Error Probability Distribution and Density Functions for Rayleigh and Rician Fading Channels with Diversity

This paper analyzes the distribution and density functions of the probability of error for Rayleigh and Rician fading channels with diversity. An expression for the signal-to-noise ratio is derived for an asynchronous CDMA (A-CDMA) system with diversity. The error probability distribution and density functions are derived and plotted for different mean energy-to-noise ratios.

Spectrum Efficiency of Weibull Fading Channels for Various Adaptation Policies with and Without Diversity Combining

In this paper, closed-form expressions for the capacities per unit bandwidth (spectrum efficiency) of Weibull fading channels are derived and plotted for (a) Switch and Stay Combining diversity case and (b) no diversity case for adaptation policies like: (i) Optimal Power and Rate Adaptation policy, (ii) Optimal Rate Adaptation with constant transmit power policy, (iii) Channel Inversion with Fixed Rate policy, and (iv) Truncated Channel Inversion policy. In addition, spectrum efficiency expressions for asymptotic approximations, upper bounds, approximations for low and high SNR cases are derived for the cases with and without diversity. The probability density function of capacity, and the complementary cumulative distribution function of capacity are derived and plotted from the moment generating function for the cases with and without diversity. Optimal power and rate adaptation policy provides the highest capacity and optimal rate adaptation with constant transmit power policy provides the highest capacity penalty over other policies for the no diversity and SSC diversity cases. Numerical results for spectrum efficiency are plotted for all adaptation policies with and without diversity.     

FSK With Limiter-Discriminator Detector in Nakagami Fading Channels With and Without Selection Combining

In this paper, an equation for the error probability of M-ary frequency shift keying with limiter-discriminator detection in Nakagami fading channels for arbitrary m is derived. The authors do the same for selection combining with L diversity channels for integer m and for switch and stay combining with two diversity channels for m=1 (Rayleigh fading). The error probability for various values of m, L, frequency deviation, and filter bandwidth is computed     

Capacity and Error Probability for Maximal Ratio Combining Reception over Correlated Nakagami Fading Channels

In this paper, the capacity and error probability of maximal ratio combining (MRC) reception are considered for different modulation schemes over correlated Nakagami fading channels. Based on an equivalent scalar additive white Gaussian noise (AWGN) channel, we derive the characteristic function (CF) and the probability density function (PDF) of the signal to noise ratio for MRC reception over Nakagami fading channels. Using these CF and PDF results, closed form error probability and capacity expressions are obtained for PSK, PAM and QAM modulation. Wei Li received his Ph.D. degree in Electrical and Computer Engineering from the University of Victoria in 2004. He is now a Post-doctoral Research Fellow in the Department of Electrical and Computer Engineering at the University of Victoria. He is a Member of the IEEE. His research interests include ultra-wideband system, spread spectrum communications, diversity for wireless communications, and cellular communication systems. Hao Zhang was born in Jiangsu, China, in 1975. He received his Bachelor Degree in Telecom Engineering and Industrial Management from Shanghai Jiaotong University, China in 1994, his MBA from New York Institute of Technology, USA in 2001, and his Ph.D. in Electrical and Computer Engineering from the University of Victoria, Canada in 2004. His research interests include ultra-wideband radio systems, MIMO wireless systems, and spectrum communications. From 1994 to 1997, he was the Assistant President of ICO(China) Global Communication Company. He was the Founder and CEO of Beijing Parco Co., Ltd. from 1998 to 2000. In 2000, he joined Microsoft Canada as a Software Engineer, and was Chief Engineer at Dream Access Information Technology, Canada from 2001 to 2002. He is currently an Adjunct Assistant Professor in the Department of Electrical and Computer Engineering at the University of Victoria. T. Aaron Gulliver received the Ph.D. degree in Electrical and Computer Engineering from the University of Victoria, Victoria, BC, Canada in 1989. From 1989 to 1991 he was employed as a Defence Scientist at Defence Research Establishment Ottawa, Ottawa, ON, Canada. He has held academic positions at Carleton University, Ottawa, and the University of Canterbury, Christchurch, New Zealand. He joined the University of Victoria in 1999 and is a Professor in the Department of Electrical and Computer Engineering. He is a Senior Member of the IEEE and a member of the Association of Professional Engineers of Ontario, Canada. In 2002, he became a Fellow of the Engineering Institute of Canada. His research interests include information theory and communication theory, algebraic coding theory, cryptography, construction of optimal codes, turbo codes, spread spectrum communications, space-time coding and ultra wideband communications.     

General Order Selection Combining for Nakagami and Weibull Fading Channels

In this paper, some analytical results for general order selection (GOS) over independent but not necessarily identically distributed (i.n.d.) Weibull and Nakagami fading channels are presented. The Weibull fading parameters are assumed to be equal whereas the Nakagami fading parameters are assumed to be integer-valued. It is shown that the pdf of the q-th order statistic can be expressed as a linear combination of Weibull and Nakagami pdf's respectively. Closed-form expressions for the moment generating functions and general moments are derived. In addition, exact closed-form expressions for the symbol error rate are obtained for a number of modulation schemes. Numerical results show that for the same average channel gains, the performance on i.n.d. channels may be better or worse than on i.i.d. channels.     

