On the BEP analysis of M-QAM and R-QAM under cascaded double η-μ, κ-μ or α-μ fading channels

doi:10.1016/j.dsp.2020.102870

Digital Signal Processing

Volume 107, December 2020, 102870

https://doi.org/10.1016/j.dsp.2020.102870 Get rights and content

Abstract

This article presents new expressions for the bit error probability (BEP) of the M-ary quadrature amplitude modulation (M-QAM) and rectangular QAM (R-QAM) scheme in a frequency non-selective channel model with cascaded double fading, characterized in the links by the distributions η-μ, κ-μ or α-μ. In this paper, the probability density function (PDF) of the signal-to-noise ratio (SNR) of the double link is written in terms of an integral of the product of the PDF of the envelope of the first link by the PDF of the SNR of the second link. With this procedure, expressions for the BEP of the M-QAM and R-QAM scheme, with and without a maximum ratio combiner (MRC) associated, are obtained by means of the general expressions of Cho and Yoon, calculated with the help of the Gauss-Laguerre quadrature. The new expressions are written in terms of the integral of the moment generating function (MGF) of the instantaneous SNR from the second link, which are simpler than using expressions involving infinite series in terms of special functions like hypergeometric functions and the Meijer G-function. In addition, insights regarding the expressions obtained and comments about their complexity are provided. Several curves for the BEP are presented, for different parameters that characterize the channel and different numbers of branches of the MRC. All analytical expressions obtained in this paper are corroborated by computer simulations performed with the Monte Carlo method.

Introduction

Channel models with cascaded fading have been considered for different communication systems. They can be used to characterize, for example, environments in which propagation is aided by one or more than one relay station (dual-hop or multi-hop systems), which capture signals from a propagation direction and redirect them to the receiving antenna. These cascaded fading models have also been considered to characterize keyhole channels. In this context, depending on the particularities of each urban environment, the statistics of the signal received through the cascaded fading are characterized by different probability distributions.

In [1], a vehicular network with a dual-hop relay station is analyzed under non-uniform shadowed double Nakagami-m, with transmitters of different powers. The analysis is performed considering non-uniform sources of co-channel interference for three relaying schemes, namely dynamic decode and forward (DDF), dynamic quantize map and forward (DQMF) and static quantize map and forward (SQMF). The authors also present the analysis of the optimal sensing and transmission time of the vehicles in the three relaying schemes, in terms of the conditions of the channels between vehicles and extend the results to multiple relay stations.

In [2], the double Rayleigh fading is also considered in the context of communication between vehicles. In the work, the communication link between two vehicles is established with the aid of a vehicle that functions as a relay station. According to the authors, the fading of the first link, from the source to the relay station, can be characterized either by the Rayleigh distribution or by the product distribution of two Rayleigh variables, while the second link is characterized only by the double Rayleigh fading. The performance of the links is evaluated by the symbol error probability (SEP), for the 4-ary quadrature amplitude modulation (4-QAM) and binary phase shift keying (BPSK) modulation schemes, and by the outage probability. This double fading model is also considered in [3] in the characterization of channels established between vehicles for the evaluation of average secrecy capacity (ASC). The double Rayleigh fading is also analyzed in the context of the dual-hop cognitive inter-vehicular relay-assisted communication system. In [4], exact and approximate analytical expressions are derived for the outage probability, corroborated by computer simulations performed with the Monte Carlo method, considering relay selection (RS) policies and two distinct fading scenarios.

In [5], the authors derive expressions for outage probability, ergodic capacity and SEP from an approximation obtained for the PDF of the cascaded fading envelope characterized, in each link, by the Nakagami and Gamma distributions. With the PDF approximation of the cascaded double fading envelope, accomplished by the Hankel's expansion of the modified Bessel function of second species, the expressions of the ergodic capacity are presented in terms of the sum of the Meijer G-function.

The Nakagami distribution is also considered in [6] in the characterization of N cascaded links with the total fading represented by the product of N independent Nakagami random variables. The authors contextualize the fading model by assessing the secrecy outage probability (SOP) and the probability of strictly positive secrecy capacity (SPSC). The obtained expressions are all written in terms of the Meijer G-function for one and two eavesdroppers in the communication system.

In [7], statistics such as level crossing rate (LCR) and average duration of fading (ADF) are calculated and presented together with the performance evaluation, by SEP, of the reception of modulated signals in M-ary phase-shift keying (M-PSK) and M-ary quadrature amplitude modulation (M-QAM) under double Hoyt fading. The authors present the analysis in the scenario of links established between vehicles and present theoretical curves of LCR, ADF and the cumulative distribution function (CDF) of the double fading envelope. The theoretical curves are corroborated by measurements, showing that this type of double fading is really typical in the links established between vehicles.

