Joint Training and Reflection Pattern Optimization for Non-Ideal RIS-Aided Multiuser Systems

We consider the RIS-assisted uplink multiuser wireless communication system in Fig. 1, which consists of a BS with $L$ receive antennas, an RIS with $M$ reflecting elements, and $K$ single-antenna UEs. Let $\mathbf{G}\in\mathbb{C}^{L\times M}$, $\mathbf{h}_{r,k}\in\mathbb{C}^{M\times 1}$, $\mathbf{h}_{d,k}\in\mathbb{C}^{L\times 1}$ denote the channel between the RIS and the BS, the channel between the $k$-th UE and the RIS, and the channel between the $k$-th UE and the BS, respectively, which are assumed to follow the Rayleigh fading model. Denote the stacked channels of the UEs-RIS link and the UEs-BS link by $\mathbf{H}_{r}\triangleq[\mathbf{h}_{r,1},\cdots,\mathbf{h}_{r,K}]\in\mathbb{C}^{M\times K}$ and $\mathbf{H}_{d}\triangleq[\mathbf{h}_{d,1},\cdots,\mathbf{h}_{d,K}]\in\mathbb{C}^{L\times K}$, respectively. Moreover, the reflection coefficient matrix at the non-ideal RIS is denoted by $\mathbf{\Phi}=\text{diag}\left\{[\phi_{1},\cdots,\phi_{M}]^{T}\right\}$, with $\phi_{m}$ denoting the reflection coefficient of the $m$-th element at the RIS.

To characterize the RIS phase shifts, we utilize the parallel resonant circuit-based model considered in [40]. More specifically, the equivalent impedance of the $m$-th reflecting element, $m\in\mathcal{M}\triangleq\{1,\cdots,M\}$, is expressed as

where $C_{m}$, $R_{m}$, $L_{1}$, and $L_{2}$ stand for the effective capacitance, the effective resistance, the bottom layer inductance, and the top layer inductance of the parallel resonant circuit, respectively, and $\omega$ represents the angular frequency of the incident signal. Then, the reflection coefficient of the $m$-th element of the RIS is given by

where $\beta_{m}\in[0,1]$ and $\theta_{m}\in[0,2\pi]$ stand for the reflection amplitude and the phase shift, respectively, and $Z_{0}$ denotes the free space impedance. Different from the commonly used unit modulus phase shift model, the amplitude and the phase of $\phi_{m}$ in (2) are coupled. Specifically, according to [40], the reflection amplitude $\beta_{m}$ of the $m$-th element can be expressed as the following function of the corresponding phase shift $\theta_{m}$:

where the values of the constants $\beta_{\text{min}}\geq 0$, $\delta\geq 0$, and $\alpha\geq 0$ depend on the specific circuit parameters in (1). Fig. 2 illustrates an exemplified behaviour of the model in (3), where $\alpha=1.6$ and $\delta=0.43\pi$.

Let us focus on the channel estimation for the considered system. An RIS cannot transmit training signals or perform channel estimation. Therefore, we cannot estimate $\mathbf{H}_{r}$ or $\mathbf{G}$ directly. Nevertheless, it is possible to estimate the cascaded channel of the reflected link at the BS.

We consider a block fading channel model where each block is divided into training-based channel estimation and data transmission stages. By adopting the channel estimation protocol in [28, 31], as shown in Fig. 3 at the top of the next page, we further divide the training stage into $B$ subframes, each of which consists of $\tau$ symbols, where $\tau\geq K$ and $B\geq M+1$. The RIS coefficients are kept fixed within each subframe and vary for different subframes. The users transmit the same training signals periodically in different subframes. Mathematically, in the $b$-th subframe, $b\in\mathcal{B}\triangleq\{1,\cdots,B\}$, the received signals of $\tau$ consecutive symbols at the BS, denoted by $\mathbf{Y}(b)\in\mathbb{C}^{L\times\tau}$, is given by

where $\mathbf{\Phi}(b)=\text{diag}\left\{[\phi_{1}(b),\cdots,\phi_{M}(b)]^{T}\right\}$ represents the fixed RIS reflection coefficients in subframe $b$ and $\mathbf{X}$ denotes the stacked training symbols of $K$ UEs, which is expressed as