Combining Pilot-Symbol-Aided Techniques for Fading Estimation and Diversity Reception in Multipath Fading Channels

A novel pilot-symbol-aided (PSA) fading estimation technique that combines the estimates from a conventional PSA technique and a bandwidth-efficient PSA technique to achieve better performances is proposed for digital signals in multipath fading channels. The conventional technique has better performances at low signal-to-noise ratios (SNRs), while the bandwidth-efficient technique is superior at high SNRs. Monte Carlo computer simulation has been used to assess the effects of the proposed combining technique on the bit-error-rate (BER) performances of 16-ary quadrature-amplitude-modulation (16QAM), with and without two-branch diversity reception, in a flat Rayleigh fading channel. Results have shown that the combining technique has the advantages of both of the conventional technique and the bandwidth-efficient technique and is more preferred for use with diversity reception. Man-Hung Ng received a BSc degree in Computer Studies from City University of Hong Kong in 1991. He worked as a computer programmer in Hong Kong from 1991 to 1995. In 1996, he obtained a MSc degree in Communication and Radio Engineering from King's College London. He joined the University of Hong Kong as a research assistant in 1997, and completed a PhD degree in mobile communications in 2001. He joined Lucent Technologies N.S. UK in 2001 and is now a principal standards engineer. Sing-Wai Cheung received the BSc degree in Electrical and Electronic Engineering from Middlesex University, U.K. in 1982 and the PhD degree from Loughborough University of Technology, U.K. in 1986. From 1986–1988, he was a post-doctorate research assistant in the Communications Research Group of King's College, London University. During 1988–1990, he was with the Radio and Satellite Communications Division in British Telecom Research Laboratories (now British Telecom Laboratories). He joined the Department of Electrical and Electronic Engineering at the University of Hong Kong in 1990 and is now an Associate Professor. He contributes regularly courses on mobile and satellite communications systems. His current research interests include modulation, coding, fading compensation and diversity and MIMO for mobile and satellite communications systems.     

Average LCR and AFD for SC Diversity Over Correlated Weibull Fading Channels

The analytic methods of average LCR (Level Crossing Rate) and AFD (Average Fading Duration) are applied in the analysis for the SC (Selection Combining) diversity, which works over the correlated Weibull statistics channel. The dual branch SC diversity was used to prove the accuracy of the derived theoretical formula in this paper. We adopt different probability density functions (pdf) and cumulate distribution functions (CDF) to show the Moreover, the numerical results obtained from computer are verified to the published researches.     

A Closed-Form Approximation for the Error Probability of BPSK Fading Channels

This letter presents a simple closed-form expression to evaluate the error probability of binary fully-interleaved fading channels. The proposed expression does not require a numerical Laplace transform inversion, numerical integration or similar techniques, and captures the role of the relevant system parameters in the overall error performance. The expression has the same asymptotic behavior as the Bhattacharyya (Chernoff)-union bound but closes the gap with the simulation results. Its precision is numerically validated for coded and uncoded transmission over generic Nakagami-m fading channels.     

A Unified Approach to Computing Error Probabilities of Diversity Combining Schemes over Correlated Fading Channels

The average bit-error rate performance of one-stage and two-stage diversity combining schemes operating over correlated fading channels is investigated. Two channel models that can significantly simplify the performance analysis are considered. In particular, a linear correlation channel model having equal branch variances can be decorrelated at the receiver, so that the branches become independent. It is shown that, in general, employing diversity combining schemes for decorrelated or orthogonalized branches can recover some of the diversity gain lost due to the branch correlations. This is observed, for example, for the case of hybrid selection/maximum ratio combining operating over decorrelated and orthogonalized non-zero mean Gaussian fading channels. Furthermore, a fading amplitude channel model is proposed assuming vector norm superposition of the impinging plane waves. This channel model is well-suited for the performance analysis of maximum ratio and equal gain combining schemes operating over correlated fading channels. Finally, the average bit error rates of several diversity combining schemes are evaluated analytically using the Prony approximation method as well as using computer simulation.     

Nakagami衰落环境下合并接收的平均错误概率

本文对最大比值合并,等增益合并及选择性合并等合并技术进行了归纳与总结,给出了各种调制方法在Nakagami信道下的错误概率的精确表达式或近似表达式。同时还给出了相应的计算方法。对这类问题研究中的主要理论工具进行了总结与分析。本文最后还讨论了这方面值得进一步研究的一些问题。     

Selective Diversity for Rayleigh Fading Channels With a Feedback Link

A feedback scheme for regulating error probability on slowly fading communication channels is described and analyzed. In this selective diversity system, only the bestRofLavailable diversity subchannels are used, the selection being revised from moment to moment. Comparison with conventional diversity transmission on an 8-channel system shows a performance improvement ranging from 20.8 to 9.5 dB, where the low figure corresponds to constant power operation of the transmitter and a typical value of round-trip delay. A novel feature of selective diversity is that, for moderate SNR, the error probability decreases to zero as a strong inverse power ofL, the order of diversity, which contrasts with behavior of conventional systems in which increasingLcauses the error probability to decrease to a nonzero lower bound. The implication is that the figures quoted above can be improved indefinitely by increasing the order of diversity.     

