- Open Access
The impact of spatial correlation on the statistical properties of the capacity of nakagami-m channels with MRC and EGC
© Rafiq et al; licensee Springer. 2011
- Received: 3 March 2011
- Accepted: 30 September 2011
- Published: 30 September 2011
In this article, we have studied the statistical properties of the instantaneous channel capacitya of spatially correlated Nakagami-m channels for two different diversity combining methods, namely maximal ratio combining (MRC) and equal gain combining (EGC). Specifically, using the statistical properties of the instantaneous signal-to-noise ratio, we have derived the analytical expressions for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LCR), and average duration of fades (ADF) of the instantaneous channel capacity. The obtained results are studied for different values of the number of diversity branches and for different values of the receiver antennas separation controlling the spatial correlation in the diversity branches. It is observed that an increase in the spatial correlation in the diversity branches of an MRC system increases the variance as well as the LCR of the instantaneous channel capacity, while the ADF of the channel capacity decreases. On the other hand, when EGC is employed, an increase in the spatial correlation decreases the mean channel capacity, while the ADF of the instantaneous channel capacity increases. The presented results are very helpful to optimize the design of the receiver of wireless communication systems that employ spatial diversity combining techniques. Moreover, provided that the feedback channel is available, the transmitter can make use of the information regarding the statistics of the instantaneous channel capacity by choosing the right modulation, coding, transmission rate, and power to achieve the capacity of the wireless channelb.
- Probability Density Function
- Spatial Correlation
- Channel Capacity
- Maximal Ratio Combine
- Joint Probability Density Function
The performance of mobile communication systems is greatly affected by the multipath fading phenomenon. In order to mitigate the effects of fading, spatial diversity combining is widely accepted to be an effective method [1, 2]. In spatial diversity combining, such as MRC and EGC, the received signals in different diversity branches are combined in such a way that results in an increased overall received SNR . Hence, the system throughput increases, and therefore, the performance of the mobile communication system improves. It is commonly assumed that the received signals in diversity branches are uncorrelated. This assumption is acceptable if the receiver antennas separation is far more than the carrier wavelength of the received signal . However, due to the scarcity of space on small mobile devices, this requirement cannot always be fulfilled. Thus, due to the spatial geometry of the receiver antenna array, the receiver antennas are spatially correlated. It is widely reported in the literature that the spatial correlation has a significant influence on the performance of mobile communication systems employing diversity combining techniques (see, e.g., [4–6], and the references therein).
There exists a large number of statistical models for describing the statistics of the received radio signal. Among these channel models, the Rayleigh , Rice  and lognormal [9, 10] models are of prime importance due to which they have been thoroughly investigated in the literature. Numerous papers have been published so far dealing with the performance and the capacity analysis of wireless communication systems employing diversity combining techniques in Rayleigh and Rice channels (e.g., [6, 11, 12]). However, in recent years the Nakagami-m channel model  has gained considerable attention due to its good fitness to experimental data and mathematically tractable form [14, 15]. Moreover, the Nakagami-m channel model can be used to study scenarios where the fading is more (or less) severe than the Rayleigh fading. The generality of this model can also be observed from the fact that it inherently incorporates the Rayleigh and one-sided Gaussian models as special cases. For Nakagami-m channels, results pertaining to the statistical analysis of the signal envelope at the combiner output in a diversity combining system, assuming spatially uncorrelated diversity branches, can be found in . For such systems, statistical analysis of the instantaneous channel capacity has also been presented in . Moreover, when using EGC, the system performance analysis is reported in . In addition, a large number of articles can also be found in the literature that study Nakagami-m channels in systems with spatially correlated diversity branches [5, 19–24]. Furthermore, the instantaneous capacity of spatially correlated Nakagami-m multiple-input multiple-output (MIMO) channels has also been investigated in . However, to the best of the authors' knowledge, there is still a gap of information regarding the statistical analysis of the instantaneous capacity of spatially correlated Nakagami-m channels with MRC and EGC. Specifically, second-order statistical properties, such as the LCR and the ADF, of the instantaneous capacity of spatially correlated Nakagami-m channels with MRC or EGC have not been investigated in the literature. The aim of this paper is to fill this gap in information.
