I Introduction
As a promising technique for supporting skyrocket data rate in the fifthgeneration (5G) cellular networks, millimeter wave (mmWave) communications have received an increasing attention due to the large available bandwidth at mmWave frequencies [1]. In recent years, research on channel modeling for mmWave wireless communications has been intense both in industry and academia [2, 3]. Based on recent smallscale fading measurements of the GHz outdoor millimeterwave channels [3], the fluctuating tworay (FTR) fading model has been proposed as a versatile model that can provide a much better fit than the legacy Rician fading model.
However, one of the fundamental challenges of mmWave communication is the susceptibility to blockage effects. Thus, hindrances still occur due to the existence of buildings, trees, cars, and even human body. To address this problem, a typical solution is to add new supplementary links. For example, the amplifyandforward (AF) relay can be introduced in areas to receive the weak signal and then amplify and retransmit it toward the destination [4]. Alternatively, reconfigurable intelligent surfaces (RISs), comprised of many reflecting elements, have recently drawn significant attention due to their superior capability in manipulating electromagnetic waves [5]. Taking advantage of cheap and nearly passive RIS attached in facades of buildings, signals from the base station (BS) can be retransmitted along desired directions by tuning their phases shifts, thereby leveraging the lineofsight (LoS) components between the RIS and users to maintain good communication quality.
Obviously, RIS and relay operate in different mechanisms to provide supplementary links. With the help of RIS, the propagation environment can be improved because of extremely low power consumption without introducing additional noise, but the incident signal at the reflector array is reflected without being amplified. Thus, whether the RIS is more economical and efficient than the relay is still controversial. In [6], a comparison between RIS and an ideal fullduplex relay was made and it was found that large energy efficiency gains by using an RIS, but the setup is not representative for a typical relay. Besides, authors in [7] made a fair comparison between RISaided transmission and conventional decodeandforward (DF) relaying, with the purpose of determining how large an RIS needs to be to outperform conventional relaying, and it is found that a large number of reflecting elements are needed to outperform the DF relaying in terms of minimizing the total transmit power and maximizing the energy efficiency. However, previous works did not build on a versatile statistical channel model that well characterizes wireless propagation in mmWave communications to derive performance metrics.
In this paper, we aim to answer the significant question “How can a RIS outperform AF relaying over realistic mmWave channels?”. Using the FTR fading channel model, we derive novel exact expressions to analyze the system performance for both systems. The main contributions of this paper are summarized as follows:

First, we derive the exact probability density function (PDF), cumulative distribution function (CDF), and generalized moment generating function of a product of independent but not identically distributed (i.n.i.d.) FTR random variables (RVs) and the sum of product of FTR RVs. These statistical characteristics are useful in many communication scenarios, such as multihop wireless communication systems
[8] and keyhole channels of multipleinput multipleoutput (MIMO) systems [9]. 
To provide a fair comparison between RISaided transmission and AF relaying, we propose an optimal power allocation scheme for AF relay systems, and we further use a binary search tree method to obtain optimal phases at the RIS. Besides, the convergence of the proposed phase optimization method is investigated.

We derive a novel generic single integral expression for the CDF of the endtoend signaltonoise ratio (SNR) of AF systems by considering not identically distributed hops and hardware impairments. For the cases of nonideal and ideal hardware, we obtain the exact PDF and CDF expressions of the endtoend SNR.

Assuming ideal transceiver hardware, we provide a fair comparison between RISaided system and AF relay aided communication system. The exact outage probability (OP) and average bit error probability (ABEP) expressions are derived for both scenarios to obtain important engineering insights. It is interesting to find that the RISaided system can achieve the same OP and ABEP as the AF relay system with less reflecting elements if the transmit power is low. More importantly, as the channel conditions improve, the RISaided system achieves more ABEP reduction than the AF relay aided system having the same transmit power.
The remainder of the paper is organized as follows. In Section II, we briefly introduce the RISaided and AF relay systems, and derive exact statistics for the endtoend SNR of both systems. Section III presents the optimal phase shift of the RIS’s reflector array and the optimal power allocation scheme for the AF relay system. Performance metrics of two systems, such as OP and ABEP, are carried out in Section IV. In Section V, numerical results accompanied with MonteCarlo simulations are presented. Finally, Section VI concludes this paper.
Ii System Model And Preliminaries
Iia System Description
We focus on a single cell of a mmWave wireless communication system and analyze two scenarios: when RIS is adopted and when AF relay is used, as shown in Fig. 1 and Fig. 2 respectively. To characterize the performance of the considered mmWave communication system, we assume that the global channel state information (CSI) is perfectly known at the BS, RIS and relay.^{1}^{1}1An efficient CSI acquisition method for RISaided mmWave networks has been proposed [10]. Besides, the results in this paper serve as theoretical performance upper bounds for the considered system. In Fig. 1, the RIS is set up between the BS and the user, comprising reflector elements arranged in a uniform array. In addition, the reflector elements are configurable and programmable via an RIS controller. In Fig. 2, a halfduplex AF relay is deployed at the same location as the RIS.^{2}^{2}2The RISaided system has been compared with the classic DF relaying in deterministic flatfading channel where all channel gains have the same squared magnitude [7]. We consider the classic repetition coded AF relaying protocol [11] where the transmission is divided into two equal phases.
