Composite Phase PDF in Gamma Shadowed Nakagami Fading Channel

This paper presents a new Fourier series form probability density function (PDF) for the phase of the received signal over gamma shadowed Nakagami-m fading channel with additive Gaussian noise. This PDF is further used to evaluate the average symbol-error probability of M-ary CPSK. An alternative new closed-form expression for bit error-rate of binary CPSK is also derived and numerically compared with the corresponding result based on the Fourier series form phase PDF. We show that the two expressions are indeed numerically equivalent.     

Probability of error analysis of predetection generalized selection combining receivers with correlated unbalanced Nakagami branches

In this paper we determine the probability of error of a predetection generalized selection combining (GSC) receiver with correlated and unbalanced diversity branches in a Nakagami‐m multipath fading channel. We start by finding the joint probability density function (PDF) of the decision variables. This involves the derivation of the joint PDF of the L largest random variables (L maxima) of an input population of N > L correlated nonidentically distributed random variables, based on the statistics of the input population. The results obtained are then used in the derivation of the error probabilities of noncoherent FSK (NCFSK) and DPSK receivers. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

SEP calculations for coherent M‐ary FSK in different fading channels with MRC diversity

In this paper, the authors derive symbol error probability (SEP) expressions for coherent M‐ary frequency shift keying (MFSK) modulation schemes in multipath fading channels. The multipath or small‐scale fading process is assumed to be slow and frequency non‐selective. In addition, the channel is also subjected to the usual degradation caused by the additive white Gaussian noise (AWGN). Different small‐scale fading statistics such as Rayleigh, Rician (Nakagami‐n), Hoyt (Nakagami‐q), and Nakagami‐m have been considered to portray diverse wireless environments. Further, to mitigate fading effects through space diversity, the receiver front‐end is assumed to be equipped with multiple antennas. Independent and identically distributed (IID) as well as uncorrelated signal replicas received through all these antennas are combined with a linear combiner before successive demodulation. As the detection is coherent in nature and thus involves phase estimation, optimum phase‐coherent combining algorithms, such as predetection maximal ratio combining (MRC), may be used without any added complexity to the receiver. In the current text, utilizing the alternate expressions for integer powers (1≤n≤4) of Gaussian Q function, SEP values of coherent MFSK are obtained through moment generating function (MGF) approach for all the fading models (with or without MRC diversity) described above. The derived end expressions are composed of finite range integrals, which can be numerically computed with ease, dispenses with the need of individual expressions for different M, and gives exact values up to M=5. When the constellation size becomes bigger (M≥6), the same SEP expressions provide a quite realistic approximation, much tighter than the bounds found in previous literatures. Error probabilities are graphically displayed for each fading model with different values of constellation size M, diversity order L, and for corresponding fading parameters (K, q, or m). To validate the proposed approximation method extensive Monte‐Carlo simulations were also performed, which show a close match with the analytical results deduced in the paper. Both these theoretical and simulation results offer valuable insight to assess the efficacy of relatively less studied coherent MFSK in the context of the optimum modulation choice in wireless communication. Copyright © 2010 John Wiley & Sons, Ltd.     

Quadrature space shift keying performance with dual‐hop AF relay over mixed fading

Quadrature space shift keying (QSSK) modulation combined with cooperative relaying improves the reliability in communication and enhances the overall spectral efficiency. Here, QSSK scheme is analyzed for multiple‐input multiple‐output (MIMO) wireless communication system with dual‐hop amplify‐and‐forward (AF) relaying systems over asymmetric mixed Rayleigh/Rician and symmetric Nakagami‐m/Nakagami‐m fading channels. Analytical expressions for cumulative distribution function (CDF) of the end‐to‐end signal‐to‐noise ratio are derived and used to evaluate the average bit error probability (ABEP) of QSSK modulation in mixed asymmetric and symmetric fading channels. The obtained ABEP expression is in the form of Whittaker function, which can be numerically evaluated using its numerical or series representation. Numerical and simulation results are presented to illustrate the impact of fading parameters on the system performance.     

Capacity limits of spectrum‐sharing systems over hyper‐fading channels

Cognitive radio (CR) with spectrum‐sharing feature is a promising technique to address the spectrum under‐utilization problem in dynamically changing environments. In this paper, the achievable capacity gain of spectrum‐sharing systems over dynamic fading environments is studied. To perform a general analysis, a theoretical fading model called hyper‐fading model that is suitable to the dynamic nature of CR channel is proposed. Closed‐form expressions of probability density function (PDF) and cumulative density function (CDF) of the signal‐to‐noise ratio (SNR) for secondary users (SUs) in spectrum‐sharing systems are derived. In addition, the capacity gains achievable with spectrum‐sharing systems in high and low power regions are obtained. The effects of different fading figures, average fading powers, interference temperatures, peak powers of secondary transmitters, and numbers of SUs on the achievable capacity are investigated. The analytical and simulation results show that the fading figure of the channel between SUs and primary base‐station (PBS), which describes the diversity of the channel, does not contribute significantly to the system performance gain. Copyright © 2011 John Wiley & Sons, Ltd.     