This paper presents the derivation and analysis of the PDF, CDF, LCR, and ADF of the instantaneous channel capacityc of spatially correlated Nakagami-m channels, for both MRC and EGC. The PDF of the channel capacity is helpful to study the mean channel capacity (or the ergodic capacity) , while the CDF of the channel capacity is useful for the derivation and analysis of the outage capacity . The mean channel capacity and the outage capacity are very widely explored by the researchers due to their importance for the system design and performance analysis. The ergodic capacity provides the information regarding the average data rate offered by a wireless link (where the average is taken over all the realizations of the channel capacity) [27, 28]. On the other hand, the outage capacity quantifies the capacity (or the data rate) that is guaranteed with a certain level of reliability [27, 28]. However, these two aforementioned statistical measures do not provide insight into the temporal behavior of the channel capacity. For example, the outage capacity is a measure of the probability of a specific percentage of capacity outage, but it does not give any information regarding the spread of the outage intervals or the rate at which these outage durations occur over the time scale. Whereas, the information regarding the temporal behavior of the channel capacity is very useful for the improvement of the system performance .
The temporal behavior of the channel capacity can be investigated with the help of the LCR and ADF of the channel capacity. The LCR of the channel capacity is a measure of the expected number of up-crossings (or down-crossings) of the channel capacity through a certain threshold level in a time interval of one second. While, the ADF of the channel capacity describes the average duration of the time over which the channel capacity is below a given level [30, 31]. A decrease in the channel capacity below a certain desired level results in a capacity outage, which in turn causes burst errors. In the past, the level-crossing and outage duration analysis have been carried out merely for the received signal envelope to study handoff algorithms in cellular networks as well as to design channel coding schemes to minimize burst errors [32, 33]. However, for systems employing multiple antennas, the authors in  have proposed to choose the channel capacity as a more pragmatic performance merit than the received signal envelope. Therein, the significance of studies pertaining to the analysis of the LCR of the channel capacity can easily be witnessed for the cross-layer optimization of overall network performance. In a similar fashion, for multi-antenna systems, the importance of investigating the ADF of the channel capacity for the burst error analysis can be argued. It is here noteworthy that the LCR and ADF of the channel capacity are the important statistical quantities that describe the dynamic nature of the channel capacity. Hence, studies pertaining to unveil the dynamics of the channel capacity are cardinal to meet the data rate requirements of future mobile communication systems.
We have analyzed the statistical properties of the channel capacity for different values of the number of diversity branches L and for different values of the receiver antennas separation δ R controlling the spatial correlation in diversity branches. For comparison purposes, we have also included the results for the mean and variance of the capacity of spatially correlated Rayleigh channels with MRC and EGC (which arise for the case when the Nakagami parameter m = 1). It is observed that for both MRC and EGC, an increase in the number of diversity branches L increases the mean channel capacity, while the variance and the ADF of the channel capacity decrease. Moreover, an increase in the severity of fading results in a decrease in the mean channel capacity; however, the variance and ADF of the channel capacity increase. It is also observed that at lower levels, the LCR is higher for channels with smaller values of the number of diversity branches L or higher severity levels of fading than for channels with larger values of L or lower severity levels of fading. We have also studied the influence of spatial correlation in the diversity branches on the statistical properties of the channel capacity. Results show that an increase in the spatial correlation in diversity branches of an MRC system increases the variance as well as the LCR of the channel capacity, while the ADF of the channel capacity decreases. On the other hand, for the case of EGC, an increase in the spatial correlation decreases the mean channel capacity, whereas the ADF of the channel capacity increases. Moreover, this effect increases the LCR of the channel capacity at lower levels. We have confirmed the correctness of the theoretical results by simulations, whereby a very good fitting is observed.
The rest of the paper is organized as follows. Section 2 gives a brief overview of the MRC and EGC schemes in Nakagami-m channels with spatially correlated diversity branches. In Section 3, we present the statistical properties of the capacity of Nakagami-m channels with MRC and EGC. Section 4 deals with the analysis and illustration of the theoretical as well as the simulation results. Finally, the conclusions are drawn in Section 5.
where μ i , l (t) (i = 1, 2, ..., 2m l ) are the underlying independent and identically distributed (i.i.d.) Gaussian processes, and m l is the parameter of the Nakagami-m distribution associated with the l th diversity branch. The Gaussian processes μ i , l (t), each with zero mean and variances , were generated using the sum-of-sinusoids method . The model parameters were calculated using the generalized method of exact Doppler spread (GMEDS1) . The number of sinusoids for the generation of the Gaussian processes μ i , l (t) was chosen to be N = 20. The SNR γ s was set to 15 dB, the parameter Ω l was assumed to be equal to , the maximum Doppler frequency fmax was 91 Hz, and the parameter was equal to unity. Finally, using (21), (3), (7), and (12), the simulation results for the statistical properties of the capacity C(t) of Nakagami-m channels with MRC and EGC were obtained.
