Outage Probability Expressions for an IRS-Assisted System with and without Source-Destination Link for the Case of Quantized Phase Shifts in $\kappa-\mu$ Fading

In this work, we study the outage probability (OP) at the destination of an intelligent reflecting surface (IRS) assisted communication system in a $\kappa-\mu$ fading environment. A practical system model that takes into account the presence of phase error due to quantization at the IRS when a) source-destination (SD) link is present and b) SD link is absent is considered. \cbendFirst, an exact expression is derived, and then we derive three simple approximations for the OP using the following approaches: (i) uni-variate dimension reduction, (ii) moment matching and, (iii) Kullback–Leibler divergence minimization. The resulting expressions for OP are simple to evaluate and quite tight even in the tail region. The validity of these approximations is demonstrated using extensive Monte Carlo simulations. \cbstartWe also study the impact of the number of bits available for quantization, the position of IRS with respect to the source and destination and the number of IRS elements on the OP for systems with and without an SD link. \cbend

Intelligent reflecting surface (IRS) assisted communication has gained much research momentum recently [1]. Reconfigurable IRS realized using arrays of passive antenna elements or scattering elements made from metamaterials can introduce specific phase shifts on the incident electromagnetic signal without any decoding, encoding, or radiofrequency processing operations [2, 3]. \cbstartAppropriate phase shifts introduced on a large number of reflector elements can create a coherent combination of their individually scattered signals, which is narrow and focused at the user [3]. This strategy is known as energy focusing [3, 4]. \cbendTraditionally, reliable communication links were achieved by implementing intelligent transmitter and receiver designs that combat signal deterioration’s introduced by the propagation environment. However, in the past few years, there has been a shift in this paradigm towards the idea of smart radio environments (SRE) where the performance gains achievable via ‘smartly’ modifying the wireless propagation channel are explored [1].

Several works in literature study the performance of SRE realized using IRS and compare them with the performance achievable using techniques like cooperative relaying, massive multiple-input multiple-output (M-MIMO), distributed antennas, backscatter communication, millimetre (mm)-wave communication, and network densification [5, 6, 7, 8, 9]. The authors of [6] show that an IRS-assisted M-MIMO system can use the same channel estimation overhead as an M-MIMO system with no IRS to achieve a higher user signal to interference and noise ratio (SINR). The authors of [10] study the performance gains achieved by an IRS-assisted M-MIMO system integrated with a non-orthogonal multiple access (NOMA) network. They also discuss the critical challenges in realizing an IRS-aided NOMA network. The key differences and similarities between an IRS-aided network and a relay network are studied by the authors of [5]. Using mathematical analysis and numerical simulations, they demonstrate that a sufficiently large IRS can outperform relay-aided systems in terms of data rate while reducing the implementation complexity. The number of reflector elements required to outperform the performance of the decode and forward (DF) relay system is studied by the authors of [11]. IRS-aided bistatic backscatter communication (BackCom) system is studied in [7]. Notably, the joint optimization of the phase shifts at the IRS and the transmit beamforming vector of the carrier emitter that minimizes the transmit power consumption is studied. Another exciting venue where IRS has useful applications are the mm-wave communication systems [8, 12, 9]. Authors of [8] proposed three different low-cost architectures based on IRS for beam index modulation schemes in mm-wave communication systems. These schemes are capable of eliminating the line-of-sight blockage of millimetre wave frequencies. \cbstart The authors of [12] modeled the small scale fading in an mm-wave communication system using the fluctuating two-ray (FTR) distribution and studied the outage probability (OP) and average bit error rate performance of both an IRS aided system and an amplify and forward (AF) relay-based system. They demonstrate the superiority of the IRS aided systems over AF systems in the mm-wave regime even with a small number of reflecting elements. \cbendThe authors of [9] also study the prospects of combating issues in mm-wave systems like severe path loss and blockage using an IRS to provide effective reflected paths and hence enhance the coverage.