Enhanced switch combining for spatial diversity with a switch window over correlated Nakagami‐m fading channels

On the basis of a mixture of the selection combining and switch‐and‐stay combining schemes, the enhanced switch combining (ESC) scheme is proposed for antenna diversity over multiple correlated Nakagami‐m fading channels, where a switch window with upper and lower switch thresholds are used. Compared with the existing select‐and‐stay combining or switch with post‐examining, the ESC scheme reduces simultaneous multiantenna observations and hence saves processing time and energy from multibranch observations, while achieving matched receiver performance. Thus, ESC also has better performance than switch‐and‐examine combining (SEC). To assess the reduction of simultaneous observations, a dual‐observation rate is defined. Moreover, the ESC unifies some well‐known switch‐based combining schemes (for example selection combining, switch‐and‐stay combining, or SEC) in the sense that, by adjusting switch thresholds, these combining schemes become different special cases of ESC. The CDF, PDF, and moment generating function of the combined signal‐to‐noise ratio for ESC are derived for general fading channels. Then, the outage probability and the average BER of different binary modulations over correlated Nakagami‐m fading channels are evaluated. Numerical results from analysis and simulation are presented to demonstrate ESC performance. Copyright © 2012 John Wiley & Sons, Ltd.     

Impact of co‐channel interference on performance of dual‐hop wireless ad hoc networks over α−μ fading channels

In this work, we investigate the performance of a dual‐hop cooperative network over α?μ fading channels with the presence of co‐channel interference (CCI) at both the relay and destination nodes. Amplify‐and‐forward (AF) relaying is considered in the relay node. The upper bound of the signal‐to‐interference‐plus‐noise ratio (SINR) of the dual‐hop relay link is used to determine the system performance. The probability density function (PDF) and the cumulative distribution function (CDF) of the upper bound of the SINR are analyzed. The system performance is determined in terms of the outage and error probabilities. Numerical results are used to present the performance analysis of the system.     

Effects of pointing errors and channel fading on the performance of free‐space optically‐preamplified PPM systems

In this paper, we study the impact of pointing errors and channel fading on the performance of free‐space, optically preamplified, M ‐ary PPM systems. We consider two types of free‐space optical links: (i) inter‐satellite links and (ii) inter‐building links. For inter‐satellite links, only pointing error is considered. Starting with a Rayleigh model for the pointing error angle, we derive analytically the PDF for the pointing error parameter and for the signal‐to‐noise ratio (SNR) per bit. For inter‐building links, we derive the density function for the SNR per bit that includes the combined effects of pointing errors and channel fading, assuming Rayleigh‐distributed pointing errors. The channel fading models considered in this study for inter‐buildings links are the log‐normal and gamma–gamma models. We provide the error probability as a function of the average SNR per bit for both types of links. To cover systems with and without forward error correction, we compute the average SNR per bit required to achieve a bit error rate of 10^?4 and 10^?9. The corresponding power penalties are computed for different symbol sizes, scintillation indexes, and pointing jitters. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of Microdiversity and Macrodiversity on Average Bit Error Probability in Gamma‐Shadowed Rician Fading Channels

In this letter, we analyze the error performance of a mobile communication system with microdiversity and macrodiversity reception in gamma‐shadowed Rician fading channels for a binary differential phase‐shift keying modulation scheme. Analytical expressions for the probability density function (PDF) and moment‐generating function (MGF) are derived. The average bit error probability can be calculated by averaging the conditional bit error probability over the PDF or using the MGF‐based approach. Numerical results are graphically presented to show the effects of macrodiversity, correlation, number of diversity branches, and severity of both fading and shadowing.     

Equal-gain diversity receiver performance in wireless channels

Performance analysis of equal-gain combining (EGC) diversity systems is notoriously difficult only more so given that the closed-form probability density function (PDF) of the EGC output is only available for dual-diversity combining in Rayleigh fading. A powerful frequency-domain approach is therefore developed in which the average error-rate integral is transformed into the frequency domain, using Parseval's theorem. Such a transformation eliminates the need for computing (or approximating) the EGC output PDF (which is unknown), but instead requires the knowledge of the corresponding characteristic function (which is readily available). The frequency-domain method also circumvents the need to perform multiple-fold convolution integral operations, usually encountered in the calculation of the PDF of the sum of the received signal amplitudes. We then derive integral expressions for the average symbol-error rate of an arbitrary two-dimensional signaling scheme, with EGC reception in Rayleigh, Rician, Nakagami-m (1960), and Nakagami-q fading channels. For practically important cases of second- and third-order diversity systems in Nakagami fading, both coherent and noncoherent detection methods for binary signaling are analyzed using the Appell hypergeometric function. A number of closed-form solutions are derived in which the results put forward by Zhang (see ibid., vol.45, p.270-73, 1997) are shown to be special cases.     