This article studies the statistical properties of the capacity of spatially correlated Nakagami-m channels with MRC and EGC. We have derived analytical expressions for the PDF, CDF, LCR, and ADF of the capacity of Nakagami-m channels with MRC and EGC. The results are studied for different values of the number of diversity branches L and receiver antennas separation δ R . It is observed that for MRC, an increase in the spatial correlation increases the variance as well as the LCR of the channel capacity; however, the ADF of the channel capacity decreases. On the other hand, when using EGC, an increase in the spatial correlation decreases the mean channel capacity, whereas the ADF of the channel capacity increases. Moreover, an increase in the spatial correlation increases the LCR of the channel capacity at only lower levels r. It is also observed that for both MRC and EGC, an increase in the number of diversity branches increases the mean channel capacity, while the variance and ADF of the channel capacity decrease. The results also show that at lower levels, the LCR is higher for channels with smaller values of the number of diversity branches L than for channels with larger values of L. The analytical findings are verified using simulations, where a very good agreement between the theoretical and simulation results was observed.
bThe scope of this paper is limited only to the derivation and analysis of the statistical properties of the instantaneous channel capacity. However, a detailed discussion on this topic can be found in, e.g., [29, 49, 50] and the references therein.
- Jakes WC, (ed): Microwave Mobile Communications. IEEE Press, Piscataway, NJ; 1994.View ArticleGoogle Scholar
- Lee WCY: Mobile Communications Engineering. 2nd edition. McGraw-Hill, New York; 1998.Google Scholar
- Salz J, Winters JH: Effect of fading correlation on adaptive arrays in digital mobile radio. IEEE Trans Veh Technol 1994, 43(4):1049-1057. 10.1109/25.330168View ArticleGoogle Scholar
- Jorswieck EA, Oechtering TJ, Boche H: Performance analysis of combining techniques with correlated diversity. Proceedings of IEEE Wireless Communications and Networking Conference, WCNC 2005 2005, 2: 849-854.View ArticleGoogle Scholar
- Lombardo P, Fedele G, Rao MM: MRC performance for binary signals in Nakagami fading with general branch correlation. IEEE Trans Commun 1999, 47(1):44-52. 10.1109/26.747812View ArticleGoogle Scholar
- Chen Y, Tellambura C: Performance analysis of L-branch equal gain combiners in equally correlated Rayleigh fading channels. IEEE Commun Lett 2004, 8(3):150-152. 10.1109/LCOMM.2004.825722View ArticleGoogle Scholar
- Young WR: Comparison of mobile radio transmission at 150, 450, 900, and 3700 MHz. Bell Syst Technol J 1952, 31: 1068-1085.View ArticleGoogle Scholar
- Rice SO: Mathematical analysis of random noise. Bell Syst Tech J 1945, 24: 46-156.MathSciNetView ArticleMATHGoogle Scholar
- Black DM, Reudink DO: Some characteristics of mobile radio propagation at 836 MHz in the Philadelphia area. IEEE Trans Veh Technol 1972, 21: 45-51.View ArticleGoogle Scholar
- Reudink DO: Comparison of radio transmission at X-band frequencies in suburban and urban areas. IEEE Trans Antenna Propag 1972, 20(4):470-473. 10.1109/TAP.1972.1140240View ArticleGoogle Scholar
- Alouini M-S, Goldsmith AJ: Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques. IEEE Trans Veh Technol 1999, 48(4):1165-1181. 10.1109/25.775366View ArticleGoogle Scholar
- Khatalin S, Fonseka JP: On the channel capacity in Rician and Hoyt fading environments with MRC diversity. IEEE Trans Veh Technol 2006, 55(1):137-141. 10.1109/TVT.2005.861205View ArticleGoogle Scholar
