Integer Fading Parameters Research Articles

Analysis of RIS-Aided Communications Over Nakagami-$m$ Fading Channels

Reconfigurable intelligent surface (RIS) has been attracted lots of attention due to the fact that it can reconfigure propagation environment with low cost for resource-constrained systems. In this paper, RIS is considered to support wireless communication system by taking both reflection path and direct path into consideration and investigate the system performance under general Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> fading. A novel analytical closed-form expression is first derived for outage probability. To get more insights, we provide simplified approximations for the high SNR regime and obtain the diversity order and coding gain in closed form. We demonstrate that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sum _{l = 1}^{N + 1} {m_{l}^*}$</tex-math></inline-formula> diversity order which can be obtained by RIS-assisted system, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is the number of reflect elements in the RIS and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${m_{l}^*}$</tex-math></inline-formula> refers to the channel fading parameters. Furthermore, we investigate the performance of ergodic achievable rate and provide tight bounds for it in closed form. Based on these results, high SNR slope is discussed. Notably, our analytical results are suitable for any integer fading parameters and arbitrary topologies, transmit power. Numerical results validate the accuracy of our analytical results and show the great performance improvement provided by the RIS-assisted system.

Alternative expressions of the PDF and CDF for Gamma, [formula omitted] and [formula omitted] shadowed distributions with applications in wireless communications

There are two approaches well known in the literature for reducing the complexity in averaged performance analyses of wireless communications systems (i.e., for alleviating the computational difficulty in averaged performance evaluations). The first method, known as the moment generating function MGF-based approach, aims to convert the instantaneous performance measure into an exponential integration or a certain sum of exponential functions of different scales. The second approach, which is the focus of this article, aims to convert the exact probability density function PDF and cumulative distribution function CDF of fading distributions into the sum of exponential functions of various scales. As such, we propose weighted sum of exponential functions as alternative closed-form approximations for the PDF and CDF of gamma distribution, η−μ distribution, and κ−μ shadowed distribution with integer fading parameters, and we present how these proposed expressions can be easily applied to the performance analysis of wireless communication systems operating over Nakagami-m fading channels, η−μ fading channels, and κ−μ shadowed fading channels.

Capacity analysis of the fluctuating double-Rayleigh with line-of-sight fading channel

The proposed research performs the capacity analysis of the wireless channel described by the fluctuating double-Rayleigh with the line-of-sight model. The closed-form analytical expressions for the conditional capacity (in the case of arbitrary noninteger fading parameter) and the ergodic capacity (in the case of the integer fading parameter) are derived in terms of the extended generalized bivariate Meijer G-function. The exact solutions are succeeded by the approximating expressions deduced for the cases of small and large ratios between the Rician K-factor and the fading parameter. The performed numeric simulation verifies the correctness of the derived results and analyzes the proposed capacity approximation quality performance.

Wireless Secrecy Under Multivariate Correlated Nakagami-m Fading

Current wireless secrecy research in the literature has mainly been performed for one wiretapper under correlated fading. In this paper, a new wireless secrecy framework for multiple wiretappers under multivariate exponentially-correlated (exp.c.) Nakagami- $m$ fading is proposed. Using the distribution of multivariate exp.c. Nakagami- $m$ fading, new, exact, and compact expressions for the ergodic secrecy capacity, and secrecy outage probability (SOP) under multiple wiretappers are obtained for an integer fading parameter $m$ . A secrecy analysis is also performed for the first time in this paper using an adaptive on/off transmission encoder under multivariate exp.c. Nakagami- $m$ fading. A secrecy analysis with three wiretappers under quadrivariate exp.c. Nakagami- $m$ fading is also given, which shows the effectiveness of the new framework. Simulation results are shown to exactly match theoretical predictions.

Performance analysis of multihop decode-and-forward relay networks with diversity in Nakagami fading channels

In this paper, we consider a multihop Decode-and-Forward (DF) relay network, in which each node receives the signals from its two preceding nodes. We assume Nakagami fading channels with integer fading parameters. In this network, the achievable diversity order is limited by the link between the source and the first relay. We derive the exact outage probability of fixed and selection relaying with spatial channel reuse, and demonstrate that applying selection relaying at the first relay is adequate to achieve full diversity. In order to derive the outage probability of this system in a Nakagami fading environment with integer fading parameters, a simple closed-form expression for the cumulative distribution function (CDF) of the sum of two independent, but not necessarily identically distributed, Erlang random variables is obtained. By approximating the outage probability for the high signal-to-noise ratio (SNR) region, we also derive the diversity order and coding gain of this selection relaying method and the fixed relaying method. The obtained asymptotic performance expressions are generalized for Nakagami fading channels with arbitrary real-valued fading parameters. Numerical and Monte Carlo results are provided which confirm the analytical expressions.

