One-sided Gaussian Research Articles

Analysis of STC-GFDM with Walsh Hadamard in generalized η−μ fading channel

ABSTRACT This manuscript presents a comprehensive analytical framework for Walsh-Hadamard Transform (WHT) precoding in Space-Time Coded Generalized Frequency Division Multiplexing (STC-GFDM) systems. The investigation specifically targets the η - μ fading channel and employs the moment generating function (MGF) approach to establish an exact closed-form expression for the Symbol Error Rate (SER). The derived expression enables the examination of one-sided Gaussian, Rayleigh, Nakagami-m and Hoyt-q fading channels as specific cases of the η - μ fading channel. Additionally, the paper also delves into the impact of crucial system parameters, including modulation order, channel fading parameters ( η and μ ) and the number of transmit and receive antennas, on the WHT-STC-GFDM system’s performance. Furthermore, the paper conducts an in-depth complexity analysis of the WHT precoder, shedding light on the computational demands associated with its implementation. In summary, this research enhances the understanding of WHT precoding in STC-GFDM systems operating in η - μ fading channels, providing valuable insights for the design and optimisation of future communication systems. The numerical results are validated through Monte-Carlo simulations.

Outage analysis for user selection based downlink cooperative NOMA network over generalized fading channels

Recently, non-orthogonal multiple access (NOMA) scheme is integrated with cooperative communication (C-NOMA) to increase the connectivity and coverage area. In this paper, we analyzed the outage performance of downlink C-NOMA system with the imperfect successive interference cancellation (I-SIC) and imperfect channel state information (I-CSI) aiding multiple users over κ−μ and η−μ independent and identically distributed fading environments. Further, a user selection scheme is adopted for downlink C-NOMA scheme for enhancing the performance of the system. To be precise, the system model considers multiple users at the destination, out of which two users are selected to serve with NOMA standards. The analytical expressions of outage probability (OP) are derived, analyzed, and compared with cooperative-orthogonal multiple access (C-OMA) scheme over generalized fading environments. Moreover, the accuracy of analytical results is validated through Monte Carlo simulation. In addition, the analytical expression of OP for the paired user is also derived, analyzed, and correlated with the C-OMA scheme. Furthermore, through κ−μ and η−μ fading distribution, the outage performance of Rayleigh, Nakagami-m, Rice, Hoyt, and one-sided Gaussian fading environment is also investigated. From the obtained results, it is evident that when compared to C-OMA scheme, the C-NOMA scheme achieves 4.4 and 6.6 percentage gain in terms of SNR for a near user and a far user, respectively at κ=1.0,μ=1.0 to attain outage of 10−1. On the other hand, for the same outage, C-NOMA achieves 4.6 and 6.7 percentage gain in terms of SNR for a near user and a far user, respectively at η=1.0,μ=1.5.

Dual-Hop Cooperative Relaying with Beamforming Under Adaptive Transmission in κ–μ Shadowed Fading Environments

In this paper, we analyze the performance of a dual-hop cooperative decode-and-forward (DF) relaying system with beamforming under different adaptive transmission techniques over κ − μ shadowed fading channels. We consider multiple antennas at the source and destination, and communication takes place via a single antenna relay. The published work in the literature emphasized the performance analysis of dual-hop DF relaying systems, in conjunction with different adaptive transmission techniques for classical fading channels. However, in a real scenario, shadowing of the line-of-sight (LoS) signal is caused by complete or partially blockage of the LoS by environmental factors such as trees, buildings, mountains, etc., therefore, transmission links may suffer from fading as well as shadowing, either concurrently or separately. Hence, the κ − μ shadowed fading model was introduced to emulate such general channel conditions. The κ − μ shadowed fading model is a general fading model that can perfectly model the fading and shadowing effects of the wireless channel in a LoS propagation environment, and it includes some classical fading models as special cases, such as κ − μ , Rician, Rician-shadowed, Nakagami- m ^ , One-sided Gaussian, and Rayleigh fading. In this work, we derive the outage probability and average capacity expressions in an analytical form for different adaptive transmission techniques: (1) optimal power and rate adaptation (OPRA); (2) optimal rate adaptation and constant transmit power (ORA); (3) channel inversion with a fixed rate (CIFR); and (4) truncated channel inversion with a fixed rate (TIFR). We evaluate the system performance for different arrangements of antennas and for different fading and shadowing parameters. The obtained analytical expressions are verified through extensive Monte Carlo simulations.