Most of the works discussed above consider performance metrics such as OP, rate and spectral efficiency to evaluate the performance of the IRS-aided communication network. Several works in the literature propose different approximations for the OP of an IRS aided system involving one source and one destination node. The authors of [13] assume the availability of a large number of reflector elements and hence use the central limiting theorem (CLT) to derive an approximate expression for the OP. Similarly, the authors of [14] also derive an approximation for the OP using CLT. Unlike the system model in [13], they assume that the direct link between the source and the destination is in a permanent outage. Such Gaussian approximations are also used to characterize metrics like ergodic capacity, secrecy outage probability in many other works, including [15, 16, 17, 18]. Similarly, Gamma approximations (using moment matching) are used for deriving approximate OP by the authors of [19, 20, 21, 22]. As mentioned by the authors of [22], the Gamma distribution is a Type-III Pearson distribution and is widely used in fitting distributions for positive random variables (RVs) [23, 24]. The authors of [15, 18, 25, 20] consider more practical IRS models, where due to hardware constraints, the possible phase shifts at the IRS elements are restricted to a finite set of discrete values. \cbstartMost of the above works consider Rayleigh fading channels for analytical tractability. However, the IRS can be deployed at a height such that a LOS link may be available with the source or the destination or both. The authors of [13] consider one such scenario where both the links with the source and the destination experience Rician fading. In this work, we consider the more general scenario where all the links experience independent $\kappa-\mu$ fading. Most of the common fading models like the Rayleigh, Rician, Nakagami$-m$ can be derived as special cases of the $\kappa-\mu$ fading model [26]. \cbend

Table I provides a brief summary of the critical literature on this topic²²2Note that the authors of [22] also use Gamma approximation. However, they consider a different path loss model for a system without an SD link and hence, neither we include [22] in Table I, nor do we compare our performance with their results.. Here, the antenna model refers to the antenna model of the source and destination pair devices. From the table, it is apparent that the study of OP considering both $b$ bit phase quantization at the IRS and an active source-destination (SD) link is not available in the open literature.

In this work, we present an exact expression and three different approximations for characterizing the OP at the destination of an IRS-assisted communication system in a $\kappa-\mu$ fading environment. \cbendHere, we consider a practical scenario where the phase shift at the IRS elements only takes a finite number of possible values owing to the quantization of the phase at the IRS. Furthermore, we evaluate the system’s performance both in the presence and absence of a direct link between the source and the destination node. Our major contributions are summarised as follows: \cbstart

• •

  We study the OP of an IRS assisted system in the presence of phase error due to quantization at the IRS in a $\kappa-\mu$ fading environment.

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  We derive an exact expression for the OP in terms of a multi-fold integral and approximate it using the uni-variate dimension reduction method.

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  Using the method of moment matching³³3The authors of [20] and [19] also uses moment matching to obtain a Gamma approximation, however in scenarios without an SD link and without quantization error, respectively. Our approximation for the received SNR is for a more general scenario, and it recovers the result in [20] as a special case., we approximate the received SNR as a Gamma RV and hence derive a simple expression for the corresponding OP.

• •

  We also derive the parameters of the Gamma distribution that has the least Kullback–Leibler (KL) divergence with the exact distribution of SNR⁴⁴4The Gamma distribution, which has minimum KL divergence with respect to the distribution of the received SNR cannot be obtained by moment matching.. We thus characterize the OP in terms of the cumulative distribution function (CDF) of the resulting Gamma RV.

• •

  We also observed the impact of the number of IRS elements, location of the IRS and the number of bits available for quantization on the OP for an IRS aided system with and without an SD link.

