Spatial Multiplexing Oriented Channel Reconfiguration in Multi-IRS Aided MIMO Systems

As shown in Fig. 1, we consider a MIMO wireless communication system assisted by $K$ IRSs, where the Tx and the Rx are equipped with $N_{t}$ and $N_{r}$ antennas, respectively. We assume that the direct Tx-Rx link is obstructed. The positions of the Tx and the Rx in a two-dimensional (2D) Cartesian coordinate system are denoted by $\mathbf{u}_{t}\in\mathbb{R}^{2\times 1}$ and $\mathbf{u}_{r}\in\mathbb{R}^{2\times 1}$, respectively. The Tx and the Rx employ uniform linear arrays (ULAs) with element spacings of $d_{t}$ and $d_{r}$, respectively. The $k$th IRS is equipped with an $M_{k}\,(M_{\mathrm{v,k}}\times M_{\mathrm{h,k}})$ elements uniform planar array (UPA), where $k\in{\mathcal{K}}\buildrel\Delta\over{=}\{1,2,\ldots,K\}$. The UPA on the $k$th IRS consists of $M_{\mathrm{v,k}}$ rows and $M_{\mathrm{h,k}}$ columns, with all spacings equal to $d_{s}$. The total available number of IRSs is $M$ and thus $\sum_{k=1}^{K}{M_{k}}\leqslant M$. The position of the $k$th IRS is denoted by $\mathbf{u}_{k}\in\mathbb{R}^{2\times 1}$.

We denote the equivalent channel from the Tx to the $k$th IRS and from the $k$th IRS to the Rx as $\mathbf{T}_{k}\in\mathbb{C}^{M_{k}\times N_{t}}$ and $\mathbf{R}_{k}\in\mathbb{C}^{N_{r}\times M_{k}}$, respectively. We assume that the distributed IRSs possess line-of-sight (LoS) paths between the Tx and the Rx. The channel from the Tx to the $k$th IRS can be expressed as

where $\rho_{\mathrm{T},k}$ denotes the complex channel gain of the Tx to the $k$th IRS link. In the array response vector, ${\Phi_{{\mathrm{T}},k}^{\mathrm{AOA}}}=2\pi d_{\mathrm{S}}\cos{\phi_{{\mathrm{T}},k}^{\mathrm{AOA}}}/\lambda$, ${\Theta_{{\mathrm{T}},k}^{\mathrm{AOA}}}=2\pi d_{\mathrm{S}}\sin{\phi_{{\mathrm{T}},k}^{\mathrm{AOA}}}\sin{\theta_{{\mathrm{T}},k}^{\mathrm{AOA}}}/\lambda$, and ${\Theta_{{\mathrm{T}},k}^{\mathrm{AOD}}}=2\pi d_{{t}}\sin{\theta_{{\mathrm{T}},k}^{\mathrm{AOD}}}/\lambda$, where ${\theta_{{\rm{T}},k}^{\rm{AOA}}}$, ${\phi_{{\rm{T}},k}^{\rm{AOA}}}$, and ${\theta_{{\rm{T}},k}^{\rm{AOD}}}$ denote the horizontal angle of arrival (AoA), the vertical AoA, and the angle of departure (AoD) of the Tx to $k$th IRS link, respectively. Furthermore, ${{\bf{a}}_{N_{t}}}\left(\cdot\right)$ and ${{\bf{a}}_{{\rm{S}},k}}\left(\cdot\right)$ represent the array response vectors at the Tx and the $k$th IRS, respectively. Hence, the array response vector of ULA can be unified by

It is worth noting that the array response for UPA can be decomposed into that of ULA as ${{\bf{a}}_{{\rm{S}},k}}\left({X,Y}\right)={{\bf{a}}_{{M_{v,k}}}}\left(X\right)\otimes{{\bf{a}}_{{M_{h,k}}}}\left(Y\right)$, where $\otimes$ is the Kronecker product.