On the Analysis of Envelope Correlation and Spectra for Rician Fading Channel

Physical fading radio channels encountered in wireless mobile communication are often modeled as a complex Gaussian process whose envelope is statistically described by Rayleigh or Rician probability distribution function (PDF). In most of the literature, the accuracy of the simulation model is estimated by comparing the simulated autocorrelation function (ACF) of inphase (or quadrature phase) component of the fading process and ACF of squared envelope with the analytical ones. In this paper, we examine the performance of a sum of sinusoid (SOS) based Rician fading channel simulator on the basis of the ACF and power spectral density (PSD) of the fading envelope. We obtained simplified approximate expressions for the autocorrelation and mean value of the fading envelope which become more accurate as the value of Rice factor increases. In the simulation, the line-of-sight (LOS) component is modeled as a zero-mean random variable with pre-chosen angle of arrival (AOA) and random initial phase. We showed that the AOA of the LOS component significantly affects the level crossing rate (LCR) and average fade duration (AFD) of the fading envelope. All simulation results are compared with the analytical results and a very good agreement between them is found.

Closed‐Form Expressions for Selection Combining System Statistics over Correlated Generalized‐K Fading Channels in the Presence of Interference

This paper considers the effects of simultaneous correlated multipath fading and shadowing on the performances of a signal‐to‐interference ratio (SIR)‐based dual‐branch selection combining (SC) diversity receiver. This analysis includes the presence of cochannel interference. A generalized fading/shadowing channel model in an interference‐limited correlated fading environment is modeled by generalized‐K distribution. Closed‐form expressions are obtained for probability density function and cumulative distribution function of the SC output SIR, as well as for the outage probability. Based on this, the influence of various fading and shadowing parameter values and the correlation level on the outage probability is examined.     

Performance of M-MRC receivers over TWDP fading channels

An expression of characteristic function of signal-to-noise ratio (SNR) for two waves with diffused power (TWDP) fading channel is derived. Using this expression, the expression for the probability density function (PDF) of the output SNR of maximal ratio combining (MRC) receiver is obtained. Expressions for the performance matrix of MRC receiver over TWDP fading channels are also deduced. PDF based approach is followed to derive expressions of outage probability and average symbol error rate for coherent and non-coherent m-ary modulation schemes. Effects of the number of branches M and the fading parameters K and Δ on the system performance are studied. The results obtained are verified by Monte Carlo simulation.     

Performance analysis of equal-gain diversity receivers over generalized Gamma fading channels

Generalized Gamma (GG) distribution is a generic model that covers many well-known fading distributions as special cases. This paper deals with the performance analysis of L-branch equal gain combining (EGC) receivers operating over GG fading channels. For these receivers and by using convergent infinite series approach, the probability of error (P_e) can be formulated in the form of an infinite series. The coefficients of P_e series can be derived by calculating complicated integrations over the fading envelope distribution. In this paper, it is shown that the required integrations for the case of GG distribution have a complex closed-form in terms of Meijer's G function, and then, a new approximation method is developed for computation of them. The proposed method only needs mean and variance of the fading envelope; hence it has low complexity and eliminates the need for calculation of complex functions. The presented numerical examples show that the developed method can approximate the required parameters and also the individual coefficients accurately and this accuracy increases with the increase of L. The proposed method is applied to analyze the probability of error performance of the L-branch EGC receiver with both coherent phase shift keying (CPSK) and frequency shift keying (CFSK) modulation schemes under different GG channel conditions. Also the effect of gain unbalance between diversity branches on the probability of error is investigated.     

Effective capacity of a correlated Nakagami‐m fading channel

The grail of next‐generation wireless networks is providing real‐time services for delay‐sensitive applications, which require that the wireless networks provide QoS guarantees. The effective capacity (EC) proposed by Wu and Negi provides a powerful tool for design of QoS provisioning mechanisms. In this paper, we intend to generalize their formula for the effective capacity of a correlated Rayleigh fading channel; specifically, we derive a closed form approximate EC formula for a special correlated Nakagami‐m fading channel, for which the inverse of the correlation coefficient matrix is tridiagonal. To verify its accuracy via simulation, we develop a Green‐matrix based approach, which allows us to analytically obtain the effective capacity (given the joint probability density function of a correlated Nakagami‐m fading channel) while being able to simulate the corresponding channel gain process. Simulation results show that our EC formula is accurate. Furthermore, to facilitate the application of the EC theory to the design of practical QoS provisioning mechanisms, we propose a simple algorithm for estimating the EC of an arbitrary correlated Nakagami‐m fading channel, given channel measurements; simulation results demonstrate the accuracy of our proposed EC estimation algorithm showing its suitability in practice. Copyright © 2011 John Wiley & Sons, Ltd.     