Opportunistic Outdated Relays Under Even-Degree-of-Freedom Generalized-Rician Fading

Using a distribution of a non-central chi-square random variable and selection combining, statistics of wireless systems employing decode-and-forward (DF) opportunistic relays (ORs) under outdated channel state information (CSI), and independent and identically distributed (i.i.d.) generalized-Rician fading environments are derived. By mathematically rewriting the Marcum $Q$ -function in its infinite-summation form, the distribution of non-central chi-square is mathematically simplified to its desirable form under different scenarios, which are then employed to compute average bit error rates for popular modulation schemes. Distributions of end-to-end system signal-to-noise ratio, useful for fifth-generation backhaul wireless communications, are also derived under integer and non-integer fading parameters. The new findings generalize existing findings employing DF ORs under i.i.d. Rayleigh fading environments. Mathematical predictions are shown to match simulation results.

Coverage Analysis of User-Centric Wireless Network in a Comprehensive Fading Environment

This letter analyzes the downlink coverage probability of user-centric wireless network in a comprehensive fading environment. The comprehensive fading environment consists of two types of fading models: 1) a long-term shadowed fading, where all signals experience an identical and independent lognormal shadowing and 2) an instantaneous fading, where the desired signal experiences $\kappa $ – $\mu $ shadowed fading with integer fading parameters and the interfering signals experience arbitrary fading. Based on the Gaussian–Hermit quadrature, a tractable and accurate expression for the serving distance with lognormal shadowing is obtained. Under a general cell association rule, a simple and generic expression for the downlink coverage probability of user-centric wireless network in the comprehensive fading scenario is successfully derived and verified through simulation. The derived expression reveals that there is no need for the coverage analysis with respect to the instantaneous interfering fading channels in terms of the independence assumption between network model and fading channels.

Impact of Relay’s Eavesdropping on Untrusted Amplify-and-Forward Networks Over Nakagami- $m$ Fading

We investigate the impact of eavesdropping on the secrecy outage performance of dual-hop amplify-and-forward networks with a single untrusted relay, in which cooperative jamming technique is employed at the destination to realize positive secrecy rate. Under Nakagami- ${m}$ fading channels, we derive tight closed-form and asymptotic expressions for the lower and upper bounds of the secrecy outage probability (SOP). Our results explicitly show that a secrecy diversity order of ${\min ({m_{1}}, {m_{2}}/{2})}$ in terms of the SOP is achieved, where ${m_{1}}$ and ${m_{2}}$ are the respective integer fading parameters of the source-relay and relay-destination links. Afterwards, we propose an asymptotically optimal power allocation (OPA) scheme that minimizes the SOP. Numerical and simulation results show that the proposed OPA offers significant improvement on the SOP performance, especially in the case of ${m_{1}> m_{2} /2}$ .

The $\kappa$- $\mu$ Shadowed Fading Model With Integer Fading Parameters

We show that the popular and general {\kappa}-{\mu} shad- owed fading model with integer fading parameters {\mu} and m can be represented as a mixture of squared Nakagami (or Gamma) distributions. Thus, its PDF and CDF can be expressed in closed-form in terms of a finite number of elementary functions (powers and exponentials). The main implications arising from such connection are then discussed, which can be summarized as: (1) the performance evaluation of communication systems operating in {\kappa}-{\mu} shadowed fading becomes as simple as if a Nakagami fading channel was assumed; (2) the {\kappa}-{\mu} shadowed distribution can be used to approximate the {\kappa}-{\mu} distribution us- ing a closed-form representation in terms of elementary functions, by choosing a sufficiently large value of m; and (3) restricting the parameters {\mu} and m to take integer values has limited impact in practice when fitting the {\kappa}-{\mu} shadowed fading model to field measurements. As an application example, the average channel capacity of communication systems operating under {\kappa}-{\mu} shadowed fading is obtained in closed-form.

