Outage Probability and Finite-SNR DMT Analysis for IRS-aided MIMO Systems: How Large IRSs Need to Be?

The main contributions of this paper are listed as follows.

• 1)

  The distribution of the MI for IRS-aided MIMO systems is first derived. Based on the result, an approximation on the outage probability over general correlated channels is obtained with only statistical CSI. To the best of the authors’ knowledge, this is the first analytical result regarding the outage probability in IRS-aided MIMO systems over general correlated channels. Numerical results validate the accuracy of the proposed method.

• 2)

  With the derived outage probability, a gradient algorithm is proposed to minimize the outage probability by optimizing the phase shifts at the IRS, assuming only statistical CSI. Numerical results show that the algorithm can efficiently decrease the outage probability.

• 3)

  A closed-form expression is obtained for the finite-SNR DMT of IRS-aided MIMO systems, which is not available in the literature. An interesting observation is that the finite-SNR DMT is highly related to the ratio between the mean and the standard deviation of the MI. The accuracy of the expression is validated by numerical results.

• 4)

  The impact of the size of the IRS on system performance is investigated. To this end, we first propose the concept called IRS efficiency to measure the efficiency of increasing the IRS size in achieving the maximum throughput. The expression of the outage probability with respect to the IRS size over uncorrelated channels is explicitly given in the high SNR regime. Based on the theoretical analysis and simulation results, we have two key observations. First, when the size of the IRS is infinitely large, the performance of the two-hop IRS system is the same as that of the single-hop link. Second, the benefit of increasing the size of the IRS saturates quickly. For example, for an IRS-aided system with 20 antennas at the transceivers over independent channels, we need to double the size of the IRS to increase the throughput from 90% to 95% of its maximum value.

The rest of this paper is organized as follows. Section II presents the system model for the IRS-aided MIMO system. Section III introduces the main results including the characterization of the distribution for the MI. The analysis and optimization of the outage probability and the finite-SNR DMT are given in Section IV. Section V discusses the effect of the IRS size on system performance. The theoretical results are validated in Section VI by numerical simulations. Finally, Section VII concludes the paper.

Notations: Bold upper case letters and bold lower case letters represent the matrix and vector, respectively. $\mathrm{Re}\left\{\cdot\right\}$ denotes the real part of a complex number. $\mathbb{P}(\cdot)$ is the probability measure. $\mathbb{C}^{N}$ and $\mathbb{C}^{M\times N}$ represent the space of $N$-dimensional vectors and the space of $M$-by-$N$ matrices, respectively. $\mathbb{A}^{H}$ represents the conjugate transpose of $\mathbb{A}$. $[\mathbb{A}]_{i,j}$ represents the $i,j$-th entry of $\mathbb{A}$. $\otimes$ denotes the element-wise product of matrices. $\operatorname{Tr}\mathbb{A}$ and $\|\mathbb{A}\|$ represent the trace and the spectral norm of $\mathbb{A}$. $\mathbb{E}$ represents the expectation operator. $\Phi(x)$ is the cumulative distribution function (CDF) of standard Gaussian distribution and $Q(\cdot)$ is the $Q$-function, where $Q(x)=1-\phi(x)$. $\mathcal{CN}$ and $\mathcal{N}$ represent the circularly complex Gaussian and real Gaussian distribution, respectively. $\xrightarrow[N\rightarrow\infty]{\mathcal{D}}$, $\xrightarrow[N\rightarrow\infty]{\mathcal{P}}$, and $\xrightarrow[N\rightarrow\infty]{{a.s.}}$ denote the convergence in distribution, the convergence in probability and the almost sure convergence, respectively. $O(\cdot)$, $o(\cdot)$, and $\Theta(\cdot)$ represent the Big-O, the Little-o, and the Big-Theta notations, respectively. Specifically, $f(n)\in O(g(n))$ if and only if there exists a positive constant $c$ and a nonnegative integer $n_{0}$ such that $f(n)\leq cg(n)$ for all $n\geq n_{0}$. $f(n)\in o(g(n))$ if and only if there exists a nonnegative integer $n_{0}$ such that $f(n)\leq cg(n)$ for all $n\geq n_{0}$ for all positive $c$. $f(n)\in\Theta(g(n))$ if and only if there exist positive $c_{1}$ and $c_{2}$ and nonnegative integer $n_{0}$ such that $c_{1}g(n)\leq f(n)\leq c_{2}g(n)$ for all $n\geq n_{0}$ [24].

