Error analysis of generalized selection combining over relay channel

Cooperative communication is a recently popular concept which allows single-antenna devices to benefit from spatial diversity. The performance analysis of cooperative communication using generalized selection combining (GSC) over independent not necessarily identically distributed Nakagami-m fading channels is presented and compared with that of the conventional maximal ratio combining (MRC) and selection combining (SC) schemes. With the aid of Padé approximants theory, new closed-form expression is derived for the moment-generating function (MGF) of the GSC output signal-to-noise ratio (SNR). MGF is an important tool for researching the system performance. In this paper, the average bit-error probability is accurately approximated using the well-known MGF approach. Numerical results show that the proposed mathematical analysis is accurate and that for the more severe fading cases, the GSC receivers are closer to the optimum MRC receivers.     

Performance evaluation of generalized selection diversity systems over Nakagami‐m fading channels

The generalized selection combining (GSC) scheme that adaptively combines a subset of M strongest paths out of L available diversity paths finds applications in several wideband receivers and broadband wireless communications. In this paper, exact closed‐form expressions for the moment generating function (MGF), the probability density function (PDF) and the cumulative density function (CDF) of the GSC(M, L) output signal‐to‐noise ratio (SNR) in independent and identically distributed (i.i.d) Nakagami‐m fading channels are derived while the fading index is a positive integer. These expressions hold for any M and L and provide a comprehensive framework for performance analysis including the derivation of closed‐form formulas for the average symbol error probability (ASEP) of a broad class of binary and M‐ary modulations, mean combined SNR and the outage probability of GSC(M, L) receiver structures. When the Nakagami‐m fading index is not an integer, the MGF of GSC(M, L) output SNR is derived as an (M − 1)‐fold infinite series. With this MGF, analytical expressions for both the outage probability and error rates can be readily obtained. An easily programmable recursive solution of the MGF of GSC(M, L) output SNR is also outlined for both the positive integer and noninteger fading severity index cases. Copyright © 2002 John Wiley & Sons, Ltd.     

Performance of M-PSK with GSC and EGC with Gaussian weighting errors

Using a moment-generating function (MGF)-based approach, we study the performance of M-ary phase-shift keying (M-PSK) with generalized selection combining (GSC) and equal gain combining (EGC) in fading channels (including Rayleigh, Rician, Nakagami-m, and Nakagami-q fading) with independent and identically distributed (i.i.d) branches. Analytical expressions for the error and outage probabilities, the signal-to-noise-ratio (SNR) statistics, and the channel capacity of M-PSK diversity receivers are derived, taking into account the effects of Gaussian weighting errors and all relevant system and channel parameters. Unlike the case of perfect channel-state information (CSI), the outage probability for the case of imperfect channel estimation (ICE) is not only a function of the normalized SNR with respect to the SNR threshold, but also a function of the operating SNR itself. The SNR loss of the M-PSK GSC and EGC receivers due to ICE and the relation between the receiver input and output SNRs for ICE are derived. Our results show that, even with ICE, GSC and EGC are effective in improving the output SNR and significantly reduce the error floor and the channel-capacity loss caused by ICE.     

Efficient performance evaluation for generalized selection combining on generalized fading channels

The authors propose an efficient moment generating function (MGF)-based method to evaluate the performance of generalized selection combining (GSC) over different fading channels. Employing a recently proposed method which is, however, only applicable to GSC diversity with independent and identically distributed branches, they derive a general MGF expression for the GSC output signal-to-noise ratio (SNR) for generalized fading channels, where the channel statistics in different diversity branches may be nonidentical or even distributed according to different distribution families. The resulting MGF expression is applicable to the analysis of the error probability, the outage probability, and the SNR statistics for GSC in a number of wireless communications scenarios with generalized fading. Numerical examples are presented to illustrate the application of the new analysis.     