Here, $\mathbf{x}_{k}\triangleq[x_{k}(1),\cdots,x_{k}(\tau)]^{T}\in\mathbb{C}^{\tau\times 1}$ stands for the $\tau$ training symbols of the $k$-th UE, $k\in\mathcal{K}\triangleq\{1,\cdots,K\}$. In addition, $\mathbf{Z}(b)\in\mathbb{C}^{L\times\tau}$ is the additive Gaussian white noise matrix in the $b$-th subframe, and $\mathbf{g}_{m}\in\mathbb{C}^{L\times 1}$ and $\mathbf{h}_{r,m}\in\mathbb{C}^{K\times 1}$ stand for the $m$-th column of $\mathbf{G}$ and $\mathbf{H}_{r}^{H}$, respectively. To proceed, by setting $\phi_{M+1}(b)=1$ and defining

the received signal $\mathbf{Y}(b)$ in (II-B) becomes

where $\bm{\phi}(b)\triangleq[\phi_{1}(b),\cdots,\phi_{M+1}(b)]^{T}$ and $\mathbf{\Gamma}\triangleq[\mathbf{\Gamma}_{1},\cdots,\mathbf{\Gamma}_{M+1}]$ is the equivalent channel to be estimated in this work, which consists of the direct and cascaded reflected channels. Furthermore, by gathering the received signal $\mathbf{Y}(b)$ of all the $B$ subframes, we obtain

where $\mathbf{Y}\triangleq\left[\mathbf{Y}(1),\cdots,\mathbf{Y}(B)\right]\in\mathbb{C}^{L\times\tau B}$ and $\mathbf{Z}\triangleq\left[\mathbf{Z}(1),\cdots,\mathbf{Z}(B)\right]\in\mathbb{C}^{L\times\tau B}$ denote the received signals and the noises of the whole training phase, respectively, and $\mathbf{S}$ is relevant to both the training sequence and the reflection pattern, which is expressed as

Define the reflection pattern of $B$ subframes at the RIS by

each column of which represents the reflection pattern of the corresponding subframe. Then, based on the definitions in (9) and (10), we obtain the following expression for $\mathbf{S}$

where $\mathbf{\widetilde{V}}\triangleq\mathbf{V}\otimes\mathbf{I}_{K}$ and $\mathbf{\widetilde{X}}\triangleq\mathbf{I}_{B}\otimes\mathbf{X}$ are of size $[(M+1)K]\times BK$ and $KB\times\tau B$, respectively.

The goal of channel estimation for the considered RIS-aided multiuser system is to recover the channel matrix $\mathbf{\Gamma}$ based on the knowledge of the received signal matrix $\mathbf{Y}$ and a properly designed matrix $\mathbf{S}$. From (8), we obtain the LS and the LMMSE estimates of $\mathbf{\Gamma}$ as follows [41]:

where $\mathbf{S}^{\dagger}=\mathbf{S}^{H}(\mathbf{S}\mathbf{S}^{H})^{-1}$ denotes the pseudoinverse of $\mathbf{S}$, $\mathbf{R}_{\mathbf{\Gamma}}=\mathbb{E}\{\mathbf{\Gamma}^{H}\mathbf{\Gamma}\}$ is the correlation matrix of $\mathbf{\Gamma}$, and $\sigma^{2}$ denotes the noise variance. The estimation errors corresponding to (12) and (13) are given by

In addition, the training overhead of the considered estimation scheme is $(M+1)K$. There exist some possible approaches to alleviate this overhead, e.g., by grouping the RIS elements [24] and by adopting the two-timescale protocol [34, 35], which will be discussed in Section VI-C.

In this paper, we aim to design the matrix $\mathbf{S}$, i.e., the training sequence $\mathbf{X}$ at the UE side and the reflection coefficient $\mathbf{V}$ at the non-ideal RIS, such that the channel estimation MSEs $J_{\text{LS}}$ or $J_{\text{LMMSE}}$ are minimized. The considered joint optimization problem is formulated as

where $J(\mathbf{X},\mathbf{V})$ is given by (14) for the LS estimator and by (15) for the LMMSE estimator. $\|\mathbf{x}_{k}\|^{2}\leq P_{k}$ stands for the transmit power constraint of UE $k$ and $P_{k}$ is the corresponding power budget. $[\mathbf{V}]_{m,n}\in\mathcal{F}$ means that the entry $[\mathbf{V}]_{m,n}$ satisfies the phase shift constraint given in (3). $[\mathbf{V}]_{M+1,n}=1$ follows from the definition $\phi_{M+1}(b)=1$.

