1 Analysis of IRS-Assisted Downlink Wireless Networks over Generalized Fading

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Analysis of IRS-Assisted Downlink Wireless

Networks over Generalized Fading

Yunli LI, and Young Jin CHUN Member, IEEE

Abstract

Future wireless networks are expected to provide high spectral efﬁciency, low hardware cost, and scalable

connectivity. An appealing option to meet these requirements is the intelligent reﬂective surface (IRS), which

guarantees a smart propagation environment by adjusting the phase shift and direction of received signals. However,

the composite channel of IRS-assisted wireless networks, which is composed of a direct link and cascaded link

aided by the IRS, has made it challenging to carry out system design and analysis. This motivates us to ﬁnd

tractable and accurate channel modeling methods to model multiple types of channels. To this end, we adopt

mixture Gamma distributions to model the direct link, the cascaded link, and the mixture channel. Moreover, this

channel modeling method can be applied to various transmission environments with an arbitrary type of fading

as the underlying fading of each link. Additionally, a uniﬁed stochastic geometric framework is introduced based

on this tractable channel model. First, we derived distributions of the cascaded link and the mixture channel

by proving multipliability and quadratic form of mixture Gamma distributed channels. Then, we carried out a

stochastic geometric analysis of the system performance of the IRS-assisted wireless network with the proposed

channel modeling method. Our simulation shows that the mixture Gamma distributed approximation method

guarantees high accuracy and promotes the feasibility of system performance analysis of IRS-assisted networks

with complicated propagation environments, especially with a generalized fading model. Furthermore, the proposed

analytical framework provides positive insights into the system design regarding reliability and efﬁciency.

Index Terms

intelligent reﬂective surface, mixture Gamma distribution, cascaded channel, mixture channel, generalized

fading, stochastic geometry.

This work was supported in part by the Early Career Scheme under Project 9048208 established under the University Grant Committee

(UGC) of the Hong Kong Special Administrative Region (HKSAR), China; in part by the City University of Hong Kong (CityU), Startup

Grant 7200618; in part by the CityU, Strategic Research Grant 7005467; and in part by the RMGS Donation Grant 9229080. (Corresponding

author: Y. J. Chun).

Y. L. Li and Y. J. Chun are with the Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China. Y. J. Chun

is also with the Center for Internet of Things, City University of Hong Kong Dongguan Research Institute, Dongguan 523000, China.

(e-mail: yunlili2-c@my.cityu.edu.hk; yjchun@cityu.edu.hk)

arXiv:2210.02717v1 [cs.NI] 6 Oct 2022

I. INTRODUCTION

For 6G wireless communication, such as Terahertz (THz) systems, transformative solutions to a fully

connected world are expected to drive the surge for accommodating the complicated propagation environ-

ment, boosting spectral efﬁciency and providing high reliability. When the 6G system mitigating to higher

frequency, these requirements are huge challenges due to fast attenuation and weak penetration [1]. One

promising approach that emerged recently is the notion of an intelligent communication environment

(ICE). ICE is able to control the propagation environment to adapt to the complicated propagation

environment, enhance the reliability, and enlarge the coverage cost-effectively [2].

Various technologies have been proposed to achieve ICE, and one popular and practical solution is

the intelligent reﬂective surface [3], which is also known as reconﬁgurable intelligent surface (RIS) [4],

and large-scale intelligent surface (LIS) [5]. The IRS consists of a massive number of passive reﬂective

elements on its planar surface and a control part that adjusts each element’s phase shift and direction.

In contrast to traditional RF chains, the passive IRS elements only reﬂect signals without additional

active processing, which facilitates the IRS to be deployed easily and cost-efﬁciently. It is worth noting

that the passive-IRS potentially achieves a quantum leap improvement for self-interference and noise

ampliﬁcation compared to active relays and surfaces. In other words, IRS is a revolutionary technology

that can achieve high spectrum and energy efﬁciency communications with low costs [6]. Based on these

advantages, we will adopt passive-IRS in the sequel.

A. Related works

Spurred by the massive popularity of IRS, considerable researches have been undertaken in the latest

decades regarding each aspects of IRS. There are relatively sufﬁcient works about the link-level analysis

of IRS-assisted wireless communication systems [6]. In [7], the direct link from Base Station (BS) to

User Equipment (UE) was modeled as Rayleigh fading while links aided by IRS were modeled as

Rician fading, and the IRS worked with quasi-static phase shift design. In [8], the authors analyzed the

network performance of IRS-assisted two-way communications between two users over Rayleigh fading

by approximating the double Rayleigh fading with a Gamma distribution through moment matching.

