On the performance of DF-based multi-hop system over α − κ − μ and α − κ − μ-extreme fading channels

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Abstract

Several emerging application scenarios for the multi-hop wireless networking have been introduced in 5G and beyond networks, to provide ultra-low latency and much higher data rate to the end users. In this work, an integrated performance evaluation of a decode-and-forward (DF) multi-hop wireless communication system is undertaken over the non-linear generalized ακμ and ακμ-Extreme fading models. Novel closed-form expressions for the amount of fading (AoF), channel capacity under various adaptive transmission techniques are derived. Additionally, decoding error probability for short packet communication is derived and analyzed. The provided expressions are very useful in designing the channel/link quality for practical multi-hop communication networks. The effects of the fading parameters and the number of hops on the system performance are demonstrated. The correctness of the proposed solutions has been corroborated by comparing them with Monte Carlo simulation results.

Introduction

While wireless communication will be the core technique, a direct communication between the transmitter and the receiver is faced with many limitations. In particular, communication over long distances is either possible through prohibitively high transmission power or using multi-hope relaying techniques. The use of multi-hope relaying increases the network coverage area with high achievable data rate. Several emerging application scenarios for the multi-hop wireless networking have been found in 5G and beyond technologies. For example, in device-to-device communication, the transmitted information reaches its final destination via several intermediate devices, known as relays [1]. DF-based multi-hop systems have several other applications in ad-hoc networks, sensor networks and microwave links among many others [2]. The main idea behind multi-hope communication is that the channel from a source node to a destination node is split into multiple possibly shorter links by using some intermediate nodes. Based on the nature and complexity of the relaying system, multi-hop communication systems can be classified as (i) decode-and-forward (DF) and (ii) amplify-and-forward [1]. For a DF-based multi-hop system, the relaying node receives the encoded signal, decodes it to regenerate the original symbol and forwards the newly encoded signal to the next node [2]. The modeling of mobile radio channel is a challenging task as the wireless signal is subjected to various hurdles/corruptions in its propagation path from the transmitter to the receiver [3]. Accurate characterization of the wireless channel is therefore of paramount importance for the realistic assessment of wireless system performance.

Several statistical generalized distributions like αηκμ, αημ, ακμ, αμ, ημ and κμ etc. are developed in the literature to model small-scale fluctuations of the fading channel envelope. Recently, the effective capacity analysis over αηκμ fading channel has been proposed as a generalized solution for almost all of the multi-path fading channels [4]. The secrecy performance over correlated αμ distribution has been proposed in [5], while the channel capacity and energy detection performance over αημ distribution have been presented in [6] and [7]. The κμ and ακμ are the widely accepted models, which accounts for the signal propagation in line-of-sight (LOS) communications. In [8], the error probability for an uplink communication over non-homogeneous κμ generalized fading channels is investigated. Different performance metrics over shadowed κμ composite fading channel is discussed in [9], [10] and [11]. Motivated by the pioneering works in [12] and [13] the generalized ακμ and ακμ-Extreme fading models were proposed in [14] and [15]. The importance of these models lies in the fact, that these channels can accommodate most of the practical fading models as special cases. In particular, when α=2, the ακμ distribution reduces to the κμ distribution [12] and for κ0, the αμ is obtained [16]. From the κμ and αμ, models, one can obtain other distributions including the Nakagami-m, Weibull, Rice, one-sided Gaussian and Rayleigh models for particular values of α, κ or μ. In [17], the performance of a dual-branch selection combining (SC) over ακμ fading channels is evaluated. The spectrum sensing capability of a cooperative system over ακμ and ακμ-Extreme fading channels is investigated in [18]. The work in [19] and [20] have studied the energy detector performance over generalized shadowed composite fading distributions. The ακμ-Extreme fading model is derived from the ακμ distribution by allowing μ0 and κ in order to better characterize small-scale variations of mobile radio propagation under non-linear severe multi-path fading effects usually encountered in enclosed environments [15]. For α=2, the ακμ-Extreme distribution reduces to a linear severe fading model, the κμ-Extreme distribution [13].