- Nakagami M: The m -distribution: a general formula of intensity distribution of rapid fading. In Statistical Methods in Radio Wave Propagation. Edited by: Hoffman WG. Pergamon Press, Oxford, UK; 1960.Google Scholar
- Choi SH, Smith P, Allen B, Malik WQ, Shafi M: Severely fading MIMO channels: models and mutual information. In Proceedings of IEEE International Conference on Communications, ICC 2007 Edited by: Glasgow, UK. 2007, 4628-4633.View ArticleGoogle Scholar
- Yacoub MD, Bautista JEV, de Rezende Guedes LG: On higher order statistics of the Nakagami- m distribution. IEEE Trans Veh Technol 1999, 48(3):790-794.View ArticleGoogle Scholar
- Yacoub MD, da Silva CRCM, Bautista JEV: Second-order statistics for diversity-combining techniques in Nakagami-fading channels. IEEE Trans Veh Technol 2001, 50(6):1464-1470. 10.1109/25.966577View ArticleGoogle Scholar
- Rafiq G, Kontorovich V, Pätzold M: The influence of severity of fading on the statistical properties of the capacity of Nakagami- m channels with MRC and EGC. Proceedings of 2010 European Wireless Conference, EW 2010 2010, 406-410.Google Scholar
- Samimi H, Azmi P: An approximate analytical framework for performance analysis of equal gain combining technique over independent Nakagami, Rician and Weibull fading channels. Wireless Pers Commun 2007, 43(4):1399-1408. 10.1007/s11277-007-9314-zView ArticleGoogle Scholar
- Aalo VA: Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment. IEEE Trans Commun 1995, 43(8):2360-2369. 10.1109/26.403769View ArticleGoogle Scholar
- Zhang QT: Exact analysis of postdetection combining for DPSK and NFSK systems over arbitrarily correlated Nakagami channels. IEEE Trans Commun 1998, 46(11):1459-1467. 10.1109/26.729390View ArticleGoogle Scholar
- Zhang QT: Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix. IEEE Trans Veh Technol 1999, 48(4):1141-1150. 10.1109/25.775363View ArticleGoogle Scholar
- Mun C, Kang C, Park H: Approximation of SNR statistics for MRC diversity systems in arbitrarily correlated Nakagami fading channels. 1999, 35(4):266-267.Google Scholar
- Win MZ, Winters JH: Exact error probability expressions for MRC in correlated Nakagami channels with unequal fading parameters and branch powers. Proceedings of IEEE Global Telecommunications Conference, GLOBECOM 1999 1999, 5: 2331-2335.Google Scholar
- Khatalin S, Fonseka JP: Capacity of correlated Nakagami- m fading channels with diversity combining techniques. IEEE Trans Veh Technol 2006, 55(1):142-150. 10.1109/TVT.2005.861206View ArticleMATHGoogle Scholar
- Rafiq G, Kontorovich V, Pätzold M: On the statistical properties of the capacity of the spatially correlated Nakagami-m MIMO channels. In Proceedings of IEEE 67th Vehicular Technology Conference, IEEE VTC 2008-Spring. Marina Bay, Singapore; 2008:500-506.View ArticleGoogle Scholar
- Proakis J, Salehi M: Digital Communications. 5th edition. McGraw-Hill, New York; 2008.Google Scholar
- Gesbert D, Akhtar J: Breaking the barriers of Shannon's capacity: an overview of MIMO wireless systems. Telenor's Journal Telektronikk: Information theory and its applications 2002, 1.2002: 53-64.Google Scholar
- Paulraj AJ, Nabar RU, Gore DA: Introduction to Space-Time Wireless Communications. Cambridge University Press, Cambridge, UK; 2003.Google Scholar
- Luccini M, Shami A, Primak S: Cross-layer optimization of network performance over multiple-input multiple-output wireless mobile channels. 2010, 4(6):683-696.Google Scholar
- Hogstad BO, Pätzold M: Capacity studies of MIMO models based on the geometrical one-ring scattering model. In Proceedings of 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC 2004. Volume 3. Barcelona, Spain; 2004:1613-1617.Google Scholar