Unified error probability analysis for generalized selectioncombining in Nakagami fading channels

We study generalized selection combining (GSC) schemes in independent Nakagami fading channels, where N diversity branches with the largest instantaneous signal-to-noise ratios (SNRs) are selected from the total of L (N⩽L) branches and then coherently or noncoherently combined. We propose two different techniques to derive the moment generating function (MGF) expressions for the GSC output SNR in generalized Nakagami fading channels, where there are distinct and noninteger fading severity parameters, as well as different average SNRs in different diversity branches. For arbitrary fading severity parameter m_k, k=1, ···L, the MGF expression is given in a summation of N-dimensional definite integrals with the limits independent of SNR or channel parameters, and therefore can be evaluated very efficiently with numerical methods. Furthermore, for integer m_k closed-form MGF expressions are derived. Specializations of our results to Rayleigh channels and independent identically distributed (i.i.d.) Nakagami channels are presented, which are either new or equivalent to previously published results. Using the newly derived MGF expression, we provide a unified error probability analysis for many coherent and noncoherent modulation/detection schemes     

Approximate BER performance of generalized selection combining inNakagami-m fading

The error performance of generalized selection combining (GSC), which optimally combines the K highest signal-to-noise ratio (SNR) signals out of L total diversity signals, in Nakagami-m fading was recently evaluated using moment generating function (MGF) of the GSC output SNR. However, no single closed-form expression for the MGF exists for arbitrary K and L. In fact, a closed-form expression for the MGF is possible only for an individual combination of K and L. In this letter, a single closed-form expression for approximating the MGF is, therefore, derived and employed in evaluating the approximate error performance. Although the approximation is only applicable for GSC with K being a factor of L, it nonetheless achieves a high degree of accuracy     

Performance analysis of TCM with generalized selection combining on Rayleigh fading channels

We derive the computational cutoff rate, R/sub 0/, for coherent trellis-coded modulation (TCM) schemes on independent identically distributed (i.i.d.) Rayleigh fading channels with (K, L) generalized selection combining (GSC) diversity, which combines the K paths with the largest instantaneous signal-to-noise ratios (SNR) among the L available diversity paths. The cutoff rate is shown to be a simple function of the moment generating function (MGF) of the SNR at the output of the (K, L) GSC receiver. We also derive the union bound on the bit error probability of TCM schemes with (K, L) GSC in the form of a simple, finite integral. The effectiveness of this bound is verified through simulations.     

An MGF-based performance analysis of generalized selectioncombining over Rayleigh fading channels

Using the notion of the “spacing” between ordered exponential random variables, a performance analysis of the generalized selection combining (GSC) diversity scheme over Rayleigh fading channels is presented and compared with that of the conventional maximal-ratio combining and selection combining schemes. Starting with the moment generating function (MGF) of the GSC output signal-to-noise ratio (SNR), we derive closed-form expressions for the average combined SNR, outage probability, and average error probability of a wide variety of modulation schemes operating over independently, identically distributed (i.i.d.) diversity paths. Because of their simple form, these expressions readily allow numerical evaluation for cases of practical interest. The results are also extended to the case of non-i.i.d. diversity paths     

Analysis of Generalized Selection Diversity Systems in Wireless Channels

Motivated by practical considerations in the design of low-complexity receiver structures for wideband cellular code division multiple access, millimeter wave, and ultrawideband communications, the study on the generalized selection combining (GSC) receiver that adaptively combines a subset of M "strongest" paths out of L available paths has intensified over the past few years. This paper derives concise analytical expressions for the moment generating function (MGF) of the GSC(M,L) output signal-to-noise ratio when the fading statistics are independent and identically distributed. The novelty of this mathematical framework in computing the MGF relies on the fact that it allows all common multipath fading channel models (Rayleigh, Rician, Nakagami-m, and Nakagami-q) to be treated in a unified sense. It also leads to a much more computationally efficient formula than those available in the literature and is valid for any combinations of M and L values. Using these newly derived MGFs, a unified error probability analysis for many coherent and noncoherent digital-modulation/detection schemes in a myriad of fading environments was provided     

Exact Error Rate Analysis of Output-Threshold Generalized Selection Combining (OT-GSC)

Generalized selection combining (GSC) is one of the most widely investigated low-complexity diversity techniques. Recently, output-threshold GSC (OT-GSC) was proposed as a power-saving variant of traditional GSC scheme. In this paper, we present an exact performance analysis of OT-GSC over Rayleigh fading channels. In particular, we derive the cumulative distribution function (CDF), probability density function (PDF) and moment generation function (MGF) of the combined SNR with OT-GSC. Then, we apply these results to study the outage probability and average error rate performance of OT-GSC. This analysis allows for a thorough and accurate comparison of OT-GSC with other variants of GSC, such as minimum selection GSC.     