On the average of the product of two Gaussian Q functions over η − μ and κ − μ fading channels using MRC diversity reception

In this study, the authors propose a simple and tighter approximation to an integral representing the average of the product of two Gaussian Q functions over parametric η - μ and κ - μ fading channels. This facilitates the symbol error probability computation of most of the coherent digital modulation techniques over η - μ and κ - μ fading distributions. In order to improve the error performance of these digital modulation schemes, maximal-ratio combining (MRC) diversity reception is also employed. Moreover, since η - μ and κ - μ are generic fading scenarios, the widely used popular fading models like Rayleigh, one-sided Gaussian, Nakagami-m, Nakagami-q and Rician can be unified under one umbrella. This study also gives an insight of the asymptotic approximations of the proposed integrals. In addition, the analytical framework is validated with the help of Monte Carlo simulations, which enables an easy practical implementation of the proposed work.

Non-Linear EH Relaying in Delay-Transmission Mode over n-µ Fading Channels

Energy harvesting is a technique to harvest energy from RF (radio frequency) waves. The RF signals have the ability to convey energy and information concurrently. The EH in coop-erative relaying systems may increase the capacity and coverage of wireless networks. In this work, we study a dual-hop (two-hop) relaying. This system has three nodes: a source, a relay, and a destination. The source and destination have multiple antennas. We account a non-linear EH model and TSR (time-switching-based relaying) protocol at the single-antenna relay node. We evaluate the system performance over η−µ fading channels. With a saturation threshold, a non-linear EH receiver restrains the harvested power. In the TSR protocol, the relay changes mode between the EH and information processing, by which a fraction of time is used with each process. The fading model η−µ incorporates some fading models as notable cases, viz., Nakagami- m, One-sided Gaussian, Nakagami-q (Hoyt), and Rayleigh. The system performance is analyzed in terms of the average capacity and throughput for different saturation threshold power levels, divers antennas arrangements, and different parameter values of η and µ.

An Extension of the <inline-formula> <tex-math notation="LaTeX">$\kappa$</tex-math> </inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu$</tex-math> </inline-formula> Shadowed Fading Model: Statistical Characterization and Applications

We here introduce an extension and natural generalization of both the \kappa-\mu$\,$shadowed and the classical Beckmann fading models: the Fluctuating Beckmann (FB) fading model. This new model considers the clustering of multipath waves on which the line-of-sight (LoS) components randomly fluctuate, together with the effect of in-phase/quadrature power imbalance in the LoS and non-LoS components. Thus, it unifies a variety of important fading distributions as the one-sided Gaussian, Rayleigh, Nakagami-m, Rician, \kappa-\mu, \eta-\mu, \eta-\kappa, Beckmann, Rician shadowed and the \kappa-\mu$\,$shadowed distribution. The chief probability functions of the FB fading model, namely probability density function, cumulative distribution function and moment generating function are derived. The second-order statistics such as the level crossing rate and the average fade duration are also analyzed. These results can be used to derive some performance metrics of interest of wireless communication systems operating over FB fading channels.

A Strong Entropy Power Inequality

When one of the random summands is Gaussian, we sharpen the entropy power inequality (EPI) in terms of the strong data processing function for Gaussian channels. Among other consequences, this ‘strong’ EPI generalizes the vector extension of Costa’s EPI to non-Gaussian channels in a precise sense. This leads to a new reverse EPI and, as a corollary, sharpens Stam’s uncertainty principle relating entropy power and Fisher information (or, equivalently, Gross’ logarithmic Sobolev inequality). Applications to network information theory are also given, including a short self-contained proof of the rate region for the two-encoder quadratic Gaussian source coding problem and a new outer bound for the one-sided Gaussian interference channel.

Statistical Properties of Double Hoyt Fading With Applications to the Performance Analysis of Wireless Communication Systems

In this paper, we investigate the statistical properties of double Hoyt fading channels, where the overall received signal is determined by the product of two statistically independent but not necessarily identically distributed single Hoyt processes. Finite-range integral expressions are first derived for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LCR), and average duration of fades of the envelope fading process. A closed-form approximate solution is also deduced for the LCR by making use of the Laplace approximation theorem. Applying the derived PDF of the double Hoyt channel, we then provide analytical expressions for the average symbol error probability of both coherent M-ary phase-shift keying and square M-ary quadrature amplitude modulation schemes. It is shown that all the obtained theoretical results include those that are already known for double Rayleigh channels as a special case. In addition, simplified expressions for the Hoyt $\times $ Rayleigh, Rayleigh $\times $ one-sided Gaussian, and double one-sided Gaussian channels are presented. Moreover, the applicableness of the proposed model to measured real-world propagation channels is examined and discussed by comparing the derived CDF and LCR with published measurement data collected in inter-vehicular propagation environments. Numerical and simulation results are also provided to confirm the validity of the derivations.