Novel results for the performance of single and double stages cognitive radio systems through Nakagami-m fading and log-normal shadowing

Previously reported results presented approximate expressions for the performance of cognitive radio systems through Nakagami fading and log-normal shadowing. In this article, exact expressions for the Receiver Operating Characteristics of cognitive radio system are derived analytically for a single stage and a double stage system. By single stage, it is assumed that the secondary users are close to the fusion centre therefore, they can be considered directly coupled to it. For the second stage system the secondary users are considered far from the fusion centre, thus another communication link is assumed between the secondary users and the fusion centre with similar characteristics as the first link. In both scenarios Nakagami fading and log-normal shadowing are assumed. Results for integer and non-integer Nakagami fading parameter `m' are obtained. Depicted results show that better performance is obtained with the increase of the number of secondary users and increase of `m'. Deterioration due to shadowing is also indicated.

Outage Probability of TDBC Protocol in Multiuser Two-Way Relay Systems with Nakagami-m Fading

This paper considers a multiuser two-way relay system with time division broadcast (TDBC) protocol, where multiple end-sources compete to exchange information with another end-source through the help of a single half-duplex amplify-and-forward (AF) relay. We derive a tight closed-form lower bound for the system outage probability over Nakagami-m fading channels with integer fading parameters. An asymptotic expression for the outage probability in high signal-to-noise ratio (SNR) regime is also obtained. Numerical results corroborate the theoretical analysis.

Moments Based Framework for Performance Analysis of One-Way/Two-Way CSI-Assisted AF Relaying

When analyzing system performance of conventional one-way relaying or advanced two-way relaying, these two techniques are always dealt with separately and, thus, their performance cannot be compared efficiently. Moreover, for ease of mathematical tractability, channels considered in such studies are generally assumed to be subject to Rayleigh fading or to be Nakagami-m channels with integer fading parameters, which is impractical in typical urban environments. In this paper, we propose a unified moments-based framework for general performance analysis of channel-state-information (CSI) assisted amplify-and-forward (AF) relaying systems. The framework is applicable to both one-way and two-way relaying over arbitrary Nakagami-m fading channels, and it includes previously reported results as special cases. Specifically, the mathematical framework is firstly developed under the umbrella of the weighted harmonic mean of two Gamma-distributed variables in conjunction with the theory of Padé approximants. Then, general expressions for the received signal-to-noise ratios of the users in one-way/two-way relaying systems and the corresponding moments, moment generation function, and cumulative density function are established. Subsequently, the mathematical framework is applied to analyze, compare, and gain insights into system performance of one-way and two-way relaying techniques, in terms of outage probability, average symbol error probability, and achievable data rate. All analytical results are corroborated by simulation results as well as previously reported results whenever available, and they are shown to be efficient tools to evaluate and compare system performance of one-way and two-way relaying.

Asymptotic Performance Analysis of Multihop Relaying with Co-Channel Interference in Nakagami-m Fading Channels

In this letter, the asymptotic performance of channel state information-assisted amplify-and-forward multihop relaying with co-channel interference (CCI) at both relays and destination over Nakagami-m fading channels is investigated. Assuming integer fading parameter m for source → relay → destination links and arbitrary fading parameter m for CCI links, we derive closed-form expressions for bit error rate of the system at high signal-to-interference-plus-noise regime and characterize the diversity order and coding gain achieved by the system. Simulation results are provided to validate the analysis.

Exact Performance Analysis of Dual-Hop Semi-Blind AF Relaying over Arbitrary Nakagami-m Fading Channels

Relay transmission is promising for future wireless systems due to its significant cooperative diversity gain. The performance of dual-hop semi-blind amplify-and-forward (AF) relaying systems was extensively investigated, for transmissions over Rayleigh fading channels or Nakagami-m fading channels with integer fading parameter. For the general Nakagami-m fading with arbitrary m values, the exact closed-form system performance analysis is more challenging. In this paper, we explicitly derive the moment generation function (MGF), probability density function (PDF) and moments of the end-to-end signal-to-noise ratio (SNR) over arbitrary Nakagami-m fading channels with semi-blind AF relay. With these results, the system performance evaluation in terms of outage probability, average symbol error probability, ergodic capacity and diversity order, is conducted. The analysis developed in this paper applies to any semi-blind AF relaying systems with fixed relay gain, and two major strategies for computing the relay gain are compared in terms of system performance. All analytical results are corroborated by simulation results and they are shown to be efficient tools to evaluate system performance.