We consider a system consisting of one source node ($\mathbf{S}$) communicating with one destination node ($\mathbf{D}$) using an $\mathbf{IRS}$ with $N$ reflector elements as shown in Fig. 1. Here $\mathbf{S}$ and $\mathbf{D}$ are both equipped with a single antenna each. Furthermore, we assume that the distance between $\mathbf{IRS}$ and $\mathbf{S/D}$ is large enough such that all elements of the $\mathbf{IRS}$ are at the same distance from $\mathbf{S/D}$. Let, $\mathbf{h}^{SR}\in\mathbb{C}^{N\times 1}$, $\mathbf{h}^{RD}\in\mathbb{C}^{N\times 1}$ and ${h}^{SD}\in\mathbb{C}^{1}$ denote the small-scale fading channel coefficients of the $\mathbf{S}$ to $\mathbf{IRS}$, $\mathbf{IRS}$ to $\mathbf{D}$ and $\mathbf{S}$ to $\mathbf{D}$ link respectively. \cbstartIt is assumed that all the channels experience independent $\kappa-\mu$ fading, i.e $[\mathbf{h}^{AB}]_{n}$ follows the $\kappa-\mu$ distribution with parameters $\kappa_{AB}$ and $\mu_{AB}$ where $A,B\in\{S,R,D\}$. Also the total power of each channel, i.e., $\mathbb{E}\left[|\mathbf{h}^{AB}|^{2}\right]=d_{AB}^{-\beta}:=\hat{t}^{2}_{AB}$ Let $\alpha$ and $\theta_{n}$ represent amplitude coefficient and phase shift introduced by the $n$-th $\mathbf{IRS}$ element, respectively. The signal received at node $\mathbf{D}$ is then given by, $y=\sqrt{p}\left({h}^{SD}+\alpha\left(\mathbf{h}^{SR}\right)^{T}\mathbf{\Theta}\mathbf{h}^{RD}\right)s+w,$ where $\mathbf{\Theta}=\ \text{diag}\left(e^{j\theta_{1}},...,e^{j\theta_{N}}\right)$, $p$ is the transmit power, $s$ is the transmitted signal with $\mathbb{E}$[$|s|^{2}$]=1 and $w$ is the AWGN with noise power $\sigma^{2}$. The SNR at the node $\mathbf{D}$ of the IRS-supported network is then given by $\gamma_{IRS}=\gamma_{s}\Big{|}{h}^{SD}+\alpha\left(\mathbf{h}^{SR}\right)^{T}\mathbf{\Theta}\mathbf{h}^{RD}\Big{|}^{2},$ where $\gamma_{s}=\frac{p}{\sigma^{2}}$. To achieve maximum SNR at $\mathbf{D}$, the phase-shift of the $n$-th IRS element needs to be selected as follows,

Note that such a choice of phase shifts that maximise the SNR is very common in literature and is also used by the authors of [18, 20, 11]. \cbendLet $b$ be the number of bits used to represent the phase. Then the set of all possible phase shifts at each of the IRS element is given by $\{0,\frac{2\pi}{2^{b}},\cdots,\frac{(2^{b}-1)2\pi}{2^{b}}\}$ [18]. Hence, $\theta_{n}^{opt}$ may not be always available and the exact phases shift at the $n$-th IRS element can be represented as $\theta_{n}$ = $\theta_{n}^{opt}$ + $\Phi_{n}$, where $\Phi_{n}$ denotes the phase error at the $n^{th}$ reflector. Note that, $-2^{-b}\pi\leq\Phi_{n}\leq 2^{-b}\pi$ and we model $\Phi_{n}\sim\mathcal{U}[-2^{-b}\pi,2^{-b}\pi]$, similar to the authors of [29], [18]. Here, $\mathcal{U}[a,b]$ represents the uniform distribution over the support $[a,b]$. Thus, the expression for SNR incorporating the phase error term is given as follows:

In the next section, we derive an exact expression and then three simple approximations for the CDF of $\gamma_{IRS}$ in (2) and hence the OP at $\mathbf{D}$. \cbstartNote that, our system model is similar to [18], however we consider the more general case of $\kappa-\mu$ fading here. Furthermore, the focus in [18] was the ergodic capacity and hence they do not derive the CDF of $\gamma_{IRS}$. \cbend

Outage at a node is the phenomenon of the instantaneous SNR falling below a particular threshold, say $\gamma$. The OP at node $\mathbf{D}$ can be evaluated as,

Note that $\gamma_{IRS}$ is the square of the absolute value of the sum of a $\kappa-\mu$ RV and a sum of i.i.d. double $\kappa-\mu$ RVs [30] each scaled by the exponential of a uniform RV. The PDF of a double $\kappa-\mu$ RV can be expressed in terms of a double infinite summation of terms involving the Meijer G-functions [30, (7)]. Hence the distribution of the sum of $N$ such scaled double $\kappa-\mu$ RVs has a very complicated expression[28, 19]. \cbendThis makes the characterization of the exact distribution of the OP a mathematically intractable task. So far, there were various kinds of approximations for OP (as shown in Table I) proposed in the open literature. \cbstartHowever, to the best of our knowledge, none of them considered the most general case, i.e., the presence of an SD link, b-bit quantization error and $\kappa-\mu$ fading channels. \cbendHence, in the subsequent subsections, we consider the most general case and present an exact expression and three different simple approximations for (3).