Similar to the Tx to the $k$th IRS link, the channel matrix from the $k$th IRS to the Rx can be expressed as

where $\rho_{\mathrm{R},k}$ denotes the complex channel gain of the $k$th IRS to the Rx link. For notational convenience, in the sequel we substitute $\mathbf{a}_{\mathrm{S},k,\mathrm{T}}$ and $\mathbf{a}_{\mathrm{S},k,\mathrm{R}}$ for $\mathbf{a}_{\mathrm{S},k}\left(\Phi_{\mathrm{T},k}^{\mathrm{AOA}},\Theta_{\mathrm{T},k}^{\mathrm{AOA}}\right)$ and $\mathbf{a}_{\mathrm{S},k}\left(\Phi_{\mathrm{R},\mathrm{k}}^{\mathrm{AOD}},\Theta_{\mathrm{R},k}^{\mathrm{AOD}}\right)$, respectively. Besides, we define $\rho_{k}\triangleq\rho_{\mathrm{T},k}\rho_{\mathrm{R},k}$ and we denote $\mathbf{\Phi}_{k}=\mathrm{diag}(e^{j\phi_{k,1}},e^{j\phi_{k,2}},\cdots,e^{j\phi_{k,M_{k}}})$ as the passive beamforming matrix of the $k$th IRS, where $\phi_{k,m_{k}}$ is the phase of the $m_{k}$th element on the $k$th IRS, $m_{k}\in\mathcal{M}_{k}\triangleq\left\{1,2,…,M_{k}\right\}$, $k\in\mathcal{K}$, and $\phi_{k,m_{k}}\in\left[0,2\pi\right)$. As such, the effective Tx-IRS-Rx MIMO channel aided by $K$ IRSs is given by $\mathbf{H}=\sum_{k=1}^{K}{\mathbf{R}_{k}\mathbf{\Phi}_{k}\mathbf{T}_{k}}$.

Let $f\left(\mathbf{\Phi}_{k}\right)\triangleq\mathbf{a}_{\mathrm{S},k,\mathrm{R}}^{H}\mathbf{\Phi}_{k}\mathbf{a}_{\mathrm{S},k,\mathrm{T}}$, $\omega_{k}=\angle(\rho_{k}f\left(\mathbf{\Phi}_{k}\right))$, $\mathbf{A}_{\mathrm{T}}=\frac{1}{\sqrt{N_{t}}}[e^{j\omega_{1}}\mathbf{a}_{N_{t}}\left(\Theta_{\mathrm{T},1}^{\mathrm{AOD}}\right),...,e^{j\omega_{K}}\mathbf{a}_{N_{t}}\left(\Theta_{\mathrm{T},K}^{\mathrm{AOD}}\right)]$, and $\mathbf{A}_{\mathrm{R}}=\frac{1}{\sqrt{N_{r}}}[\mathbf{a}_{N_{r}}\left(\Theta_{\mathrm{R},1}^{\mathrm{AOA}}\right),...,\mathbf{a}_{N_{r}}\left(\Theta_{\mathrm{R},K}^{\mathrm{AOA}}\right)]$. Thus, the composite channel consisting of $K$ IRSs can be simplified as

where $\mathbf{\Sigma}=\sqrt{N_{t}N_{r}}\mathrm{diag}\left(|\rho_{1}f\left(\mathbf{\Phi}_{1}\right)|,...,|\rho_{K}f\left(\mathbf{\Phi}_{K}\right)|\right)$. As such, the received signal $\mathbf{y}\in\mathbb{C}^{N_{r}\times 1}$ at the Rx is given by $\mathbf{y}=\mathbf{Hx}+\mathbf{z}$, where $\mathbf{x}\in\mathbb{C}^{N_{t}\times 1}$ denotes the transmitted signal vector and $\mathbf{z}\sim\mathcal{CN}(0,\sigma^{2}\mathbf{I}_{N_{r}})$ is the additive white Gaussian noise at the Rx with power $\sigma^{2}$.