Regarding the generalized probability distributions, in [8] a channel is considered with double fading characterized by the double generalized Gamma distribution. According to the authors, this distribution is suitable for modeling non-homogeneous fading, commonly observed in links between vehicles. In this context, different statistical metrics, asymptotic expressions, amount of fading (AOF), outage probability and spatial diversity gain analysis are presented. This Gamma-Gamma model is also considered in [9] to characterize variations in the intensity of signals received in wireless optical communications links, caused by atmospheric turbulence and alignment errors in the transmitter and receiver positions. The authors evaluate the performance of the link using the BEP calculated for the M-PSK modulation scheme in a reception system with a maximum ratio combiner (MRC) associated with the receiver's optical sensor array. In [9], performance evaluation metrics, such as BEP and outage probability, are expressed in terms of the Meijer G-function.

In [10], the K distribution, characterized by three parameters, is considered for the analysis of N cascaded links by means of metrics such as outage probability and BEP for different digital modulation schemes. In [11] a fading model characterized by the product of N independent Nakagami random variables is considered to evaluate the outage probability and the BEP of modulation schemes. In [12] a study is carried out of the impact of the cascaded Rayleigh fading in a system with diversity established by the use of space-time lattice codes.

In [13], the McKay-Meijer G shadowed fading model is analyzed in terms of metrics like BEP and capacity, since the fading model is suitable for scenarios such as indoor communication and free space optical. The authors present new expressions for BEP, under different modulation schemes, and for the channel capacity, considering different adaptive schemes. The expressions are derived considering the product of M links and the channel subject to the generalized Gaussian noise (GGD). All expressions derived in [13] are corroborated by computer simulations.

In the present article, exact BEP expressions are presented for the M-QAM and rectangular QAM (R-QAM) modulation scheme, in links subject to cascaded double η-μ, κ-μ or α-μ fading. All analytical expressions are corroborated by computer simulations performed with the Monte Carlo method. The PDF of the instantaneous signal to noise ratio (SNR) of the total link is written in this paper as an integral of the product of the PDF of the fading envelope of the first link by the PDF of the instantaneous SNR of the second link. Thus, the BEP expressions can be written in terms of finite sums involving the integral, in the interval $[0, π / 2]$ , of the moment generating function (MGF) of the instantaneous SNR of the second link. According to the authors' knowledge, expressions for the BEP of the M-QAM and R-QAM scheme, with and without an MRC associated, obtained by means of the general expressions of Cho and Yoon [14] under these cascaded double fading models are new.

The following contributions can then be summarized:

1.
A new expression for the BEP of the M-QAM and R-QAM scheme, under cascaded double η-μ, κ-μ or α-μ fading, in terms of finite sums involving the integral of the MGF of the instantaneous SNR of the second link.
2.
A new expression for the BEP of the M-QAM and R-QAM scheme, considering cascaded double η-μ, κ-μ or α-μ fading, for a receiver with an MRC associated.

The remaining of the paper is organized as follows. Section 2 addresses related works. Section 3 presents the mathematical characterization of the problem. In Section 4, BEP expressions are presented for the M-QAM and R-QAM modulation scheme, with the channel under additive white Gaussian noise (AWGN) and cascaded double η-μ, κ-μ or α-μ fading without the use of spatial diversity in reception. In Section 5, BEP expressions are presented considering an MRC associated. In Section 6 and Section 7, respectively, considerations about correlated cascaded double fading and computational complexity of the BEP expressions are provided. Section 8 presents numerical evaluation of the new BEP expressions corroborated by Monte Carlo simulations. Section 9 presents the conclusions.

Section snippets

Related works

In [15], the cascaded α-μ fading is characterized and contextualized for the problem of performance analysis of wireless communication systems in terms of reliability and security. Reliability is characterized by analyzing parameters such as AOF, outage probability, average channel capacity and SEP. The security analysis of the link is carried out by means of the secrecy outage probability, the probability of non-zero secrecy capacity and the average secrecy capacity.

In the same context of the

Problem characterization

The received signal considered in this article, in a signaling interval of duration $T_{s}$ , can be written at the receiver matched filter output as $r (t) = β s (t) + n (t),$ in which $β = β_{1} β_{2}$ is the cascaded frequency non-selective fading, represented by the continuous and independent random variables $β_{1}$ and $β_{2}$ , and $n (t)$ is the AWGN noise. In Fig. 1, a block diagram of the received signal observed in a signaling interval is presented.