Consider a point-to-point downlink MIMO communication system, where an IRS is deployed to establish favorable communication links for the user equipment (UE) that would otherwise be blocked. There are $M$ antennas at the basestation (BS) and $N$ antennas at the UE, and the number of elements at the IRS is $L$. Given the line-of-sight (LoS) path is blocked, the received signal $\mathbb{y}\in\mathbb{C}^{N}$ is given by

where $\mathbb{s}\in\mathbb{C}^{M}$ denotes the transmitted signal with unit average transmit power, i.e., $\mathbb{E}|s_{i}|^{2}=1,i=1,2,...,M$, and $\mathbb{n}\in\mathbb{C}^{N}\sim\mathcal{CN}(0,\sigma^{2}\mathbb{I}_{N})$ represents the additive white Gaussian noise (AWGN) with variance $\sigma^{2}$. $\mathbb{H}_{2}\in\mathbb{C}^{L\times M}$ and $\mathbb{H}_{1}\in\mathbb{C}^{N\times L}$ represent the channel matrices from the BS to the IRS and from the IRS to the UE, respectively. $\mathbb{\Psi}\in\mathbb{C}^{L\times L}=\mathrm{diag}(\psi_{1},\psi_{2},...,\psi_{L})=\mathrm{diag}(e^{\jmath{\theta_{1}}},e^{\jmath{\theta_{2}}},...,e^{\jmath{\theta_{L}}})$ with $\theta_{i}\in[0,2\pi),~{}i=1,2,...,L$, denotes the phase shifts imposed by the IRS, and we define $\bm{\theta}=(\theta_{1},\theta_{2},...,\theta_{L})$. In this paper, we consider the general Rayleigh model with

where $\mathbb{R}_{i}$ and $\mathbb{T}_{i},~{}i=1,2$, are four positive semi-definite correlation matrices. $\mathbb{R}_{1}$ and $\mathbb{T}_{2}$ denote the correlation matrices of receive and transmit antennas, respectively. $\mathbb{T}_{1}$ and $\mathbb{R}_{2}$ denote the transmit and receive correlation matrices of the IRS, respectively. $\mathbb{X}\in\mathbb{C}^{N\times L}$ and $\mathbb{Y}\in\mathbb{C}^{L\times M}$ are two independent and identically distributed (i.i.d.) Gaussian random matrices, whose entries follow $\mathcal{CN}(0,\frac{1}{L})$ and $\mathcal{CN}(0,\frac{1}{M})$, respectively. We assume that statistical CSI, i.e., correlation matrices of the channel, is available. To obtain the statistical CSI, the samples of the separate channels are needed. In practice, we can estimate the IRS-UE channel and the BS-IRS channel separately by the methods proposed in [25], [26] and then estimate the corresponding channel covariance matrices based on the techniques proposed in [27], [28].

The MI of the IRS-aided MIMO system is given by

where $\rho=\frac{P}{M\sigma^{2}}$ with $P$ denoting the total transmit power. Based on the distribution of the MI, we can evaluate the performance of IRS-aided MIMO systems with different metrics. For example, the average throughput can be determined by the EMI as $\mathbb{E}I(\rho)$. On the other hand, the reliability of the system can be measured by the outage probability, which, for a preset transmission rate $R$, can be written as

The EMI of IRS-aided MIMO systems has been obtained in the literature [11]. In the following, we first investigate the distribution of $I(\rho)$ and then utilize the result to analyze the outage probability and the finite-SNR DMT of IRS-aided MIMO systems.

The results of this paper are developed based on the following assumptions.

Assumption 1. $0<\lim\inf\limits_{M\geq 1}\frac{M}{L}\leq\frac{M}{L}\leq\lim\sup\limits_{M\geq 1}\frac{M}{L}<\infty$, $0<\lim\inf\limits_{M\geq 1}\frac{M}{N}\leq\frac{M}{N}\leq\lim\sup\limits_{M\geq 1}\frac{M}{N}<\infty$.

Assumption 2. $\lim\sup\limits_{M\geq 1}\|\mathbb{R}_{i}\|<\infty$, $\lim\sup\limits_{M\geq 1}\|\mathbb{T}_{i}\|<\infty$, $i=1,2$ [29] [30].

Assumption 3. $\inf\limits_{M\geq 1}\frac{1}{M}\operatorname{Tr}\mathbb{R}_{1}>0$, $\inf\limits_{M\geq 1}\frac{1}{M}\operatorname{Tr}\mathbb{T}_{2}>0$, $\inf\limits_{M\geq 1}\frac{1}{M}\operatorname{Tr}\mathbb{T}_{1}\mathbb{\Psi}\mathbb{R}_{2}\mathbb{\Psi}^{H}>0$ [31] [32].

A.1 is the asymptotic regime considered for the large-scale system, where the dimensions of the system ($M$, $N$, and $L$) grow to infinity at the same paces. A.2 and A.3 restrict the rank of the correlation matrices so that the extremely low-rank case, i.e., the ranks of the correlation matrices do not increase with the number of antennas, will not occur.