Solving the problem in (II-C) is difficult due to the coupled variables $\mathbf{X}$ and $\mathbf{V}$ in the objective function and the realistic constraint for the reflection coefficient. We develop efficient algorithms to tackle the problems for the LS and LMMSE channel estimators in the following two sections, respectively.

Although the training matrix and the reflection pattern are coupled in the objective function $J_{\text{LS}}$ in problem (II-C), as stated in the following theorem, we can determine the optimal training matrix in a closed form for the LS channel estimator.

It follows from Theorem 1 that any orthogonal training matrix is optimal for the LS channel estimator, which conforms to the results in MIMO communications[41] and ideal RIS-assisted communication systems[27]. Herein, we adopt a simple discrete Fourier transform (DFT)-based training matrix for estimating the uplink multiuser channel. Specifically, the training sequence of each UE is expressed as

where $\mathbf{d}_{k}$ denotes the $k$-th column of the $\tau\times\tau$ DFT matrix.

In this subsection, we consider the optimization with respect to the RIS reflection pattern under the considered non-ideal phase shift constraint, which, according to the proof of Theorem 1, takes the form:

Although the variable $\mathbf{X}$ has been eliminated, it is still nontrivial to find the optimal solution of problem (III-B) mainly due to the intricate phase shift constraint. In fact, although it has been proved in [27] that the optimal reflection pattern minimizing the MSE of the LS channel estimator is an orthogonal matrix, e.g., the DFT matrix or the Hadamard matrix, for ideal unit-modulus RIS phase shifts, this conclusion does not necessarily hold for the non-ideal phase shift constraints considered in this work. Hence, to design the non-ideal RIS reflection pattern for the LS channel estimator, we develop an MM-based algorithm.

The basic idea of the MM algorithm is to find a series of simple surrogate problems whose objective functions are locally approximated to the original objective function in each iteration, and then iteratively solve the surrogate problems until convergence [42]. To do this, we first derive an appropriate surrogate function with a more tractable form to locally approximate the objective function of problem (III-B), as given in the following proposition.

The major advantage of constructing the surrogate function in (20) is that the intractable inversion operation in the objective function of (III-B) is removed and we can perform the minimization of (20) by optimizing each entry of $\mathbf{V}$ simultaneously. Specifically, replacing the objective function in (III-B) with $f\left(\mathbf{V};\mathbf{V}_{0}\right)$, we obtain the following problem

In particular, we show that the optimal solution to problem (III-B) can be determined in an element-wise manner.

With the solution of the RIS reflection pattern given in (23), the proposed MM-based algorithm for the LS channel estimator is summarized in Algorithm 1. Moreover, we show the convergence of the MM algorithm in the following theorem.

We first perform the optimization of $\mathbf{X}$ with a fixed $\mathbf{V}$. Substituting $\mathbf{S}=\mathbf{\widetilde{V}}\mathbf{\widetilde{X}}$ into $g(\mathbf{S};\mathbf{S}_{0})$ and omitting the constant terms, the training sequence design under the per-UE transmitter power constraints is formulated as

where $\mathbf{\widetilde{V}}_{0}=\mathbf{V}_{0}\otimes\mathbf{I}_{K}$ and $\mathbf{V}_{0}$ represents the fixed reflection pattern. Problem (IV-A) can be formulated into a convex problem with respect to the block-diagonal matrix $\mathbf{\widetilde{X}}$, as follows:

and then solved. However, directly solving (IV-A) can be computationally inefficient due to the large dimension of $\mathbf{\widetilde{X}}$. Therefore, we propose an MM-based solution for problem (IV-A), which only requires calculating closed-form expressions iteratively and results in a lower computational complexity.

Specifically, we employ the following upper bound to the first term in the objective function, according to [42, eq. (26)]:

where $\lambda_{2}\mathbf{I}\succeq(\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H})^{T}\otimes\mathbf{\widetilde{V}}^{H}_{0}\mathbf{R}_{\mathbf{\Gamma}}\mathbf{\widetilde{V}}_{0}$ and $\mathbf{\widetilde{X}}_{0}$ is the solution of $\mathbf{\widetilde{X}}$ in the previous iteration. For simplicity, the value of $\lambda_{2}$ can be set to $\lambda_{2}=\lambda_{\text{max}}((\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H})^{T}\otimes\mathbf{\widetilde{V}}^{H}_{0}\mathbf{R}_{\mathbf{\Gamma}}\mathbf{\widetilde{V}}_{0})=\lambda_{\text{max}}(\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H})\lambda_{\text{max}}(\mathbf{\widetilde{V}}_{0}^{H}\mathbf{R}_{\mathbf{\Gamma}}\mathbf{\widetilde{V}}_{0})$.

