel{\u0304}{\gamma}}$ and δ_{ k } = 0(k > 0).
Weibull multipath fading channel
Weibull fading, based on the Weibull distribution, is a simple statistical model of fading both in indoor and outdoor wireless communications.
and $\Phi \left(p,l\right)=\frac{p}{l},\frac{p+1}{l},\dots ,\frac{p+l1}{l}$ and l,k are the two small positive integers coprime such that $\frac{l}{k}=\frac{\beta}{2}.$
This formulas generalizes the capacity obtained for Rayleigh multipath fading channel (13) by setting $\beta =2(k=l=1,\lambda =\frac{1}{L})$, and the Shannon capacity of Weibull fading channel (one path) given by Sagias et al. for L = 1[6].
The parameters of Figure 11
β  1.5  2  2.8 
l  3  1  7 
k  4  1  5 
As the number β increases, as the fading disappears (expression (33)), and the capacity (40) becomes closer to that of the AWGN for four Rake receiver fingers and $\beta \u2a7e2.8$.
The Figure11b depict the comparison between the analytical and simulation expressions by generating 2000 Weibulldistributed random numbers. It is shown that the approximated Shannon capacity given in (40) is very closed to the exact one.
Conclusion
In this article, we have derived the closedform expression of the channel capacity over Nakagamim, Rician and Rayleigh multipath flat fading channel with Lbranch MRC receiver and an tight approximation for Weibull case in terms of Meijer Gfunctions. The DSCDMA cellular system case was also considered. The results were validated using MonteCarlo Simulations for all multipath fading channels studied. The plotted curves show that as L increases, the capacity approaches the AWGN channel capacity. Furthermore, four branches are sufficient, in all cases, to eliminate the fading effect and then to achieve the maximum capacity.
Appendix 1
 (i)
Left loop beginning at −∞ + λi and ending at −∞ + δi for λ < δ encircling once all the poles −b _{ j }−l (for j = 1…,m,l = 0,1,…) leaving the poles 1−a _{ j } + l(for j = 1…,n,l = 0,1,…) to the right.
 (ii)
Right loop beginning at ∞ + λi and ending at ∞ + δi for λ > δ encircling once all the poles 1−a _{ j } + l (for j = 1…,m,l = 0,1,…) leaving the poles b _{ j }−l (for j = 1…,n,l = 0,1,…) to the left.
 (iii)A line with indentation beginning at λ−i ∞ ending at λ + i ∞ (for some real λ) separating the poles of the integrand g like the cases (i) and (ii). We refer to [29, 30] for the existence of G(z).

For any x ([23], /07.34.03.0228.01),${e}^{x}={G}_{0,1}^{1,0}\left[x\left\begin{array}{c}0\end{array}\right.\right]$(42)

For any x ([23], /07.34.03.0456.01),$ln\left(1+x\right)=\underset{2,2}{\overset{1,2}{G}}\left[x\left\begin{array}{c}1,1\\ 1,0\end{array}\right.\right]$(43)

For −Π < argx ≤ 0 the modified Bessel function of the first kind ([23], /07.34.03.0230.01),${I}_{n}\left(x\right)={i}^{n}{G}_{0,2}^{1,0}\left[\frac{{x}^{2}}{4}\left\begin{array}{c}n/2,n/2\end{array}\right.\right]$(44)

The Mellin transform of a Gfunction is ([23], /07.34.21.0009.01),$\underset{0}{\overset{\infty}{\int}}{x}^{s1}{G}_{p,q}^{m,n}\left[\mathrm{yx}\left\begin{array}{c}\left({a}_{p}\right)\\ \left({b}_{q}\right)\end{array}\right.\right]\mathrm{dx}={y}^{s}g\left(s\right)$(45)

The Mellin transform of a product of two Gfunctions is ([23], /07.34.21.0011.01),[28],$\begin{array}{cc}\underset{0}{\overset{\infty}{\int}}\hfill & {x}^{\alpha 1}{G}_{u,v}^{s,t}\left[\mathrm{\sigma x}\left\begin{array}{c}\left({c}_{u}\right)\\ \left({d}_{v}\right)\end{array}\right.\right]{G}_{p,q}^{m,n}\left[\mathrm{\omega x}\left\begin{array}{c}\left({a}_{p}\right)\\ \left({b}_{q}\right)\end{array}\right.\right]\mathrm{dx}\hfill \\ ={\sigma}^{\alpha}{G}_{p+v,q+u}^{m+t,n+s}\left[\frac{\omega}{\sigma}\left\begin{array}{c}\left({a}_{n}\right),1\alpha {d}_{1},\dots ,1\alpha {d}_{v},{a}_{n+1},\dots ,{a}_{p}\\ \left({b}_{m}\right),1\alpha {c}_{1},\dots ,1\alpha {c}_{v},{b}_{m+1},\dots ,{b}_{q}\end{array}\right.\right]\hfill \end{array}$(46)

Declarations
Authors’ Affiliations
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