Performance evaluation of triple-branch GSC diversity receivers over generalized-K fading channels

In this letter, a detailed performance analysis of generalized selection combining GSC(2,3) receivers operating over independent but not necessarily identically distributed (n.i.d.) generalized-K (KG) fading channels is presented. For this class of receivers, a novel closed-form expression for the moments of the output signal-to-noise ratio (SNR) is derived. This result can be afterwards used to evaluate the outage probability and the average symbol error probability of different signal constellations. Various performance evaluation results are also presented and compared to equivalent simulation ones.     

Diversity reception over generalized-K (KG) fading channels

A detailed performance analysis for the most important diversity receivers operating over a composite fading channel modeled by the generalized-K (Kg) distribution is presented. The Kg distribution has been recently considered as a generic and versatile distribution for the accurate modeling of a great variety of short term fading in conjunction with long term fading (shadowing) channel conditions. For this relatively new composite fading model, expressions for important statistical metrics of maximal ratio combining (MRC), equal gain combining (EGC), selection combining (SC) and switch and stay combining (SSC) diversity receivers are derived. Using these expressions and by considering independent but not necessarily identical distributed fading channel conditions, performance criteria, such as average output signal-to-noise ratio, amount of fading and outage probability are obtained in closed form. Moreover, following the moments generating function (MGF) based approach for MRC and SSC receivers, and the Pade approximants method for SC and EGC receivers, the average bit error probability is studied. The proposed mathematical analysis is complemented by various performance evaluation results which demonstrate the accuracy of the theoretical approach.     

A Trivariate Nakagami-m Distribution with Arbitrary Covariance Matrix and Applications to Generalized-Selection Diversity Receivers

This paper deals with a trivariate Nakagami-m distribution derived from the diagonal elements of a Wishart matrix. For this distribution, infinite series representations for its probability density and cumulative distribution functions are derived having an arbitrary covariance matrix and integer-order fading parameters. Moreover, upper bounds on the error resulting from truncating the infinite series are obtained. Based on the derived formulas, the performance of triple-branch generalized selection combining (GSC) receivers is analyzed. For this type of receivers, the outage and the average bit error probability for a variety of modulation schemes are analytically obtained. The performance of GSC receivers is compared to that of conventional selection and maximal-ratio diversity schemes. In order to check the accuracy and convergence of the derived formulas, various performance evaluation results are presented and compared to equivalent simulation ones.     

On the performance analysis of wireless receiver using generalized-gamma fading model

In this paper, we provide a unified analysis for wireless system over generalized fading channels that is modeled by the two parameter generalized gamma model. This model is versatile enough to represent short-term fading such as Weibull, Nakagami-m, or Rayleigh as well as shadowing. The performance measures such as the amount of fading, average bit error rate, and signal outage are considered for analysis. With the aid of moment generating function (MGF) approach and Padé approximation (PA) technique, outage probability and average bit error rate have been evaluated for a variety of modulation formats. We first use the PA technique to find a simple way to evaluate compact rational expressions for the MGF of output signal-to-noise ratio, unlike previously derived intricate expressions in terms of Fox’s H and MeijerG functions. Using these rational expressions, we evaluate the performance of wireless receivers under a range of representative channel fading conditions. Our results are validated through computer simulations, which shows perfect match.     

Performance analysis of a class of GSC receivers over nonidentical Weibull fading channels

The performance of a class of generalized-selection combining (GSC) receivers operating over independent but nonidentically distributed Weibull fading channels is studied. We consider the case where the two branches with the largest instantaneous signal-to-noise ratio (SNR), from a total of L available, GSC(2, L) are selected. By introducing a novel property for the product of moments of ordered Weibull random variables, convenient closed form expressions for the moments of the GSC(2,L) output SNR are derived. Using these expressions, important performance criteria, such as average output SNR and amount of fading, are obtained in closed form. Furthermore, employing the Pade/spl acute/ approximants theory and the moment-generating function approach, outage and bit-error rate performance are studied. An attempt is also made to identify the equivalency between the Weibull and the Rice fading channel, which is typically used to model the mobile satellite channel. We present various numerical performance evaluation results for different modulation formats and channel conditions. These results are complemented by equivalent computer simulated results which validate the accuracy of the proposed analysis.     