A schematic diagram with the system model adopted in this paper is presented

Analysis of BEP without spatial diversity

In this section, (15) and (16) are evaluated by means of the Gauss-Laguerre quadrature, which is considered for the improper integral in the range from zero to infinite. The $P_{b}$ expression is then written in terms of the integral, in the interval $[0, π / 2]$ , of the MGF expression of the instantaneous SNR of the second link. Taking into account the PDFs of the three fading distributions considered, the improper integral must be written according to the structure $\int_{0}^{\infty} x^{a} f (x) e^{- x} d x \approx \sum_{l = 1}^{N} ω_{l} \cdot f (x_{l}),$ in which

Analysis of BEP with spatial diversity

In the case of considering an MRC with L branches, the instantaneous SNR at the combiner output can be written as $γ = \sum_{l = 1}^{L} γ_{l},$ in which $γ_{l}$ represents the instantaneous SNR in the l-th branch. Thus, the BEP can be written as $P_{b} = \int_{0}^{\infty} \dots \int_{0}^{\infty} f_{Γ_{1}, \dots, Γ_{L}} (γ_{1}, \dots, γ_{L}) P (e | \sum_{l = 1}^{L} γ_{l}) d γ_{1}, \dots, d γ_{L},$ in which $Γ_{i}$ , with 1 ≤ i ≤ L, represents the instantaneous SNR in the link established between the transmitter and the i-th antenna element associated with the MRC combiner.

Considering that the antenna elements of the combiner are

Considerations about correlated cascaded double fading

Although only decorrelated fading is considered in this study, the analysis for correlated fading can be extended to both the MRC receiver and selection diversity (SD) receiver. For developments involving correlated fading, one can see for instance references [26] and [27]. The aforementioned analysis to an MRC with two branches, for example, can start from the joint PDF of the envelope for cascaded double fading in the receiver's branches, written as $f_{V_{1}, V_{2}} (v_{1}, v_{2}) = \int_{0}^{\infty} \frac{1}{r_{1}^{2}} f_{R_{1}} (r_{1}) f_{R_{2}, R_{3}} (\frac{v_{1}}{r_{1}}, \frac{v_{2}}{r_{1}})$

Considerations about computational complexity

Table 1 presents a complexity analysis of the BEP expressions calculated in this paper for the M-QAM scheme, with and without spatial diversity. The complexity analysis is concerned with the time complexity of performing computations on a multitape Turing machine. In this analysis, the constants $C_{η - μ}$ , $C_{κ - μ}$ and $C_{α - μ}$ , as well as the zeros $x_{k}$ of the Gauss-Laguerre polynomial and the coefficients $ω_{k}$ can be obtained from tables or previously calculated and stored in a vector and therefore are not

Results

In this section, the analytical expressions for the BEP are corroborated by simulations performed with the Monte Carlo method, considering the cascaded double η-μ, κ-μ or α-μ fading, for 1 × 10⁶ transmitted bits and considering perfect CSI. For the generation of the random variables of the cascaded distributions, spatial-temporally uncorrelated random processes characterized by the η-μ, κ-μ or α-μ distribution are considered. Thus, for each link, random variables modeled by the η-μ, κ-μ or α-μ

Conclusions

Expressions for the bit error probability (BEP) of the M-ary quadrature amplitude modulation (M-QAM) or rectangular QAM (R-QAM) scheme, with and without spatial diversity, considering the channel subject to cascaded double η-μ, κ-μ or α-μ fading, are determined in this paper by means of the general expressions of Cho and Yoon, for the BEP of QAM digital modulation under additive white Gaussian noise (AWGN). In the work, the expressions obtained are new and deduced from the fact that the

CRediT authorship contribution statement

Hugerles Silva and Wamberto Queiroz carried out the mathematical development of the paper, and simulations were done by Danilo Almeida. Francisco Madeiro and Arnaldo S. R. Oliveira performed the technical review and analysis of the results. Hugerles Silva, Wamberto Queiroz, Arnaldo S. R. Oliveira and Francisco Madeiro wrote the paper. All authors read and approved the final manuscript.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was funded in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – Finance Code 001, by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by the European Regional Development Fund (FEDER), through the Competitiveness and Internationalization Operational Programme (COMPETE 2020) of the Portugal 2020 framework [Project 5G-PERFECTA with Nr. 038190 (POCI-01-0247-FEDER-038190)].