Neutron star cooling and the rp process in thermonuclear X-ray bursts

When the upper layer of an accreting neutron star experiences a thermonuclear runaway of helium and hydrogen, it exhibits an X-ray burst of a few keV with a cool-down phase of typically 1~minute. When there is a surplus of hydrogen, hydrogen fusion is expected to simmer during that same minute due to the rp process, which consists of rapid proton captures and slow beta-decays of proton-rich isotopes. We have analyzed the high-quality light curves of 1254 X-ray bursts, obtained with the Proportional Counter Array on the Rossi X-ray Timing Explorer between 1996 and 2012, to systematically study the cooling and rp process. This is a follow-up of a study on a selection of 37 bursts from systems that lack hydrogen and show only cooling during the bursts. We find that the bolometric light curves are well described by the combination of a power law and a one-sided Gaussian. The power-law decay index is between 1.3 and 2.1 and similar to that for the 37-bursts sample. There are individual bursters with a narrower range. The Gaussian is detected in half of all bursts, with a typical standard deviation of 50~s and a fluence ranging up to 60% of the total fluence. The Gaussian appears consistent with being due to the rp process. The Gaussian fluence fraction suggests that the layer where the rp process is active is underabundant in H by a factor of at least five with respect to cosmic abundances. Ninety-four percent of all bursts from ultracompact X-ray binaries lack the Gaussian component, and the remaining 6% are marginal detections. This is consistent with a hydrogen deficiency in these binaries. We find no clear correlation between the power law and Gaussian light-curve components.

A Comprehensive Analysis of 5G Heterogeneous Cellular Systems Operating Over $\kappa$ – $\mu$ Shadowed Fading Channels

Emerging cellular technologies such as those proposed for use in 5G communications will accommodate a wide range of usage scenarios with diverse link requirements. This will include the necessity to operate over a versatile set of wireless channels ranging from indoor to outdoor, from line-of-sight (LOS) to non-LOS, and from circularly symmetric scattering to environments which promote the clustering of scattered multipath waves. Unfortunately, many of the conventional fading models adopted in the literature to develop network models lack the flexibility to account for such disparate signal propagation mechanisms. To bridge the gap between theory and practical channels, we consider $\kappa$-$\mu$ shadowed fading, which contains as special cases, the majority of the linear fading models proposed in the open literature, including Rayleigh, Rician, Nakagami-m, Nakagami-q, One-sided Gaussian, $\kappa$-$\mu$, $\eta$-$\mu$, and Rician shadowed to name but a few. In particular, we apply an orthogonal expansion to represent the $\kappa$-$\mu$ shadowed fading distribution as a simplified series expression. Then using the series expressions with stochastic geometry, we propose an analytic framework to evaluate the average of an arbitrary function of the SINR over $\kappa$-$\mu$ shadowed fading channels. Using the proposed method, we evaluate the spectral efficiency, moments of the SINR, bit error probability and outage probability of a $K$-tier HetNet with $K$ classes of BSs, differing in terms of the transmit power, BS density, shadowing characteristics and small-scale fading. Building upon these results, we provide important new insights into the network performance of these emerging wireless applications while considering a diverse range of fading conditions and link qualities.

A Stochastic Geometric Analysis of Device-to-Device Communications Operating Over Generalized Fading Channels

Device-to-device (D2D) communications are now considered an integral part of future 5G networks, which will enable direct communication between user equipments and achieve higher throughputs than conventional cellular networks, but with the increased potential for co-channel interference. The physical channels, which constitute D2D communications, can be expected to be complex in nature, experiencing both line-of-sight (LOS) and non-LOS conditions across closely located D2D pairs. In addition to this, given the diverse range of operating environments, they may also be subject to clustering of the scattered multipath contribution, i.e. , propagation characteristics which are quite dissimilar to conventional Rayleigh fading environments. To address these challenges, we consider two recently proposed generalized fading models, namely $\kappa$ - $\mu$ and $\eta$ - $\mu$ , to characterize the fading behavior in D2D communications. Together, these models encompass many of the most widely utilized fading models in the literature such as Rayleigh, Rice (Nakagami- $n$ ), Nakagami- $m$ , Hoyt (Nakagami- $q$ ), and One-sided Gaussian. Using stochastic geometry, we evaluate the spectral efficiency and outage probability of D2D networks under generalized fading conditions and present new insights into the tradeoffs between the reliability, rate, and mode selection. Through numerical evaluations, we also investigate the performance gains of D2D networks and demonstrate their superiority over traditional cellular networks.