Given the eigenvalue decompositions $\mathbb{R}=\mathbb{U}_{R}\mathbb{R}_{1}\mathbb{U}_{R}^{H}$, $\mathbb{S}=\mathbb{U}_{S}\mathbb{T}^{\frac{1}{2}}_{1}\mathbb{\Psi}\mathbb{R}_{2}\mathbb{\Psi}^{H}\mathbb{T}^{\frac{1}{2}}_{1}\mathbb{U}_{S}^{H}$, $\mathbb{T}=\mathbb{U}_{T}\mathbb{T}_{2}\mathbb{U}_{T}^{H}$, where $\mathbb{R}$, $\mathbb{S}$, and $\mathbb{T}$ are diagonal matrices, and the singular value decomposition (SVD) $\mathbb{T}^{\frac{1}{2}}_{1}\mathbb{\Psi}\mathbb{R}_{2}^{\frac{1}{2}}=\mathbb{U}_{S}^{H}\mathbb{S}^{\frac{1}{2}}\mathbb{V}_{S}$, the MI in (3) can be written as

where step $a$ follows by plugging (2) into (3). Step $b$ is obtained by plugging in the eigenvalue decompositions. Step $c$ follows from $\mathbb{X}^{\prime}=\mathbb{U}_{R}\mathbb{X}\mathbb{U}_{S}^{H}$, $\mathbb{Y}^{\prime}=\mathbb{V}_{S}\mathbb{Y}\mathbb{U}_{T}^{H}$, and $\det(\mathbb{I}+\mathbb{A}\mathbb{B})=\det(\mathbb{I}+\mathbb{B}\mathbb{A})$. Due to the unitary invariant attributes of Gaussian random matrices, a Gaussian matrix $\mathbb{G}$ is equivalent to $\mathbb{G}^{\prime}=\mathbb{U}\mathbb{G}\mathbb{V}$ statistically, where $\mathbb{U}$ and $\mathbb{V}$ are any unitary matrices. Therefore, $\mathbb{X}$ ($\mathbb{Y}$) and $\mathbb{X}^{\prime}$ ($\mathbb{Y}^{\prime}$) are statistically equivalent so that we will not differentiate them in the following. Thus, the equivalent channel matrix can be given by

The following theorem gives an approximation for the EMI.

Next, we investigate the fluctuation of $I(\rho)$. The challenge arises from the fact that the effective channel is the product of two random matrices. There are some related results in the literature. The CLT of linear spectral statistics for $F$-matrices was given in [38]. In [39], the CLT of linear spectral statistics with general non-Gaussian entries was given for the product of two i.i.d. random matrices, which was also investigated in [19] by a free probability approach for Gaussian random matrices. The authors of [19] gave the CLT for the MI of double-Rayleigh channels when $M=N$. However, the CLT for the MI over correlated channels has not been considered and is one of the main contributions of this paper. Based on the CLT, we also investigate the outage probability for IRS-aided MIMO systems using large RMT.

To characterize the distribution of the MI, we first prove the Gaussianity of the MI.

To better illustrate the impact of the size of the IRS, we further consider the special case where the channels are i.i.d., i.e., $\mathbb{R}=\mathbb{I}_{N},\mathbb{S}=\mathbb{I}_{L},\mathbb{T}=\mathbb{I}_{N}$, which is referred to as the double-Rayleigh model [19]. To derive the CLT, we first determine the EMI.

Proposition 1 and Proposition 2 indicate that the mean and variance of the MI are determined by the root $g$ of the cubic equation (16), whose coefficients are related to $\tau$ and SNR $\rho$.

As a direct result of Theorem 2, the approximation of the outage probability is given by the following theorem.

The above result is different from [20] in the sense that both $\mathbb{R}_{1}$ and $\mathbb{T}_{2}$ are not restricted to be an identity matrix, indicating that the proposed method can handle the general correlated channels. In fact, the result in Theorem 3 is also applicable for double-scattering channels [33] [40].

Given statistical CSI, the optimization problem for the outage probability can be formulated as

where $\psi_{l}=\exp(\jmath\theta_{l})$. The challenges arise from two aspects: (1) Evaluation of the objective function; (2) The non-convexity of the problem caused by the uni-modular constraints of $\psi_{i}$. The first one can be resolved by approximating the outage probability using (19) in Theorem 3. Given the SNR $\rho$, the mean $\overline{I}(\rho)$ and variance $V(\rho)$ are functions of the phase shifts $\psi_{l},l=1,2,...,L$, so we use the notations $\overline{I}(\mathbb{\Psi})$ and $V(\mathbb{\Psi})$ instead. Thus, $\mathcal{P}1$ can be rewritten as

Although the parameters $\delta$, $g$, and $\overline{g}$ are coupled with $\theta_{l}$ as explained in Remark 1, the partial derivatives of the objective function with respect to $\theta_{l}$ can be given in a closed form. Therefore, $\mathcal{P}2$ can be solved using the gradient descent method. Specifically, in each iteration, the update of $\theta_{l}$ is obtained by searching in the negative gradient direction, until the value of the objective function converges to a stationary point.