Performance analysis of cooperative network over Nakagami and Rician fading channels

This paper presents an analysis on the performance of single‐relay and multiple fixed‐relay cooperative network. The relay nodes operate in amplify‐and‐forward (AF) mode and transmit the signal through orthogonal channels. We consider maximal‐ratio combining at the destination to get the spatial diversity by adding the received signals coherently. The closed‐form moment‐generating function (MGF) for the total equivalent signal‐to‐noise ratio (SNR) is derived. The exact expressions of symbol‐error rate, outage capacity, and outage probability are obtained using the closed‐form MGF for single‐relay and multiple‐relay cooperative network with M‐ary phase shift keying (M‐PSK) and M‐ary quadrature amplitude modulation (M‐QAM) over independent and non‐identical Nakagami‐m channels and Rician fading channels. The approximated closed‐form expression of ergodic capacity is derived for both Nakagami‐m and Rician fading channels. The performance of the system is analyzed at various relay locations. The theoretical results are then compared with the simulation results obtained for binary PSK, quadrature PSK, and 16‐QAM modulation schemes to verify the analysis. Here, the expressions derived can be easily and more efficiently used to compute the performance parameters than doing Monte Carlo simulations. It is shown that cooperation is significant only for low K values for Rician by plotting cooperation gain versus K. The results show that the cooperative network performs best when the relay is located in the middle of source to destination link, at lower SNR values, and the performance of the system is worst if the relay is located closer to the source than to the destination. Copyright © 2013 John Wiley & Sons, Ltd.     

Error probability and channel capacity analysis of wireless system over inverse gamma shadowed fading channel with selection diversity

In this paper, we present an analytical framework to analyse the error probability and the channel capacity of the inverse gamma (I‐Gamma) shadowed fading channel. First, the work discusses the utility of the I‐Gamma over log‐normal (LN) and gamma fading models where the closeness of I‐Gamma with other existing shadowing models is carried out. Utilising the probability density function (PDF) of the I‐Gamma shadowed fading channel, various metrics of the communication system, namely, the average symbol error probability (SEP), the channel capacity under optimal rate adaptation (ORA), channel inversion with fixed rate (CIFR), and truncated CIFR (TIFR) are derived. Further, the work is extended to derive a novel selection combining (SC) PDF, and the analytical results for the SEP and the channel capacity of SC diversity are presented. Furthermore, we also provide simpler asymptotic expressions for the average SEP. In addition, the simplified high and low signal‐to‐noise‐ratio (SNR) solutions to channel capacity are also provided. The derived mathematical formulations have been endorsed by comparing with Monte Carlo simulations.     

An approximate analytical framework for performance analysis of equal gain combining technique over independent Nakagami,Rician and Weibull fading channels

In this paper, an approximate analytical method for performance analysis of equal gain combiner (EGC) receiver over independent Nakagami and Rician fading channels is presented. We use a convergent infinite series approach which makes it possible to describe the probability of error of EGC receiver in the form of an infinite series. In this paper, we develop a new approximation method for computation of the required coefficients in this series which lets us to derive simple analytical closed-form expressions with good accuracy compared with the exact results existing in the literatures. Our proposed approximation method only needs the mean and the variance of the fading envelope, which are known for various fading distributions, and hence, bypasses the required integration over the fading envelope distribution while computing the required coefficients. This feature lets us to extend our approximation method for performance analysis of EGC receiver over independent Weibull fading channels where the required integration has not any closed-form or tabulated solutions. To give an application of our developed method, we analyze the probability of error of an EGC receiver for binary, coherent PSK (CPSK) modulation over independent Nakagami, Rician and weibull fading channels and study the effect of the fading conditions on the system performance.     

Performance of wireless multihop communications systems with cooperative diversity over fading channels

We present closed‐form bounds for the performance of wireless multihop communications systems with cooperative diversity over Nakagami‐m fading channels. The end‐to‐end signal‐to‐noise ratio is formulated and upper bounded by using the inequality between harmonic and geometric means of positive random variables. Closed‐form expression is derived for the moment‐generating function and is used to obtain lower bounds for the average error probability. Numerical results are compared with computer simulations showing the tightness of the proposed bounds. Copyright © 2007 John Wiley & Sons, Ltd.