New results for Shannon capacity over generalized multipath fading channels with MRC diversity
© El Bouanani et al.; licensee Springer. 2012
Received: 17 December 2011
Accepted: 12 September 2012
Published: 7 November 2012
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.
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.
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 the instantaneous signal-to-noise ratio (SNR) per symbol at the output of MRC receiver in correlated Nakagami-m fading was derived in terms of Fox’s H and Gamma functions 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. In, 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 Fox’s H and Meijer functions[13, 14]. Recently, a novel expression for the BER of modulations and Shannon capacity over generalized-K and Nakagami-m fading channels in terms of Meijer G-function and its generalization 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 DS-CDMA system with MRC-Rake receiver was investigated in.
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.
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
x i , y i , b i 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,
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.
represents the zero-mean, unit-variance Gaussian noise added to the i th transmitted code symbol of the k th user,
is the fading amplitude associated to the k th user, the l th path and the i th symbol (the 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),
From (4) and (1), we can see that the equivalent fading amplitude in both DS-CDMA 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 I0 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).
where 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 are Rayleigh, Rician, Nakagami-m or Weibull distributed random variables (RVs).
Rayleigh multipath fading channel
where Xj 1 (in-phase component) and Xj 2 (quadrature component) are independent normally distributed RVs with mean 0 and the same variance σ2.
where denotes the gamma function.
For single-path Rayleigh fading channel (L = 1),the expression (13) is exactly that found by Sagias et al. 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 IL−1(.) is a modified Bessel function of the first kind of order L⏱−⏱1.
This formula generalizes the capacity expression founded by Sagias et al. 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 and then we find the above expression (13).
Average energies of the multipath fading (LOS and non LOS)
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.
Nakagami-m 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
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 Nakagami-m distributed of the same average energy 2σ2. Since the distribution of the of two independent gamma RVs of parameters (α1β) and (α2β) is a gamma distribution of parameters (α1 + α2β), the RV is a gamma distribution of parameters and mean 2L σ2.
Practically, for 3-path 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), 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 and l,k are the two small positive integers co-prime such that
This formulas generalizes the capacity obtained for Rayleigh multipath fading channel (13) by setting , and the Shannon capacity of Weibull fading channel (one path) given by Sagias et al. for L = 1.
The parameters of Figure 11
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 .
The Figure11b depict the comparison between the analytical and simulation expressions by generating 2000 Weibull-distributed random numbers. It is shown that the approximated Shannon capacity given in (40) is very closed to the exact one.
In this article, we have derived the closed-form expression of the channel capacity over Nakagami-m, Rician and Rayleigh multipath flat fading channel with L-branch MRC receiver and an tight approximation for Weibull case in terms of Meijer G-functions. The DS-CDMA cellular system case was also considered. The results were validated using Monte-Carlo 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.
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.
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).
- Shannon CE: A mathematical theory of communication. Bell Syst. Tech. J 1948, 27: 379-423; 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 M-S: 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 diversity-combining techniques. IEEE Trans. Veh. Technol 1999, 48: 1165-1181. 10.1109/25.775366View ArticleGoogle Scholar
- Mallik RK, Win MZ, Shao JW, Alouini M-S, Goldsmith AJ: Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading. IEEE Trans. Wirel. Commun 2004, 3(4):1124-1133. 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):97-99. 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: 1063-1067. 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: Closed-form statistics for the sum of squared Nakagami-m variates and its applications. IEEE Trans. Commun 2004, 54(8):1353-1359.View ArticleGoogle Scholar
- Ansari IS, Yilmaz F, Alouini M-S, Kucur O: New results on the sum of gamma random variates with application to the performance of wireless communication systems over Nakagami-m 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):649-668. 10.1007/s11277-009-9837-6View ArticleGoogle Scholar
- Khatalin S, Fonseka JP: Capacity of correlated Nakagami-m fading channels with diversity combining techniques. IEEE Trans. Veh. Commun 2006, 55(1):142-150. 10.1109/TVT.2005.861206View ArticleMATHGoogle Scholar
- Di Renzo M, Graziosi F, Santucci F: Channel capacity over generalized fading channels: a novel MGF-based approach for performance analysis and design of wireless communication systems. IEEE Trans. Veh. Technol 2010, 59(1):127-149.View ArticleGoogle Scholar
- Yilmaz F, Alouini M-S: A unified MGF-based capacity analysis of diversity combiners over generalized fading channels. IEEE Trans. Commun 2012, 60(3):862-875. Available at http://arxiv.org/abs/1012.2596View ArticleGoogle Scholar
- Hai NT, Yakubovich SB: The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory. World Scientific, Singapore; 1992.View ArticleMATHGoogle Scholar
- Ansari IS, Al-Ahmadi S, Yilmaz F, Alouini M-S, Yanikomeroglu H: A new formula for the BER of binary modulations with dual-branch selection over generalized-k composite fading channels. IEEE Trans. Comm 2011, 59: 1291-1303.View ArticleGoogle Scholar
- Xia M, Xing C, Chung Y, Aissa S: Exact performance analysis of dual-hop semi-blind AF relaying over arbitrary Nakagami-m fading channels. IEEE Trans. Wirel. Commun 2011, 10(10):3449-3459.View ArticleGoogle Scholar
- Cideciyan RD, Eleftheriou E, Rupf M: Concatenated Reed-Solomon/convolutional coding for data transmission in CDMA-Based cellular systems. IEEE Trans. Commun 1997, 45: 1291-1303. 10.1109/26.634693View ArticleGoogle Scholar
- Stuber GL: Principles of Mobile Communications. Kluwer Academic Publishers, Mass; 1996.View ArticleGoogle Scholar
- Proakis JG: Digital Communications. McGraw-Hill, 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 m-Distribution, a general formula of intensity of rapid fading. In Statistical Methods in Radio Wave Propagation. Edited by: Hoffman WC. New York:Pergamon Press, London; 1960:3-36.View ArticleGoogle Scholar
- Moschopoulos PG: The distribution of the sum of independent Gamma random variables. Annals Inst. Stat. Math. Part A 1985, 37: 541-544. 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):614-616. 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; 1990:212-224.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. McGraw-Hill, New York; 1953.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.