The introduction of IRSs enables the creation of a controllable scattering channel environment, potentially establishing favorable conditions for multi-stream transmission. By properly positioning IRSs so that $\mathbf{A}_{\mathrm{T}}$ and $\mathbf{A}_{\mathrm{R}}$ are orthogonal matrices (i.e., $\mathbf{A}_{\mathrm{R}}^{H}\mathbf{A}_{\mathrm{R}}=\mathbf{I}$ and $\mathbf{A}_{\mathrm{T}}^{H}\mathbf{A}_{\mathrm{T}}=\mathbf{I}$), the equation (4) naturally admits a form of the singular value decomposition (SVD). In this scenario, the structures of the transmitter’s precoding and the receiver’s combiner can be explicitly derived as $\mathbf{A}_{\mathrm{T}}$ and $\mathbf{A}_{\mathrm{R}}^{H}$, respectively. This configuration allows for the minimization of inter-stream interference while simplifying the transceiver design.

We aim to maximize the SE of the multi-IRS aided system by optimizing the IRS beamforming, IRS placement, elements allocation, and transmit covariance matrix. The corresponding optimization problem is formulated as

where $\mathbf{Q}=\mathbb{E}\{\mathbf{x}\mathbf{x}^{H}\}$ denotes the transmit covariance matrix and $P$ denotes the maximum transmit power of the Tx.

First, to satisfy the constraint (5c), IRSs should be positioned along the discrete fourier transform (DFT) directions of both the Tx and the Rx. This arrangement satisfies the requirement for the AoA and AoD discretization:

When the positions of the Tx and the Rx as well as $(\Theta_{\mathrm{T},k}^{\mathrm{AOD}},\Theta_{\mathrm{R},k}^{\mathrm{AOA}})$ are given, we can determine the corresponding IRS position $\mathbf{u}_{k}$. However, obtaining the optimal IRS positions requires an exhaustive enumeration of all combinations due to (6), which results in high computation complexity. To address this problem, we employ a greedy algorithm that searches for the position with the maximum channel gain at each iteration. Besides, we define $\mathcal{P}_{1}\triangleq\left\{\mathcal{S}_{1},\mathcal{S}_{2},…,\mathcal{S}_{L}\right\}$ that collects all the possible IRS candidate positions and their corresponding Tx-IRS-Rx channel gain, where $L$ denotes the maximum number of different sets in $\mathcal{P}_{1}$, $\mathcal{S}_{l}\triangleq\left(\theta_{l},\varphi_{l},\tau_{l}\right)$ denotes the IRS position where the AoD at the Tx is $\theta_{l}$ and the AoA at the Rx is $\varphi_{l}$ with the corresponding Tx-IRS-Rx channel gain $\tau_{l}$, $\theta_{l}\in\mathcal{A}_{1}$, $\varphi_{l}\in\mathcal{A}_{2}$, $l\in\mathcal{L}\triangleq\left\{1,2,…,L\right\}$, and $\mathcal{S}_{i}\neq\mathcal{S}_{j},i\neq j,i,j\in\mathcal{L}$. Then, the process of the greedy search can be described by

Therefore, we execute this greedy search process from $k=0$ to $K-1$ and successfully obtain the positions for $K$ IRSs.

By employing the orthogonal placement, the $k$th singular value of $\mathbf{H}$ is the non-zero singular value of $\mathbf{R}_{k}\mathbf{\Phi}_{k}\mathbf{T}_{k}$. And $\mathbf{Q}$ in (5) can be derived as $\mathbf{Q}=\mathbf{A}_{\mathrm{T}}\mathrm{diag}\left(p_{1},p_{2},…,p_{K}\right)\mathbf{A}_{\mathrm{T}}^{H}$, where $p_{k}$ is the transmit power allocated to the $k$th sub-channel that will be optimized in subsection B. Thus, the SE of the $K$ IRS system can be rewritten as

where $\chi_{k}=|\rho_{k}|^{2}/\sigma^{2}$. Due to the fact that maximizing the SE can be achieved by independently maximizing each $f(\mathbf{\Phi}_{k})$, we employ the following passive beamforming structure to optimize each $f(\mathbf{\Phi}_{k})$:

where $(\mathbf{t}_{k})_{m_{k}}$ is the $m_{k}$th element of $\mathbf{a}_{N_{t}}^{H}\left(\Theta_{\mathrm{T},k}^{\mathrm{AOD}}\right)$ and $(\mathbf{r}_{k})_{m_{k}}$ is the $m_{k}$th element of $\mathbf{a}_{N_{r}}\left(\Theta_{\mathrm{R},k}^{\mathrm{AOA}}\right)$. By employing the above configuration, $f\left(\mathbf{\Phi}_{k}\right)$ can be maximized. Under the optimal $\mathbf{\Phi}_{k}^{*}$, we have $f\left(\mathbf{\Phi}_{k}^{*}\right)=M_{k}$.

To better illustrate the spatial multiplexing gain offered by multiple IRSs, we unveil the condition that double-IRS outperforms single-IRS in the following proposition. Besides, we define $R_{1}=\log_{2}\left(1+\chi PM^{2}\right)$ is the maximum achievable rate with a single IRS comprising $M$ elements and $R_{2}=\underset{p_{1},p_{2}}{\max}\left(\log_{2}\left(1+p_{1}\eta_{1}\right)+\log_{2}\left(1+p_{2}\eta_{2}\right)\right)$ with $p_{1}+p_{2}=P$ and $\eta_{k}=\chi_{k}M^{2}/4$ is the maximum achievable rate with two IRSs, each comprising $M_{1}=M_{2}=M/2$ elements under orthogonal placement conditions.

Proposition 1 theoretically illustrates that, under specific conditions and with appropriate allocation strategies, the performance of a double IRS system can surpass that of a single IRS configuration. Furthermore, we aim to fully exploit the spatial multiplexing potential of multi-IRS architectures through appropriate resource allocation strategies, which motivates us to investigate the joint power and elements allocation algorithm. Next, we jointly optimize the IRS elements and power allocation under the obtained IRS placement and beamforming. To this end, problem (5) is reduced to

First, to address the integer constraint in (5d), we relax the discrete value $M_{k}$ to its continuous counterpart $\tilde{M}_{k}$. Consequently, the objective function of problem (11) can be relaxed to

However, (12) is non-convex due to the coupling of $p_{k}$ and $\tilde{M}_{k}$. To tackle this difficulty, we introduce auxiliary variables $l_{k}$. Problem $(\mathrm{P1})$ can be reformulated as the following optimization problem:

Problem (13) is still intractable hard to solve due to (13a). To address the non-convexity of constraint (13a), we employ the successive convex approximation (SCA) technique with appropriate variable substitutions. Thus, we introduce auxiliary variables $x_{k}$ and $y_{k}$, $k\in\mathcal{K}$. Consequently, the constraints in (13a) can be reformulated as follows:

Although constraints (14) exhibit non-convex forms, their right-hand sides, specifically $\chi_{k}e^{x_{k}+y_{k}}$ and $\tilde{M}_{k}^{2}$, are convex functions with respect to their respective variables. This motivates us to apply the first-order Taylor expansion to linearize them as convex constraints given by

where $\hat{x}_{k}$, $\hat{y}_{k}$, and $\hat{M}_{k}$ are the given local points of $x_{k}$ ,$y_{k}$, and $\tilde{M}_{k}$, respectively.

As a result, problem (13) is approximated as

Problem $(\mathrm{P2})$ is a convex optimization problem, which can be solved optimally in an iterative manner by using standard solvers like CVX until convergence is achieved. As the objective value increases with each iteration and is bounded by a finite value, the solution of the original problem $(\mathrm{P1})$ is ensured to reach convergence. The integer number of reflecting elements can be determined by rounding the continuous solutions to problem $(\mathrm{P2})$. Moreover, the complexity of the greedy search algorithm in (7) is smaller than $\mathcal{O}\left(KN_{t}N_{r}\right)$ and the complexity of solving problem $(\mathrm{P2})$ via SCA is $\mathcal{O}\left(JK^{3.5}\right)$ with $J$ denoting the number of iteration while that of obtaining the optimal IRS beamforming is negligible due to the closed-form solution given in (9).