By substituting (IV-A) and removing the constant terms, we transform problem (IV-A) into

where $\mathbf{B}_{0}=\lambda_{2}\mathbf{\widetilde{X}}_{0}^{H}-\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H}\mathbf{\widetilde{X}}_{0}^{H}\mathbf{\widetilde{V}}_{0}^{H}\mathbf{R}_{\mathbf{\Gamma}}\mathbf{\widetilde{V}}_{0}+\mathbf{\Xi}_{0}\mathbf{R}_{\mathbf{\Gamma}}\mathbf{\widetilde{V}}_{0}$. By invoking the definition $\mathbf{\widetilde{X}}=\mathbf{I}_{B}\otimes\mathbf{X}$ in (5), we can readily decompose the problem in (IV-A) into $K$ subproblems, with respect to the training sequence design at each UE. Concretely, each subproblem is given by

where $\mathbf{b}_{k}\triangleq\sum_{b=1}^{B}\left[\mathbf{B}_{0}\right]^{*}_{(b-1)\tau+1:b\tau,(b-1)K+k}$. Note that (IV-A) is a convex quadratic optimization problem which can be readily tackled. In fact, there exists a closed-form optimal solution as shown in the following proposition.

By invoking (33) for all the $K$ UEs, we accomplish the optimization of $\mathbf{X}$ with a low computational cost.

Now we fix $\mathbf{X}$ and focus on the optimization of the RIS reflection pattern. According to the upper bound $g(\mathbf{S};\mathbf{S}_{0})$ in (1), we formulate the subproblem with respect to $\mathbf{V}$ as

Similar to the optimization of the training matrix tackled in the previous subsection, by employing the upper bound given in [42, eq. (26)], we obtain

and transform problem (IV-B) into

where $\mathbf{C}_{0}=\lambda_{3}\mathbf{\widetilde{V}}_{0}^{H}-\mathbf{\widetilde{X}}_{0}\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H}\mathbf{\widetilde{X}}_{0}^{H}\mathbf{\widetilde{V}}_{0}^{H}\mathbf{R}_{\mathbf{\Gamma}}+\mathbf{\widetilde{X}}_{0}\mathbf{\Xi}_{0}\mathbf{R}_{\mathbf{\Gamma}}$, $\mathbf{\widetilde{V}}_{0}$ is the solution of $\mathbf{\widetilde{V}}$ in the previous iteration, and $\lambda_{3}$ is calculated as $\lambda_{\text{max}}(\mathbf{\widetilde{X}}_{0}\mathbf{\Xi}_{0}\mathbf{\Xi}_{0}^{H}\mathbf{\widetilde{X}}_{0}^{H})\lambda_{\text{max}}\left(\mathbf{R}_{\mathbf{\Gamma}}\right)$.

By substituting $\mathbf{\widetilde{V}}\triangleq\mathbf{V}\otimes\mathbf{I}_{K}$ into the objective function of the problem in (IV-B), we obtain $\sum_{m}\sum_{n}(\lambda_{3}K|[\mathbf{V}]_{m,n}|^{2}-2\mathcal{R}\left\{c_{m,n}[\mathbf{V}]_{m,n}\right\})$, where $c_{m,n}=\sum_{k=1}^{K}\left[\mathbf{C}_{0}\right]_{(n-1)K+k,(m-1)K+k}$. Hence, we can solve problem (IV-B) in an element-wise manner. In particular, the solution to each entry of $\mathbf{V}$ in the $i$-th iteration can be obtained according to

where $\bar{\theta}_{m,n}=\mathop{\arg\min}\limits_{\theta\in[0,2\pi]}g_{m,n}(\theta)$ is derived by performing a one-dimensional search and $g_{m,n}(\theta)$ is calculated by substituting the equivalent phase shift model in (3) into $\lambda_{3}K|\beta(\theta)|^{2}-2\mathcal{R}\{c_{m,n}\beta(\theta)e^{j\theta}\}$, whose expression is given by

Based on the two proposed solutions, the LMMSE channel estimation algorithm is summarized in Algorithm 2. Regarding the convergence, it can be verified that the algorithm generates a non-increasing sequence based on the alternating optimization method that is utilized. Moreover, the objective value of the problem in (IV) has a finite lower bound. Therefore, Algorithm 2 always converges.