Performance Analysis of Dual-Hop AF Systems With Interference in Nakagami-$m$ Fading Channels

In this letter, we investigate the performance of dual-hop channel state information-assisted amplify-and-forward relaying systems over Nakagami-m fading channels in the presence of multiple interferers at the relay. Assuming integer fading parameter m, we derive closed-form expressions for the exact outage probability and accurate approximation for symbol error rate of the system. Furthermore, we look into the asymptotical high signal to noise ratio regime, and characterize the diversity order achieved by the system. All the analytical results are validated via Monte Carlo simulations.

Outage probability of multiuser transmit antenna selection/maximal-ratio combining systems over arbitrary Nakagami- m fading channels

The outage probability of multiuser diversity (MUD) in transmit antenna selection/maximal-ratio combining (TAS/MRC) systems is derived as an exact closed expression in independent and identically distributed (i.i.d.) Nakagami-m channels with an integer fading parameter. For a non-integer fading parameter, the exact outage probability is derived as a single infinite series of incomplete Gamma function. At high signal-to-noise ratios (SNRs), the analytical results deduce that the multiuser TAS/MRC systems can achieve a full diversity order equal to the product of the fading parameter, number of users, number of transmit antennas and number of receive antennas. The advantage of the total diversity gain becomes more pronounced on a severe fading channel. The achieved results provide an analytical framework for the assessment of multiuser TAS/MRC systems. All the derived expressions are verified by Monte Carlo simulations.

An efficient approximation to the correlated Nakagami-m sums and its application in equal gain diversity receivers

There are several cases in wireless communications theory where the statistics of the sum of independent or correlated Nakagami-m random variables (RVs) is necessary to be known. However, a closed-form solution to the distribution of this sum does not exist when the number of constituent RVs exceeds two, even for the special case of Rayleigh fading. In this paper, we present an efficient closed-form approximation for the distribution of the sum of arbitrary correlated Nakagami-m envelopes with identical and integer fading parameters. The distribution becomes exact for maximal correlation, while the tightness of the proposed approximation is validated statistically by using the Chi-square and the Kolmogorov-Smirnov goodness-of-fit tests. As an application, the approximation is used to study the performance of equal-gain combining (EGC) systems operating over arbitrary correlated Nakagami-m fading channels, by utilizing the available analytical results for the error-rate performance of an equivalent maximal-ratio combining (MRC) system.

A Simple and Accurate Approximation to the SEP of Rectangular QAM in Arbitrary Nakagami-m Fading Channels

Recently, Karagiannidis presented a closed-form solution for an integral which can be used to compute the average symbol error probability of general order rectangular quadrature amplitude modulation (QAM) in Nakagami-m fading channels with integer fading parameters. In this letter, using an accurate exponential bound for the Gaussian Q-function, we derive a simple approximate solution for that integral. In particular, the solution can be used to compute the average SEP of general order rectangular QAM over arbitrary Nakagami-m fading. Numerical results are presented to verify the accuracy of the solution

A closed-form solution for the distribution of the sum of Nakagami-m random phase vectors

The distribution of the modulus of the sum of random phase vectors, is of great practical importance in several applications which deal with sums of sinusoidal signals (wireless multipath transmissions, radars, optical communications, etc). In this letter, simple closed-form expressions are presented for the probability density function (PDF), the cumulative distribution function (CDF), the moment generating function (MGF) and the moments of the envelope distribution of the sum of L non-identical random Nakagami-m phase vectors with integer fading parameters. Moreover, the average over this distribution of the Gaussian Q-function and of the squared Q-function, are also presented in closed-form

Bit-error probabilities for 2- and 4-DPSK in general Nakagami fading channels with correlated Gaussian noise

Exact bit-error probability (BEP) expressions are derived for 2- and 4-ary differential phase shift keying (2- and 4-DPSK) with post-detection equal gain combining (EGC) in arbitrarily correlated slow Nakagami fading channels. Unlike previous work, the effect of noise correlation between adjacent symbol intervals is considered and shown explicitly in the BEP expressions. The characteristic function method is used to give a BEP expression in a general form for 2- and 4-DPSK. Furthermore, closed-form BEP expressions are derived for 2-DPSK with coloured noise, and for 4-DPSK with white noise and integer fading parameter m. The theoretical result is demonstrated by analysing two antenna configurations, giving some interesting findings.