A Comprehensive Framework for Performance Analysis of Dual?Hop Cooperative Wireless Systems with Fixed?Gain Relays over Generalized Fading Channels

In the present contribution, we propose a comprehensive framework for the analysis of cooperative dual-hop wireless systems over generalized fading channels, which use an amplify and forward (AF) relaying mechanism with blind and semi-blind relays. In particular, the proposed framework provides either exact results or very accurate bounds for computing the moment generating function (MGF) of the end-to-end signal-to-noise ratio (SNR) for various fading channel models typically encountered in real propagation environments. Furthermore, with the help of the MGF-based approach for performance analysis of wireless systems over fading channels, we will show that important performance indexes can be easily derived from the MGF. With respect to previous published articles on the matter, the main contribution of the paper is twofold: i) by relying on the properties of the Meijer-G function, either exact expressions or accurate bounds for the MGF of the end-to-end SNR are provided, and ii) the analysis encompasses the vast majority of fading channel models. Numerical and simulation results will be compared to substantiate the analytical derivation.     

A general moment generating function for MRC diversity over fading channels with Gaussian channel gains

For maximal ratio combining (MRC) diversity over correlated fading channels with Gaussian channel gains, we utilize unitary diagonalization of the channel covariance matrix to decorrelate the physical channels into uncorrelated virtual channels to obtain the moment generating function (MGF) of the received signal‐to‐noise ratio (SNR). The MGF thus obtained has a compact form and can be universally applied to various popular fading models. In addition to the advantage of simple derivation procedure, this general MGF can be readily modified to express various scenarios of channel power distributions as well as joint fading models. To demonstrate these advantages, we use the generalized Ricean fading as a specific example to compare our derivation and our MGF expression with an existing work in the literature. Again, we present numerical simulations for MRC reception of binary phase shift keying (BPSK) signals over Nakagami fading to compare with existing results appearing in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive blind receivers for MC‐CDMA communication systems

In this paper, we propose three adaptive blind algorithms for multiuser multicarrier code division multiple‐access systems in multipath fading channels. The proposed adaptive blind receivers are based on the property of the discreteness of the input data symbol and are updated in every symbol interval. We also use the concept that the variance of the output signal approaches to the variance of the desired signal to get the cost function. The three proposed receiver structures are the traditional finite impulse response (FIR) structure, the despreading (DES) filter structure and the generalized sidelobe canceller (GSC) structure. The advantage of the FIR filter is that the length of the filter weights does not have to be the same length as the spreading code. For the DES filter, the combination of the adaptive weight and the despreading code has the simplest structure than the other two proposed receiver structures. The constrained GSC filter is superior to the other two proposed receiver structures in the environments dominated by multiple‐access interference. By this constraint, the blind GSC filter can guarantee to converge to the desired solution. Simulation results are given to show the effectiveness and comparison of the proposed adaptive blind receivers. Copyright © 2012 John Wiley & Sons, Ltd.     

SER Performance of Amplify-and-Forward Cooperative Diversity Over Asymmetric Fading Channels

In this paper, the performance of amplify-and-forward (AF) cooperative diversity is analyzed over asymmetric fading channels. The source–relay and the relay–destination links experience Rayleigh fading while the source–destination link is subject to generalized Gamma fading. First, the probability density function (PDF) and the moment generating function (MGF) of the source–relay–destination link and the MGF of the source–destination link are derived. Then, the symbol error rate (SER) is determined based on the MGF of the total end-to-end signal-to-noise ratio (SNR). Moreover, the SER performance of N-relay assisted AF cooperative diversity is illustrated for M-ary phase shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM). Based on the derived MGF expressions, the numerical results are obtained by varying the modulation types and channel parameters for different scenarios.     

A general method for calculating error probabilities over fading channels

Signal fading is a ubiquitous problem in mobile and wireless communications. In digital systems, fading results in bit errors, and evaluating the average error rate under fairly general fading models and multichannel reception is often required. Predominantly to date, most researchers perform the averaging using the probability density function method or the moment generating function (MGF) method. This paper presents a third method, called the characteristic function (CHF) method, for calculating the average error rates and outage performance of a broad class of coherent, differentially coherent, and noncoherent communication systems, with or without diversity reception, in a myriad of fading environments. Unlike the MGF technique, the proposed CHF method (based on Parseval's theorem) enables us to unify the average error-rate analysis of different modulation formats and all commonly used predetection diversity techniques (i.e., maximal-ratio combining, equal-gain combining, selection diversity, and switched diversity) within a single common framework. The CHF method also lends itself to the averaging of the conditional error probability involving the complementary incomplete Gamma function and the confluent hypergeometric function over fading amplitudes, which heretofore resisted to a simple form. As an aside, we show some previous results as special cases of our unified framework.