More specifically, we assume that the direct transmission link between the BS and the user is blocked by trees or buildings. Thus, the RIS or relay is deployed to leverage the LoS paths to enhance the quality of received signals. In addition, the fluctuating tworay (FTR) fading model has been proposed in [3] as a generalization of the twowave with diffuse power fading model. The FTR fading model allows the constant amplitude specular waves of LoS propagation to fluctuate randomly, and incorporates ground reflections to provide a much better fit for smallscale fading measurements in mmWave communications [12]. In the following, we use the FTR fading model to illustrate the mmWave links.
IiB Exact Statistics of the EndtoEnd SNR For RISaided System
In this subsection, we present some useful statistical expressions which are useful for the performance analysis of RISaided system.
IiB1 Exact PDF and CDF of the Product of FTR RVs
The PDF and CDF of the squared FTR RV are given respectively as [12]
(1) 
(2) 
where
(3) 
(4) 
and
(5) 
where is the incomplete gamma function [13, eq. (8.350.1)], is the Legendre function of the first kind [13, eq. (8.702)], denotes the average power ratio of the dominant wave to the scattering multipath, is the fading severity parameter, and is a parameter varying from to representing the similarity of two dominant waves. In addition, the received average SNR is given by .
Substituting into (1), we can easily obtain the PDF expression of FTR RVs. Thus, let as the product of FTR RVs, where (), we can derive the exact moment, PDF and CDF expressions for , which are summarized in the following Theorems.
Theorem 1.
The moment of is given by
(6) 
Proof:
Theorem 2.
Proof:
Please refer to Appendix A.∎
Remark 1.
Although the PDF, CDF and MGF of a product of i.n.i.d. squared FTR RVs have been respectively derived in [15, eq. (7)], [15, eq. (16)] and [15, eq. (17)]. Only the parameter of each FTR RV is different from each other. In the practical mmWave communication scenario, all parameters and of each FTR RV can be different, thus the statistical characterizes obtained in Theorem 2 are more general and useful in the performance analysis.
Remark 2.
Substituting into (8) and (9), we can obtain more general statistical expressions of the product of an arbitrary number of i.n.i.d. squared FTR RVs. Besides, note that the bivariate Meijer’s function can be evaluated numerically in an efficient manner using the MATLAB program [16], and two Mathematica implementations of the single Fox’s function are provided in [17] and [18].
IiB2 Exact PDF and CDF of the Sum of Product of FTR RVs
We derive the exact PDF and CDF of the sum of product of FTR RVs in terms of multivariate Fox’s function [14, eq. (A1)], which are given in the following Theorem.
Theorem 3.
We define . Thus, the PDF and CDF of can be deduced in closedform as
(11) 
(12) 
where .
Proof:
Please refer to Appendix B.∎
Remark 3.
Although the numerical evaluation for multivariate Fox’s function is unavailable in popular mathematical packages such as MATLAB and Mathematica, its efficient implementations have been reported in recent literature.. For example, a Python implementation for the multivariable Fox’s function is presented in [19], and an efficient GPUoriented MATLAB routine for the multivariate Fox’s function is introduced in [20]. In the following, we will utilize these novel implementations to evaluate our results.
IiB3 SNR Analysis
The channel coefficients between the BS and the RIS are denoted by an vector , where the elements of are i.n.i.d. FTR RVs. However, under the FTR model, because the complex baseband voltage of a wireless channel experiencing multipath fading contains two fluctuating specular components with different phases, the elements of will have different phases. Besides, the angle of departure (AoD) and angle of arrival (AoA) of a signal will also cause phase difference [21]. Thus, can be written as
(13) 
where denotes the amplitude of the channel coefficient and is the corresponding phase. Similarly, the channel coefficients between the RIS and the user can be expressed as an vector , and the elements are also i.n.i.d. FTR RVs having different phase shifts. The channel coefficient rector can be expressed as
(14) 
where represents the amplitude of the channel coefficient and denotes the corresponding phase. Then, we focus on the downlink of the RISaided system. The signal received from the BS through the RIS for the user is given by
(15) 
where is the transmit power, is the beamforming vector satisfying , is transmit signal satisfying , denotes the additive white Gaussian noise (AWGN) at the user, , is the fixed amplitude reflection coefficient^{3}^{3}3In practice, each element of the RIS is usually designed to maximize the signal reflection. Thus, we set for simplicity. and are the phase shifts which can be optimized by the RIS controller.