Next, we compute the partial derivatives with respect to $\theta_{l}$, $l=1,2,...,L$, and we use the notation $(\cdot)^{\prime}_{l}=\frac{\partial(\cdot)}{\partial\theta_{l}}$ to represent the partial derivatives. By the chain rule, the partial derivative of $G(\mathbb{\Psi})$ with respect to $\theta_{l}$ is given by

where

$\mathbb{F}_{l}$ is defined as

where

In fact, $\mathbb{F}_{l}=(\mathbb{T}_{1}^{\frac{1}{2}}\mathbb{\Psi}\mathbb{R}_{2}\mathbb{\Psi}^{H}\mathbb{T}_{1}^{\frac{1}{2}})^{\prime}_{l}$ and the term $V^{\prime}_{l}(\rho)$ can be given by

where

The derivatives of $\gamma$’s in (27) are given as

where $(\delta^{\prime}_{l},g^{\prime}_{l},\overline{g}^{\prime}_{l})$ can be computed by the following lemma.

Next, we provide the gradient descent method. We use the Armijo-Goldstein (AG) line search method [41] to find an expected decrease of the objective function based on the local gradients. The AG condition is given in the $4$th line of Algorithm 1. The algorithm can be proved to converge to a stationary point by following the proof of Theorem 1 in [42] or Theorem 1 in [43]. The performance of the algorithm will be evaluated in Section VI.

Complexity Analysis: In each iteration, we need to compute $(\delta,g,\overline{g})$ according to the updated $\mathbb{\Phi}$ by solving (10). To avoid the computation induced by the matrix inversion, we can use the diagonal matrix $\mathbb{\Lambda}_{R}=\mathbb{U}\mathbb{R}\mathbb{U}^{H}$ to replace $\mathbb{R}$ by its eigenvalue decomposition, whose complexity is $O(N^{3})$ [44]. Similar operations can be performed on $\mathbb{S}$ and $\mathbb{T}$. By the analysis in Appendix E, the complexity of obtaining an $\varepsilon$-approximation of the solution is $N\log^{2}(\frac{1}{\varepsilon})$. Assume that the numbers of outer and inner iterations are $N_{outer}$ and $N_{inner}$, respectively. Then the total complexity of the algorithm will be $O(N_{outer}N_{inner}(N^{3}+N\log^{2}(\frac{1}{\varepsilon})))$, where $N_{outer}$ and $N_{inner}$ are determined by the convergence condition and the second-order functional attributes of $G(\mathbb{\Psi})$. As a smaller value for the objective function is obtained in each iteration, the algorithm will converge to a stationary point. With the statistical CSI, the phase shifts may not need to be updated frequently so that the overall complexity of Algorithm 1 is not high in real systems.

In this section, we investigate the finite-SNR DMT of IRS-aided systems. DMT was proposed in [21] to characterize the trade-off between diversity and multiplexing gain, which is typically referred to as the asymptotic-SNR DMT. Here, we investigate the finite-SNR DMT [22], [23], which provides more insightful information for the low and moderate SNR regimes.

By the definition of finite-SNR DMT [23], the multiplexing gain is given by

where $k=\min(L,M,N)$ and $R$ is the data rate. By approximating the outage probability using the CLT in Theorem 2, we obtain the following theorem regarding the finite-SNR DMT for IRS-aided systems in Rayleigh channels with equal power allocation among different antennas.

We first characterize the relation between the EMI and the size of the IRS. For simplicity, we assume that $M=N$ and focus on the parameter $\tau$, i.e., the ratio between the number of antennas $N$ and the size of the IRS $L$. To study the performance gain obtained by increasing the size of the IRS, we focus on the case when $\tau<1$, which means that the size of the IRS is larger than the numbers of antennas at the transceiver. Results with $\tau>1$ can be handled similarly. Inspired by the concept of massive MIMO efficiency proposed in [46], we define the concept called IRS efficiency to characterize the efficiency of increasing the IRS size to achieve higher EMI.

Given $\eta$, we can determine the minimum size of the IRS needed to achieve at least the fraction $\eta$ of the asymptotic performance. To better illustrate the impact of the IRS size, we consider independent channels and ignore the optimization of phase shifts. According to Proposition 1, for given $\rho$ and $M$, $\overline{I}(\rho)$ will be determined only by $\tau$. Next, we will investigate $\overline{I}_{\infty}(\rho)$.