Hugerles S. Silva received his B.Sc., M.Sc. and D.Sc. degrees in Electrical Engineering from the Federal University of Campina Grande (UFCG), Brazil, in 2014, 2016 and 2019, respectively. His main research interests include wireless communication, digital communication systems and wireless channel modeling.

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Danilo B.T. Almeida was born in Itabaiana, Paraíba, Brazil in 1989. He received his B.Sc. and M.Sc. Degrees in Electrical Engineering from Federal University of Campina Grande, Brazil, in 2016 and 2018, respectively. He is currently working towards the D.Sc. degree at the same university. His research interests include channel modeling estimation of parameters for wireless communication.

Wamberto J.L. de Queiroz received the M.Sc. degree from Federal University of Paraiba, Campina Grande, PB, in 2000, and the D.Sc. degree from Federal University of Campina Grande, Campina Grande, PB, in 2004, both in electrical engineering. He was an Adjunct Professor at Federal University of Ceará from 2007 to 2010 and has been with the Federal University of Campina Grande since 2010, where he is Associate Professor. His current research interests include digital communications over fading channels, channel modeling and simulations, spectrum sensing systems and estimation theory.

Arnaldo Oliveira received the Ph.D. degree on Electrical Engineering in 2007 from the University of Aveiro, Portugal. He also holds the B.Sc. and M.Sc. degrees, both on Electronics and Telecommunications, from the same university. He is currently a researcher at Telecommunications Institute - Aveiro and since 2001 he teaches computer architecture, digital systems design, programming languages and embedded systems at the University of Aveiro, where he is now an Assistant Professor. His research interests include reconfigurable digital systems, software defined radio and next generation radio access networks. He participates in several national and European funded research projects. He is the author or co-author of more than 100 journal and international conference papers.

Francisco Madeiro was born in Fortaleza, Ceará, Brazil, in 1972. He received his B.Sc., M.Sc. and D.Sc. Degrees in Electrical Engineering from Federal University of Paraíba (UFPB), Brazil, in 1995, 1998 and 2001, respectively. Since 2006 he is with University of Pernambuco (UPE), Brazil, where he is Associate Professor. His main research interests include signal processing, communication systems and computational intelligence. He was recipient of the Distinguished Award in Teaching at Polytechnic School of Pernambuco (POLI), UPE, in 2008 and 2013. He was recipient of the Distinguished Award in Research at POLI, in 2013 and 2018.

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On the BEP analysis of M-QAM and R-QAM under cascaded double η-μ, κ-μ or α-μ fading channels

Abstract

Introduction

Section snippets

Related works

Problem characterization

Analysis of BEP without spatial diversity

Analysis of BEP with spatial diversity

Considerations about correlated cascaded double fading

Considerations about computational complexity

Results

Conclusions

CRediT authorship contribution statement

Data availability

Declaration of Competing Interest

Acknowledgements

On optimal relaying strategies for VANETs over double Nakagami-m fading channels

Wirel. Netw.

Performance analysis of full-duplex vehicle-to-vehicle relay system over double-Rayleigh fading channels

Mob. Netw. Appl.

On physical layer security of double Rayleigh fading channels for vehicular communications

IEEE Wirel. Commun. Lett.

Outage performance of cognitive cooperative networks with relay selection over double-Rayleigh fading channels

IET Commun.

Applying Hankel's expansion for performance analysis in double-Nakagami (generalized-K) fading channels

IEEE Trans. Commun.

Unified analysis of secrecy capacity over N⁎Nakagami cascaded fading channel

Statistical properties of double Hoyt fading with applications to the performance analysis of wireless communication systems

IEEE Access

On the double-generalized Gamma statistics and their application to the performance analysis of V2V communications

IEEE Trans. Commun.

M-ary phase-shift keying-based single-input-multiple-output free space optical communication system with pointing errors over a gamma-gamma fading channel

Appl. Opt.

Cascaded generalised-K fading channel

IET Commun.

N⁎Nakagami: a novel stochastic model for cascaded fading channels

IEEE Trans. Commun.

Diversity analysis of space-time coding in cascaded Rayleigh fading channels

IEEE Commun. Lett.

Unified performance analysis over Mckay-Meijer G shadowed fading channel

Int. J. Electron.

On the general BER expression of one and two-dimensional amplitude modulations

IEEE Trans. Commun.

Cascaded α−μ fading channels: reliability and security analysis

IEEE Access

Cascaded $α - μ$ fading channels: reliability and security analysis