Performance Analysis of Diversity Combining Algorithms in Shadowed Fading Channels

A compound fading model incorporating short term fading and shadowing proposed recently is used to analyze the performance of wireless systems employing microscopic diversity to mitigate the effects of flat fading. This model can account for the presence of different levels of fading and shadowing and provide an analytical solution for the probability density function of the signal-to-noise ratio. Using that model, the performances of MRC and SC diversity combining algorithms were studied. The amount fading (AF) following diversity implementation was calculated and it is seen that the decline in the amount of fading is bound by the level of shadowing present, with the MRC providing a larger decrease in the amount of fading than the SC algorithm. The effect on the error rates was studied using the example of the coherent BPSK modem. Results show that the performances of wireless systems can be analyzed using the compound model for the shadowed fading channels. P.M. Shankar received his M. Sc (1972) in Physics from Kerala University, India, M. Tech (1975) in Applied Optics and Ph. D. in Electrical Engineering (1980) from Indian Institute of Technology, Delhi, India. He was a visiting scholar at the School of Electrical Engineering, University of Sydney, Australia, from 1981 to 1982. He joined Drexel University in 1982 and is currently the Allen Rothwarf Professor of Electrical and Computer Engineering. He is the author of the textbook ‘Introduction to Wireless Systems’, published by John Wiley & Sons, 2002. His research interests are in Fading Channels, Wireless communications, and Statistical signal processing for medical applications.     

Error Probability and SINR Analysis of Optimum Combining in Rician Fading

This paper considers the analysis of optimum combining systems in the presence of both co-channel interference and thermal noise. We address the cases where either the desired-user or the interferers undergo Rician fading. Exact expressions are derived for the moment generating function of the SINR which apply for arbitrary numbers of antennas and interferers. Based on these, we obtain expressions for the symbol error probability with M-PSK. For the case where the desireduser undergoes Rician fading, we also derive exact closed-form expressions for the moments of the SINR. We show that these moments are directly related to the corresponding moments of a Rayleigh system via a simple scaling parameter, which is investigated in detail. Numerical results are presented to validate the analysis and to examine the impact of Rician fading on performance.     

The Distribution of Error Probability for Rayleigh Fading and Gaussian Noise

The distribution function of the probability of error in the presence of Rayleigh fading and Gaussian noise is determined for the basic binary modulation schemes of coherent frequencyshift keying (CFSK), noncoherent frequency-shift keying (NCFSK), differential phase-shift keying (DPSK), and coherent phase-shift keying (CPSK). General expressions for the distribution function of error probability are also derived when linear maximal-ratio diversity combining is employed. Results are given for various values of average error probability and various orders of diversity.     

Performance Analysis of Linear Modulation Schemes With Generalized Diversity Combining on Rayleigh Fading Channels With Noisy Channel Estimates

Generalized diversity combining (GDC), also known as hybrid selection/maximal ratio combining or generalized selection combining, is a low-complexity diversity combining technique by which a fixed subset of a large number of available diversity channels is chosen and then combined using the rules of maximal ratio combining. In this paper, we analyze the performance of GDC on time-correlated Rayleigh fading channels with noisy channel estimates. We derive expressions for the probability of error for various linear modulation schemes with coherent detection, and discuss the conditions under which the analysis can be extended to noncoherent and differentially coherent receiver structures. Throughout the paper, using a fundamental approach to obtain the decision statistic at the combiner output, a number of new expressions for the error probabilities are obtained in a rigorous way, along with a presentation of their performance with channel estimation errors. The final expressions have roughly the same complexity of evaluation as that for the channel with only additive Gaussian noise. Our results correct various inaccuracies in the literature, and show that coherent receivers based on imperfectly estimated channel knowledge incur a significant performance loss.      

Nakagami衰落信道下多跳合作分集系统的中断率

该文在Nakagami衰落信道下研究了多跳无线网络中合作分集系统的中断性能。根据单节点和多节点中继的情况导出了合作系统再生中继时中断率的数学闭式解,并在高平均信噪比下给出了系统的分集度。理论分析和仿真研究的结果表明多跳合作系统在一定条件下可实现全路由分集,其分集度为所有路由的分集度之和,不是所有合作节点的和,但每支路由的分集度取决于该路由两跳中较小的值。     

Outage Probability Analysis of Cooperative Diversity Networks Over Weibull and Weibull-Lognormal Channels

This paper presents the analysis of outage probability for a cooperative diversity wireless network using amplify-and-forward relays over independent non-identical distributed Weibull and Weibull-lognormal fading channels for single as well as multiple relays. To reach that end, a closed-from expression for the moment-generating function of the total signal-to-noise-ratio (SNR) at the destination is derived in terms of the tabulated Meijer’s G-function. Since it is hard to determine the exact probability distribution function of the SNR, a tight lower bound approximation is proposed. Simulation results are presented that show that the outage probability lower bound tends to be tight at high SNR values thus verifying the analytical results. The results also show the potential gain of relaying on the outage probability.