Using the maximum ratio transmitting at BS, we can define as
(16) 
Accordingly, the SNR of RISaided system is given by
(17) 
Before designing the phase shift, we can theoretically obtain the maximum of with the optimal phase shift design of RIS’s reflector array as
(18) 
Corollary 1.
The PDF and CDF of the endtoend SNR for RISaided system, , can be derived as
(19) 
(20) 
IiC Exact Statistics of the EndtoEnd SNR For AF Relay System
We consider the classic AF relay protocol where the transmission is divided into two equalsized phases. The transmit power of BS is in the first phase, and one of AF relay is in the second phase. Assuming variable gain relays, the endtoend SNR of AF relay communication system is given as [11]
(21) 
where and . Eq. (21) can be tightly approximated at medium and high SNRs as [22, eq. (9)]
(22) 
For ideal hardware, i.e. and , Eq. (22
) reduces to the halfharmonic mean of
as [23](23) 
Assuming that , we can rewrite (22) as
(24) 
where and .
An integral representation for the PDF of in (22) assuming arbitrarily distributed and considering hardware impairments is given by
(25) 
To derive a generic analytical expression for the CDF of the endtoend SNR of AF relay systems with hardware impairments, we first present the following useful Lemma.
Lemma 1.
An integral representation for the CDF of in (22) assuming arbitrarily distributed and considering hardware impairments is given by
(26) 
Proof:
Please refer to Appendix C. ∎
For the special case of FTRdistributed hops, closedform expressions will be derived for both cases of nonideal and ideal hardware.
Theorem 4.
The PDF and CDF of considering hardware impairments and FTRdistributed hops can be deduced in closedform as
(27) 
(28) 
where , and
(29) 
Proof:
Please refer to Appendix D. ∎
To compare the AF relay system with the RISaided system, we consider the ideal hardware to make a fair comparison.
Corollary 2.
For the special case of ideal hardware, the PDF and CDF of can be deduced in closedform as
(30) 
(31) 
IiD Truncation Error
To show the effect of infinite series on the performance of the CDF expression of the sum of product of FTR RVs, truncation error is presented in the following. By truncating (3) with the first terms, we have
(32) 
The truncation error of the area under the with respect to the first terms is given by
(33) 
Lemma 2.
Proof:
To demonstrate the convergence of the series in (34), Table I depicts the required truncation terms for different system and channel parameters. With a satisfactory accuracy (e.g., smaller than ), only less than terms are needed for all considered cases. We also note that the number of truncation terms slightly increases with the number of RVs .
Iii Phase Shift and Power Optimization
In this section, we propose an optimal design of to maximize the SNR by exploiting the statistical CSI of the RISaided system. We also present an optimal power allocation scheme for the AF relay system.
Iiia Optimal Phase Shift Design of RIS’s Reflector Array
To characterize the fundamental performance limit of RIS, we assume that the phase shifts can be continuously varied in , while in practice they are usually selected from a finite number of discrete values to simplify the circuit implementation [24].
Based on (17), maximizing the SNR is equivalent to maximizing . Using (13) and (14), we can rewrite as
(37) 
where is the adjustable phase induced by the reflecting element of the RIS. The optimal choice of can maximizes the instantaneous SNR. Thus, the optimal satisfies
(38) 
According to [25, eq. (12)], it is easy to infer that is maximized by eliminating the channel phases (similar to cophasing in classical maximum ratio combining schemes), i.e., the optimal choice of that maximizes the instantaneous SNR is for all .
However, this solution, notably, requires that the channel phases are perfectly known at the RIS. How to perform channel estimation in RISaided system is challenging, because the RIS is assumed to be passive, as opposed to, e.g., the AF relay. Moreover, in
[21], the BS is assumed to be equipped with a large uniform linear array; therefore, the problem is transferred to measure the AoD and AoA at the RIS. However, the authors in [21] did not obtain a feasible solution to this problem. Herein, we propose a novel and simple algorithm based on the binary search tree [26]. Note that the maximum achievable expectation of the amplitude of the received signal can be expressed as(39) 
where can be calculated with the aid of (6).
Let and , and we notice that can be easily measured in the time domain and can be calculated directly using the CSI. Thus, when initially setting up the RIS, using the ideal and sent from the user, we use the binary search tree algorithm to adjust the reflection angle of the elements on the RIS one by one. For each element, we perform times of search. After searching all elements, we repeat the same operation times. The entire algorithm flow is shown in Fig. 3. This solution does not compromise on the almost passive nature of the RIS.
Combining the above discussion, we present our method to find as Algorithm 1, where denotes the remainder on division of by . Thus, with the optimal phase shift design of RIS’s reflector array, we can obtain the maximum of as (18).
To investigate the convergence of the proposed phase optimization method, we define the error of expectation as
and use the variance of
to characterize the phase error.
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