- Hogstad BO, Pätzold M: Exact closed-form expressions for the distribution, level-crossing rate, and average duration of fades of the capacity of MIMO channels. In Proceedings of 65th Semiannual Vehicular Technology Conference, IEEE VTC 2007-Spring. Dublin, Ireland; 2007:455-460.View ArticleGoogle Scholar
- Otani K, Daikoku K, Omori H: Burst error performance encountered in digital land mobile radio channel. IEEE Trans Veh Technol 1981, 30(4):156-160.View ArticleGoogle Scholar
- Vijayan R, Holtzman JM: Foundations for level crossing analysis of handoff algorithms. In Proceedings of IEEE International Conference on Communications, ICC 1993. Geneva, Switzerland; 1993:935-939.Google Scholar
- Giorgetti A, Smith PJ, Shafi M, Chiani M: MIMO capacity, level crossing rates and fades: the impact of spatial/temporal channel correlation. J Commun Netw 2003, 5(2):104-115.View ArticleGoogle Scholar
- Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 6th edition. Academic Press; 2000.MATHGoogle Scholar
- Byers GJ, Takawira F: Spatially and temporally correlated MIMO channels: modelling and capacity analysis. IEEE Trans Veh Technol 2004, 53(3):634-643. 10.1109/TVT.2004.825766View ArticleGoogle Scholar
- Alouini M-S, Abdi A, Kaveh M: Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels. IEEE Trans Veh Technol 2001, 50(6):1471-1480. 10.1109/25.966578View ArticleGoogle Scholar
- Annamalai A, Tellambura C, Bhargava VK: Equal-gain diversity receiver performance in wireless channels. IEEE Trans Commun 2000, 48(10):1732-1745. 10.1109/26.871398View ArticleGoogle Scholar
- da Costa DB, Yacoub MD, Santos Filho JCS: An improved closed-form approximation to the sum of arbitrary Nakagami- m variates. IEEE Trans Veh Technol 2008, 57(6):3854-3858.View ArticleGoogle Scholar
- Papoulis A, Pillai SU: Probability, Random Variables and Stochastic Processes. 4th edition. McGraw-Hill, New York; 2002.Google Scholar
- Foschini GJ, Gans MJ: On limits of wireless communications in a fading environment when using multiple antennas. Wireless Pers Commun 1998, 6: 311-335. 10.1023/A:1008889222784View ArticleGoogle Scholar
- Pätzold M: Mobile Fading Channels. Wiley, Chichester; 2002.View ArticleGoogle Scholar
- Pätzold M, Wang CX, Hogstad BO: Two new sum-of-sinusoids-based methods for the efficient generation of multiple uncorrelated Rayleigh fading waveforms. IEEE Trans Wirel Commun 2009, 8(6):3122-3131.View ArticleGoogle Scholar
- Fraidenraich G, Leveque O, Cioffi JM: On the MIMO channel capacity for the Nakagami- m channel. IEEE Trans Inf Theory 2008, 54(8):3752-3757.MathSciNetView ArticleMATHGoogle Scholar
- Costa N, Haykin S: Multiple-Input, Multiple-Output Channel Models: Theory and Practice. Wiley, New Jersey; 2010.View ArticleGoogle Scholar
- Telatar IE: Capacity of multi-antenna Gaussian channels. Eur Trans Telecommun Relat Technol 1999, 10: 585-595. 10.1002/ett.4460100604View ArticleGoogle Scholar
- Tulino A, Lozano A, Verdu S: Impact of antenna correlation on the capacity of multiantenna channels. IEEE Trans Inf Theory 2005, 51(7):2491-2509. 10.1109/TIT.2005.850094MathSciNetView ArticleMATHGoogle Scholar
- Kang M, Alouini SM: Capacity of MIMO Rician channels. IEEE Trans Wireless Commun 2006, 5(1):112-122.View ArticleGoogle Scholar
- Goldsmith AJ, Chua SG: Variable-rate variable-power MQAM for fading channels. IEEE Trans Commun 1997, 45(10):1218-1230. 10.1109/26.634685View ArticleGoogle Scholar
- Goldsmith AJ, Varaiya P: Capacity of fading channels with channel side information. IEEE Trans Inf Theory 1997, 43(6):1986-1992. 10.1109/18.641562MathSciNetView ArticleMATHGoogle Scholar
- Smith PJ, Garth LM, Loyka S: Exact capacity distributions for MIMO systems with small numbers of antennas. IEEE Commun Lett 2003, 7(10):481-483. 10.1109/LCOMM.2003.817318View ArticleGoogle Scholar
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