In contrast, network-level research is still scarce. In [9], the network-level performance of IRS-assisted

downlink network was analyzed over Rayleigh fading by approximating the cascaded channel as Complex

Normal (CN) distribution through Central Limit Theorem (CLT). Additionally, Gamma distribution is

introduced to approximate the received signal power. However, the existing works are focused on simple

fading models, and the channel models on the cascaded link and the mixture channel are scarce.

B. Motivation

The previous system performance analysis mainly worked on Rayleigh fading due to its simplicity and

tractability [9]. Nonetheless, given the diverse range of operating environments of 6G, they may also be

subject to clustering of scattered multipath contributions, i.e., propagation characteristics which are quite

dissimilar to conventional Rayleigh fading environments [10]. Aside from small-scale fading, large-scale

fading and random shadowing caused by obstacles in the local environment or human body movements can

impact link performance via ﬂuctuating the received signals, which can not be ignored in future wireless

communications systems, i.e., mm-Wave wireless communications and THz wireless communications

[11]. As such, it is essential to extend the analysis of IRS-assisted wireless communication systems to

generalized fading channels with novel channel modeling methods.

Moreover, the mixture channel between typical UE and its serving BS consists of two types of link:

direct link (BS→UE), and cascaded link (BS→IRS→UE). Statistical characterization of the cascaded and

mixture channel in IRS-assisted networks involves highly specialized functions, such as Fox-H or Meijer

G-function, even with the simplest Rayleigh fading on each individual links, which causes the performance

analysis of IRS-assisted wireless network to be challenging. Considerable researches have been conducted

to analyze over asymmetric cascaded channels in relay-assisted networks: mixed Rayleigh and Rician

[12], mixed Nakagami-mand Rician [13], mixed η−µand κ−µfading channels [14]. Furthermore,

there are some approximation works on a symmetric cascaded fading channel in MIMO communications:

N*Nakagami-mdistribution for Nakagami-mfading channels [15]. In addition, [16] analyzed the dual-

hop link over generalized fading channels by leveraging properties of Meijer-G function. While signiﬁcant

advances have been made by previous researches, most of the existing literature approximated the cascaded

channels by CN distribution based on CLT or modeled the channels with Meijer-G function. Besides, the

system performance analysis is mainly based on the ratio of signal power and noise power (SNR), and

ignored the interference, which is an essential part in future dense networks. Although [9] has considered

the interference effect, the channel model adopted is still approximated by CN distributions through CLT,

with Rayleigh as the underlying fading model. Therefore, an approximation model for cascaded link and

mixture channel with high accuracy for generalized fading models, is critical for evaluating IRS-assisted

network system performance metrics of interest, especially for B5G and 6G.

C. Contributions

Motivated by the above, we emphasize addressing the modeling of cascaded link, mixture channel,

and system-level performance analysis for IRS-assisted wireless networks in this work. We extend the

research from Rayleigh fading to arbitrary underlying fading types, such as Nakagami-m, Rician, κ-µ,

and κ-µshadowed fading, which is a generalized channel modeling method ﬁtting to various networks.

We also evaluated the performance metrics with a uniform stochastic geometric framework. The main

contributions of this work are summarized as below:

1) First and foremost, modeling the channel gain tractably for the cascaded link and mixture channel

with high accuracy is essential for the analysis of IRS-assisted networks. In this work, we introduced

a general channel modeling method for multiple types of channels in IRS-assisted networks utilizing

the multipliability and quadratic form of the mixture Gamma distribution. Thus, we approximated the

direct channel, cascaded channel and mixture channel by mixture Gamma distributions with accuracy

less than 10−5. This mixture Gamma channel modeling method works for arbitrary underlying fading

and includes single channel, double channel, and mixture channel as a special case.

2) Then, we derived the distribution of conditional received signal power, and Laplace transform of

the aggregated interference using stochastic geometry under three operation modes: a) one IRS

is associated with the typical UE and performs beamforming whilst other related IRSs randomly

scattering the received signals; b) all related IRSs randomly scatter signals to the typical UE without

beamforming; c) there is no related IRS, and the whole network works as a traditional network.