Secrecy Capacity Analysis Over $\kappa $ – $\mu $ Fading Channels: Theory and Applications

In this paper, we consider the transmission of confidential information over a $\kappa $ – $\mu $ fading channel in the presence of an eavesdropper who also experiences $\kappa $ – $\mu $ fading. In particular, we obtain novel analytical solutions for the probability of strictly positive secrecy capacity (SPSC) and a lower bound of secure outage probability (SOP $^{L}$ ) for independent and non-identically distributed channel coefficients without parameter constraints. We also provide a closed-form expression for the probability of SPSC when the $\mu $ parameter is assumed to take positive integer values. Monte-Carlo simulations are performed to verify the derived results. The versatility of the $\kappa $ – $\mu $ fading model means that the results presented in this paper can be used to determine the probability of SPSC and SOP $^{L}$ for a large number of other fading scenarios, such as Rayleigh, Rice (Nakagami- $n$ ), Nakagami- $m$ , One-Sided Gaussian, and mixtures of these common fading models. In addition, due to the duality of the analysis of secrecy capacity and co-channel interference (CCI), the results presented here will have immediate applicability in the analysis of outage probability in wireless systems affected by CCI and background noise (BN). To demonstrate the efficacy of the novel formulations proposed here, we use the derived equations to provide a useful insight into the probability of SPSC and SOP $^{L}$ for a range of emerging wireless applications, such as cellular device-to-device, peer-to-peer, vehicle-to-vehicle, and body centric communications using data obtained from real channel measurements.

Ratio of Products of Variates

Exact, closed-form expressions for the probability density function and the cumulative distribution function for the ratio of the products of an arbitrary number of independent and non-identically distributed $\alpha$ - $\mu$ variates are derived. The only restriction placed on the participating distributions is that there must be a rational relationship between their $\alpha$ parameters, which, in practical terms, represents no impairment. The expressions are given in terms of the Meijer G-function and, alternatively, in terms of a finite sum of hypergeometric functions. These results can be used to investigate the performance of wireless communication systems in a variety of realistic propagation environments in which the numerator and denominator products might be used to represent the signal and the interference. In addition, the results given comprise those for the ratio of the products of arbitrary combinations of other useful distributions such as one-sided Gaussian, negative exponential, Rayleigh, Weibull, Gamma, and Nakagami- $m$ . Simulation is used to confirm the results. An application example is given in order to illustrate the use of the formulations.

Cooperative Dual-Hop Wireless Communication Systems With Beamforming Over <inline-formula> <tex-math notation="LaTeX">$\eta-\mu$</tex-math></inline-formula> Fading Channels

The end-to-end performance of cooperative beamforming in dual-hop relaying systems operating over η - μ fading channels is investigated. In the analysis, the source-destination node pair is equipped with multiple antennas, whereas the fixedgain relay has a single antenna. The η - μ fading channel has Nakagami-m, Rayleigh, one-sided Gaussian, and Nakagami-q (Hoyt) channels as special cases. In this paper, novel statistical expressions for the generalized moments of the end-to-end signal- to-noise ratio (SNR), cumulative density function (cdf), probability density function (pdf), and moment-generating function (mgf) are derived and evaluated. Furthermore, the obtained results are used to derive new and simple expressions for the pdf, the outage probability, and the mgf of the end-to-end SNR over Nakagami-m fading channels. In addition, we show that the derived expression for parameter C over the Nakagami-m fading channel is incorrect, and hence, it leads to inaccurate results. Also, we study the impact of the fading parameters η and μ on the end-to-end system performance. Simulation results show that, in contrast to the fading parameter η, μ has a significant impact on the system performance.

Product of $\alpha$– $\mu$ Variates

Exact closed-form expressions for the probability density function and the cumulative distribution function for the product of an arbitrary number of independent non-identically distributed $\alpha$ – $\mu$ variates are derived. The expressions are given in terms of the Meijer G-function, and in terms of a finite sum of hypergeometric functions, and can be easily computed by means of well-known mathematical software tools. As a consequence of the generality of the $\alpha$ – $\mu$ distribution as well as of the results, these statistics comprise those for the product of arbitrary combinations of other useful distributions such as One-Sided Gaussian, Negative Exponential, Rayleigh, Weibull, Gamma, and Nakagami- $m$ . As an application example, the outage probability for the cascade channel is presented.