The performance of DF-based relaying wireless systems and networks has been extensively investigated under several fading distributions. The channel capacity of a multi-hop relaying system over Rayleigh fading channels has been studied in [21] whereas the end-to-end performance of a DF relaying system over Ricean and Nakagami-m fading channels is explored in [22]. The authors in [23] investigated the bit-error-rate performance of a DF multi-hop cognitive networks over Rayleigh fading channel. The performance of multi-hop relaying in industrial wireless sensor networks over Nakagami-m fading channels is presented in [24]. The work in [25] has studied the system performance of multi-hop relay network in a quasi-static slow fading environment. Recently the outage probability and bit error rate performance of a DF-based multi-hop system over the generalized fading channels has been performed in [26]. Although there are many works related to the study of DF-based multi-hop relaying systems over different fading channels. None of the previous works has investigated the end-to-end channel capacity and decoding error probability over ακμ and ακμ-Extreme fading channels, which represent our key contribution. It is important to study the performance metrics of the multi-hop wireless systems over the generalized fading models. Such investigation is essential in order to implement practical wireless system realizations with predefined performance levels. Motivated by this void, we have derived the closed-form mathematical expressions for amount of fading (AoF), channel capacity under various adaptive transmission schemes and decoding error probability for short packet communication for end-to-end DF relaying system over ακμ and ακμ-Extreme fading channels. The expressions derived here are presented in a simple manner to optimize clarity and readability. Furthermore, the results produced here are generalized expressions and valid for other fading distributions as special cases.

The rest of the paper is organized as follows: The system and channel model are described in Section 2. In Section 3, we derived the closed-form expressions of performance metrics like AoF, decoding error probability and channel capacity under different transmission policy for the DF-based multi-hop wireless system over ακμ and ακμ-Extreme fading channels. The asymptotic series expansion of decoding error probability is presented in Section 4. Performance evaluation results and discussions are provided in Section 5. Finally, we round up the paper in Section 6 with conclusions and possible future considerations.

Section snippets

System and channel models

We consider a wireless relay network in which a source node S transmits its data packets to the destination node D through n1 relay devices operating over ακμ and ακμ-Extreme fading channels. The n+1 nodes (S, D, and the intermediate relay) are assumed to be sufficiently and equally spaced. Hence, a device can only receive the transmitted data of its previous-hop adjacent node. Intermediate terminals always perform hard decisions on the received symbols before forwarding them to their

Amount of fading

The second-order AoF is a useful performance metric that can be used to account for the severity of fading. AoF is defined as the ratio of the variance to the square average SNR per symbol, AoFγ(2)Var(γ)/γ¯2. That is,AoFγ(2)Eγ2Eγ2Eγ2. This can be generalized to higher-order fading as AoFγ(n)Eγn/(Eγn)1. Substitution of Eγ and Eγ2 from (8) into (12) yields after some basic simplifications the second-order AoF of γ under Case I asAoFγ=eμnκnΓ(4αn+μn)Φ˜(μn+4αn;μn;κnμn)A¯Γ(2αn+μn)2Φ

Poincare asymptotic series expansion

In this section, the Poincare asymptotic series expansion of the decoding error probability for both the ακμ and ακμ-Extreme fading models shall be discussed. The PDFs of the SNRs for the nth hop of Case I and Case II fading channels have Poincare asymptotic series expansion near the origin and is given by [26] aspγ(γ)˙Aδ(γ)+A¯Kn(θn4)ωn1k=0N1dkγtk+O(γt+N1) where O denotes the order of a term in the asymptotic series anddk=j=0k(1)jσnj(θn4)kjj!(kj)!Γ(k+μnj),tk=αn(k+μn)/21.pγe(γ)˙Ae

Numerical experiments and simulations

Though not a requirement, the performance analyses of the present model assume that the n hops of the DF-based multi-hop system are independent and identically distributed (IID) ακμ and ακμ-Extreme variates for Case I and Case II, respectively. Furthermore, we assume that all the nodes between S and D are equally spaced. Following the standard set by [2], [37], [16] and other similar works, the maximum distance between S and D is taken as nmax=5, upon normalizing by the distance of one hop.