3) Next, we introduced a uniﬁed analytical framework for the IRS-assisted network performance evalua-

tion based on the proposed mixture Gamma channel modeling method, where interested performance

metrics can be expressed as functions of the ratio of signal power and interference power plus noise

power (SINR). Furthermore, we illustrated several performance metrics, such as spectrum efﬁciency,

SINR moments, and outage probability by invoking their corresponding SINR functions.

4) Finally, we veriﬁed our channel model by Monte-Carlo simulation, which illustrated that the proposed

channel modeling method ﬁts well for multiple types of channel with high accuracy. As such, the

proposed modeling method can be applied to various wireless systems. Our analysis provides insights

on system design and further optimization of the IRS-assisted networks.

D. Organizations

The remaining paper is organized as below. In section II, we introduced the system model, association

policy, and channel models. In section III, we evaluated channel modeling method of the single link,

cascaded link, and mixture channel by proving the multipliability and quadratic form of mixture Gamma

distributed channels. In section IV, we derived the channel power gain and Laplace transforms of

the aggregated interference power under three operation modes and introduced a uniﬁed stochastic

geometric system performance analysis framework for the IRS-assisted network. In section V, we provided

simulations to verify our theoretical analysis. In section VI, we concluded the whole work.

II. SYSTEM MODEL

We consider an IRS-assisted multi-cell wireless network, where the IRSs are deployed to assist the

downlink transmission as shown in Fig. 1. The locations of BSs are modeled by an independent two

dimensional (2D) homogeneous Poisson Point Process (HPPP), denoted as ΛBwith node density λB. The

locations of IRSs and UEs are modeled as independent 2D-HPPPs, denoted as ΛIwith density λIand

ΛUwith density λU, respectively. Without loss of generality, we assume that a typical UE, denoted by

UE0, is located at the origin and each BS has an inﬁnitely backlogged queue. The channel is assumed to

be frequency-ﬂat and constant while the channel may vary over different frequency bands or time slots

[9]. To facilitate the analysis, we employ orthogonal multiple access, implying no intra-cell interference.

We summarized the common notations used in this paper in Table I.

A. BS and IRS association policy

We adopt a general association model for BS where each UE connects to the BS that provides strongest

long term received signal power without small-scale fading, denoted as BS0, which is equivalent to

connecting to the nearest BS. As such, PDF of the distance between BS0and UE0, denoted as dBU,

could be derived from the void probability of a 2D HPPP. The PDF of dBU is given by

fdBU (d) = 2πλBde−λBπd2.(1)

For the IRS association policy, we assume that at most one IRS is associated between UE0and BS0.

As [17] shows the optimal deployment location for a single associated IRS is in the vicinity of either UE0

or BS0. However, the communications suffer severe product path loss when the link distances between

nodes are too large. For this reason, we deﬁne a service area of each IRS, which is a circle with radius

D1. Further, we deﬁne an interference area, within which the not associated UEs can receive interference

signals from the IRS. The radius of this interference area is denoted as D2[9]. Since the deployment of

a large-scale centralized IRS is not practical, the association policy adopted in this work is connecting

the UE0with its nearest IRS located within service area, denoted as IRS0. Based on the distance to

UE0, the ΛIis thinned into three small point processes: the serving IRSs (denoted as ΛI,S,{IRS0}),

the interfering IRSs (denoted as ΛI,F), the noise IRSs (denoted as ΛI,N). As such, this IRS association

policy contains three operation modes in terms of the distance between UE0and its nearest IRS:

•Mode 1. If the distance between UE0and its nearest IRS is less than D1,UE0associates to its

nearest IRS.

•Mode 2. If the distance between UE0and its nearest IRS is larger than D1and less than D2,UE0

does not connect with any IRS. The IRSs, whose distance to UE0is less than D2, randomly scatter

any received signals, which contribute to the interference.

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1AnalysisofIRS-AssistedDownlinkWirelessNetworksoverGeneralizedFadingYunliLI,andYoungJinCHUNMember,IEEEAbstractFuturewirelessnetworksareexpectedtoprovidehighspectralefciency,lowhardwarecost,andscalableconnectivity.Anappealingoptiontomeettheserequirementsistheintelligentreectivesurface(IRS),whichgua...

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