Exact SINR Statistics in the Presence of Heterogeneous Interferers

We derive new results for the higher order moments of signal-to-interference-plus-noise ratio (SINR) in the presence of an arbitrary Poisson point process (PPP)-based heterogeneous interference field. The analysis leverages on a moment-generating-function (MGF) methodology, which only requires the statistics of intended signal and aggregate interference, thus eliminating the need for the exact distribution of SINR. We extend the existing results on interference statistics by deriving a generalized closed-form expression of the interference MGF considering Nakagami-m fading channels with exclusion region. In certain special cases, explicit expressions for the averages of different functions of SINR are found, which also lead to closed-form solutions for the probability distributions of aggregate interference reciprocal and signal-to-interference ratio. We prove that in such cases the effect of total PPP-based interference power on useful transmission is mathematically equivalent to the severe fluctuations from a one-sided Gaussian fading channel. As an application example, the proposed methodology is used together with stochastic geometry theory to characterize the average SINR and rate in heterogeneous cellular networks. The validity of our analytical derivations is confirmed via Monte Carlo simulations for various system settings. We show that with cellular network densification there exists a tradeoff between the average SINR and rate performance.

On the Efficient Generation ofα-κ-μandα-η-μWhite Samples with Applications

This paper is concerned with a simple and highly efficient random sequence generator for uncorrelatedα-κ-μandα-η-μvariates. The algorithm may yield an efficiency of almost 100%, and this high efficiency can be reached for all special cases such asα-μ, κ-μ, η-μ, Nakagami-m, Nakagami-q, Weibull, Hoyt, Rayleigh, Rice, Exponential, and the One-Sided Gaussian. This generator is implemented via the rejection technique and allows for arbitrary fading parameters. The goodness-of-fit is measured using the Kolmogorov-Smirnov and Anderson-Darling tests. The maximum likelihood parameter estimation for theκ-μdistribution is proposed and verified against true values of the parameters chosen in the generator. We also provide two important applications for the random sequence generator, the first one dealing with the performance assessment of a digital communication system over theα-κ-μandα-η-μfading channels and the second one dealing with the performance assessment of the spectrum sensing with energy detection over special cases of these channels. Theoretical and simulation results are compared, validating again the accuracy of the generators.

Performance Analysis of Spatial Modulation and Space-Shift Keying With Imperfect Channel Estimation Over Generalized <inline-formula> <tex-math notation="TeX">$\eta{-}\mu$</tex-math></inline-formula> Fading Channels

Novel performance analysis of spatial modulation (SM) and space-shift keying (SSK) over generalized η-μ fading channels in the presence of Gaussian imperfect channel estimation at the receiver is presented in this paper. A general expression for the pairwise error probability (PEP) is derived along with an asymptotic expression at high signal-to-noise ratio (SNR). The η-μ fading channel has Nakagami-m, Rayleigh, one-sided Gaussian, and Nakagami-q (Hoyt) channels as special cases. The derived expression for the PEP is valid for integer and noninteger values of the fading parameter μ. The impact of channel estimation errors with different η and μ values is investigated. Analytical results are sustained through Monte Carlo simulation results, and a close match is reported for a wide range of SNR and for different system parameters.

Performance analysis of generalized QAM modulation under η−μ and κ−μ fading

Abstract This paper presents new closed-form expressions for the symbol error probability (SEP) of θ-QAM modulation with maximum ratio combining (MRC) receiver under η−μ and κ−μ fading. The SEP formulae, obtained from the definite integrals of the moment generating function (MGF) of the signal-to-noise ratio (SNR) at the input of the MRC receiver, are written in terms of Lauricella functions. The numerical evaluation of the expressions is carried out for the η−μ distribution, which includes important distributions as special cases, such as Hoyt, Nakagami-m, Rayleigh, and one-sided Gaussian, as well as for the κ−μ distribution, which includes Rice, Nakagami-m, Rayleigh, and one-sided Gaussian as special cases.

Performance of dual-branch selection combining diversity systems in non-identical Nakagami- q (Hoyt) fading channels

Motivated by the low-complexity to implement a dual-branch selection combining system, which is a practical method to improve the performance of a communication system subject to fading, this work studies important performance measures for that system over independent but non-identically distributed Nakagami-q fading channels. In particular, a novel closed-form expression is derived for the moments of the output signal-to-noise ratio (SNR), which is utilised to obtain expressions for the average output SNR, and amount of fading. An expression for the outage probability is also obtained in a closed-form. Furthermore, novel expressions for the channel capacity are derived for two adaptive transmission schemes: the optimal rate adaptation with constant power and channel inversion with fixed rate. The corresponding expressions for one-sided Gaussian and Rayleigh fading are obtained as special cases of Nakagami-q fading. Finally, numerical examples are presented for illustration.