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
Shannon capacity Spectral efficiency DSCDMA Multipath fading channels Meijer Gfunctions Wireless communication Rake receiver MRC diversityIntroduction
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
References
 Shannon CE: A mathematical theory of communication. Bell Syst. Tech. J 1948, 27: 379423; 623–656.MathSciNetView ArticleMATHGoogle Scholar
 Gallager RG: Information Theory and Reliable Communication. John Wiley and Sons, Inc., New York; 1968.MATHGoogle Scholar
 Simon MK, Alouini MS: Digital Communication Over Fading Channels. Wiley, New York; 2005.Google Scholar
 Alouini MS, Goldsmith A: Capacity of Rayleigh fading channels under different adaptive transmission and diversitycombining techniques. IEEE Trans. Veh. Technol 1999, 48: 11651181. 10.1109/25.775366View ArticleGoogle Scholar
 Mallik RK, Win MZ, Shao JW, Alouini MS, Goldsmith AJ: Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading. IEEE Trans. Wirel. Commun 2004, 3(4):11241133. 10.1109/TWC.2004.830823View ArticleGoogle Scholar
 Sagias NC, Tombras GS, Karagiannidis GK: New results for the Shannon channel capacity in generalized fading channels. IEEE Commun. Lett 2005, 9(2):9799. 10.1109/LCOMM.2005.02031View ArticleGoogle Scholar
 Sagias NC, Karagiannidis GK, Zogas DA, Mathiopoulos PT, Tombras GS: Performance analysis of dual selection diversity in correlated Weibull fading channels. IEEE Trans. Commun 2004, 52: 10631067. 10.1109/TCOMM.2004.831362View ArticleGoogle Scholar
 Abramowitz M, Stegun IA: Handbook of Mathematical functions,. 55 (National Bureau of Standards  U.S. Government Printing Office, 1972)MATHGoogle Scholar
 Karagiannidis GK, Sagias NC, Tsiftsis TA: Closedform statistics for the sum of squared Nakagamim variates and its applications. IEEE Trans. Commun 2004, 54(8):13531359.View ArticleGoogle Scholar
 Ansari IS, Yilmaz F, Alouini MS, Kucur O: New results on the sum of gamma random variates with application to the performance of wireless communication systems over Nakagamim fading channels. IEEE Int. Workshop on Sig. Proc. Advan. in Wireless Comm. accepted in SPAWC’12, available in http://arxiv.org/pdf/1202.2576.pdf
 Sagias NC, Lazarakis FI, Alexandridis AA, Dangakis KP, Tombras GS: Higher order capacity statistics of diversity receivers. Wirel. Personal Commun 2011, 56(4):649668. 10.1007/s1127700998376View ArticleGoogle Scholar
 Khatalin S, Fonseka JP: Capacity of correlated Nakagamim fading channels with diversity combining techniques. IEEE Trans. Veh. Commun 2006, 55(1):142150. 10.1109/TVT.2005.861206View ArticleMATHGoogle Scholar
 Di Renzo M, Graziosi F, Santucci F: Channel capacity over generalized fading channels: a novel MGFbased approach for performance analysis and design of wireless communication systems. IEEE Trans. Veh. Technol 2010, 59(1):127149.View ArticleGoogle Scholar
 Yilmaz F, Alouini MS: A unified MGFbased capacity analysis of diversity combiners over generalized fading channels. IEEE Trans. Commun 2012, 60(3):862875. Available at http://arxiv.org/abs/1012.2596View ArticleGoogle Scholar
 Hai NT, Yakubovich SB: The Double MellinBarnes Type Integrals and their Applications to Convolution Theory. World Scientific, Singapore; 1992.View ArticleMATHGoogle Scholar
 Ansari IS, AlAhmadi S, Yilmaz F, Alouini MS, Yanikomeroglu H: A new formula for the BER of binary modulations with dualbranch selection over generalizedk composite fading channels. IEEE Trans. Comm 2011, 59: 12911303.View ArticleGoogle Scholar
 Xia M, Xing C, Chung Y, Aissa S: Exact performance analysis of dualhop semiblind AF relaying over arbitrary Nakagamim fading channels. IEEE Trans. Wirel. Commun 2011, 10(10):34493459.View ArticleGoogle Scholar
 Cideciyan RD, Eleftheriou E, Rupf M: Concatenated ReedSolomon/convolutional coding for data transmission in CDMABased cellular systems. IEEE Trans. Commun 1997, 45: 12911303. 10.1109/26.634693View ArticleGoogle Scholar
 Stuber GL: Principles of Mobile Communications. Kluwer Academic Publishers, Mass; 1996.View ArticleGoogle Scholar
 Proakis JG: Digital Communications. McGrawHill, New York; 1995.MATHGoogle Scholar
 Prudnikov AP, Brychkov YuA, Marichev OI: Integrals and Series, Volume 3: More special functions. Gordon and Breach Science Publishers, New York; 1990.MATHGoogle Scholar
 Johnson NL, Kotz S, Balakrishnan N: Continuous Univariate Distributions. John Wiley & Sons, Inc., New York; 1995.MATHGoogle Scholar
 Wolfram: The Wolfram functions site. Internet (online), http://functions.wolfram.com
 Nakagami M: The mDistribution, a general formula of intensity of rapid fading. In Statistical Methods in Radio Wave Propagation. Edited by: Hoffman WC. New York:Pergamon Press, London; 1960:336.View ArticleGoogle Scholar
 Moschopoulos PG: The distribution of the sum of independent Gamma random variables. Annals Inst. Stat. Math. Part A 1985, 37: 541544. 10.1007/BF02481123MathSciNetView ArticleMATHGoogle Scholar
 Rinne H: The Weibull distribution. CRC Press, Boca Raton; 2009.MATHGoogle Scholar
 Santos Filho JCS, Yacoub MD: Simple precise approximations to Weibull sums. IEEE Commun. Lett 2006, 10(8):614616. 10.1109/LCOMM.2006.1665128View ArticleGoogle Scholar
 Adamchik VS, Marichev OI: The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system. In Proc. Int. Conf. on Symbolic and Algebraic Computation,. Tokyo, Japan; 1990:212224.Google Scholar
 Mathai AM, Saxena RK: Generalized hypergeometric functions with applications in statistics and physical sciences. Notes of Mathematics Series No. 348, Heidelberg, Germany, (1973)Google Scholar
 Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG: Higher Transcendental Functions. McGrawHill, New York; 1953.MATHGoogle Scholar
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