Conclusions

In this paper, various performance measures of multi-hop system over generalized ακμ and ακμ-Extreme fading channels have been examined in detail. Using the PDF-based approach, we have provided the closed-form expressions for the amount of fading, channel capacity under various adaptive schemes and decoding error probability for short packet communication for an end-to-end DF-based multi-hop relaying system. The derived expressions can be used to investigate the performance of other fading

CRediT authorship contribution statement

It is stated that all the authors have equal contribution in the preparation of this manuscript.

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.

Tau Raphael Rasethuntsa received his B.Sc. in Mathematics and Statistics from the National University of Lesotho, Maseru, Lesotho in 2014 and M.Sc. in Mathematical Engineering from Istanbul Technical University, Istanbul, Turkey in 2017. He is currently pursuing his PhD at McMaster University, Hamilton, Ontario, Canada. His research interests include reliability, applied statistics as well as mathematical and statistical aspects of signal processing with special focus on performance evaluation

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  • Cited by (5)

    Tau Raphael Rasethuntsa received his B.Sc. in Mathematics and Statistics from the National University of Lesotho, Maseru, Lesotho in 2014 and M.Sc. in Mathematical Engineering from Istanbul Technical University, Istanbul, Turkey in 2017. He is currently pursuing his PhD at McMaster University, Hamilton, Ontario, Canada. His research interests include reliability, applied statistics as well as mathematical and statistical aspects of signal processing with special focus on performance evaluation of wireless communication systems.

    Manpreet Kaur received her B.Tech. in Electronics and Communication Engineering from Punjab Technical University, Punjab, India in 2005 and Master of Engineering in Electronics and Communication from Thapar University, Patiala, India in 2007. She is pursuing her PhD. from Delhi Technological University, Delhi, India. She is currently working as Member (Senior Research Staff) at Central Research Laboratory, Ghaziabad, Bharat Electronics Limited. Her research interest includes the wireless communication and cognitive radios networks.

    Sandeep Kumar received his B.Tech. in electronics and communication from Kurukshetra University, India in 2004 and Master of Engineering in Electronics and Communication from Thapar University, Patiala, India in 2007. He received his PhD. from Delhi Technological University, Delhi, India in 2018. He is currently working as Member (Senior Research Staff) at Central Research Laboratory, Bharat Electronics Limited Ghaziabad, India. His research interests include the study of wireless channels, performance modeling of fading channels and cognitive radio networks. He is also serving as a reviewer for several international journals of IEEE, Springer, Elsevier etc.

    Puspraj Singh Chauhan was born in UP, India. He received his Bachelor of Technology in Electronics and Communication Engineering from Uttar Pradesh Technical University, India in 2009, the M.Tech. degree in Digital Signal Processing from Govind Ballabh Pant Engineering College, Pauri, Uttarakhand, India in 2011. Currently he is working towards the PhD. degree in Electronics and Communication Engineering from Govind Ballabh Pant Engineering College, Pauri, Uttarakhand, India. His research interests include Wireless Communication and Propagation Channel Modeling.

    Kuldeep Singh received his M.Tech. degree in Signal Processing from Delhi University in 2006 and Ph.D. degree in Computer Vision from Delhi Technological University, India, in 2016. He is currently an Assistant Professor with the Dept. of Electronics and Comm. Engg. Malaviya National Institute of Technology, Jaipur, India. Previously, he was a Senior Scientist with the Central Research Lab, Bharat Electronics Ltd., India. He also worked as a postdoctoral fellow at University of Alberta, Canada from Oct 2017 to Apr 2018. His research interest includes Computer Vision, Deep learning applications, Crowd Behaviour Analysis, and Biometrics.

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