Double Self-Sustainable Reconfigurable Intelligent Surfaces Aided Wireless Communications

In Fig.1, we consider a multi-user MIMO communication system aided by double self-sustainable RISs. The system comprises a BS with $M$ antennas, RIS1 deployed at the BS side with ${N_{1}}$ reflection elements, RIS2 deployed at the users side with ${N_{2}}$ reflection elements, and $K$ single-antenna users. To achieve self-sustainable transmission of the RISs, the RISs are installed with energy harvesting circuit to harvest energy from the BS. The channel from the BS to RIS1, the BS to RIS2, RIS1 to RIS2, RIS1 to the $k$-th user, and RIS2 to the $k$-th user are denoted as ${{\bf{H}}_{1}}\in{\mathbb{C}^{M\times{N_{1}}}}$, ${{\bf{H}}_{2}}\in{\mathbb{C}^{M\times{N_{2}}}}$, ${\bf{D}}\in{\mathbb{C}^{{N_{1}}\times{N_{2}}}}$, ${{\bf{h}}_{1,k}}\in{\mathbb{C}^{{N_{1}}\times 1}}$, ${{\bf{h}}_{2,k}}\in{\mathbb{C}^{{N_{2}}\times 1}}$, while $k\in{\cal K}=\left\{{1,\cdots,K}\right\}$. Let ${{\bm{\theta}}_{1}}={\left[{{\theta_{1,1}},\ldots,{\theta_{1,{N_{1}}}}}\right]^{T}}$, and ${{\bm{\theta}}_{2}}={\left[{{\theta_{2,1}},\ldots,{\theta_{2,{N_{2}}}}}\right]^{T}}$ represent phase shifts of the reflection elements on RIS1 and RIS2, respectively. The transmitted signal from the BS is given by

where ${{s_{k}}}$, ${{\bf{w}}_{k}}\in{\mathbb{C}^{M\times 1}}$ represent the baseband signal and the beamforming vector sent to the $k$-th user, respectively.

In this work, the RISs adopt power splitting (PS) scheme [9], where all elements of the RISs harvest a fraction of energy from the incident signal, and the remaining energy is reflected. Let ${\beta_{1}},{\beta_{2}}$ denote amplitude coefficients of the RISs. To reduce design complexity, all elements of the RISs are assumed to have the same amplitude coefficient. Thus, the received signal at the $k$-th user is expressed as

which represents the joint channel from the BS to the $k$-th user and is composed of the double-hop channel and two single-hop channels. ${{\bm{\Theta}}_{1}}={\rm diag}\left({{{\bm{\theta}}_{1}}}\right){\mathbb{C}^{{N_{1}}\times{N_{1}}}}$, ${{\bf{\Theta}}_{2}}={\rm diag}\left({{{\bm{\theta}}_{2}}}\right){\mathbb{C}^{{N_{2}}\times{N_{2}}}}$ denote diagonal phase shift matrixes of RIS1 and RIS2, respectively. $n\sim{{\cal N}_{c}}\left({0,{\sigma^{2}}}\right)$ is the noise vector at the $k$-th user, where ${{\sigma^{2}}}$ is the noise power [5]. Let ${{{\bf{\tilde{D}}}}_{k}}{\bf{=D}}{\rm diag}\left({{{\bf{h}}_{2,k}}}\right)=\left[{{{{\bf{\tilde{d}}}}_{1,k}},\cdots,{{{\bf{\tilde{d}}}}_{{N_{2}},k}}}\right]$. ${{\bf{R}}_{\lambda,k}}={{\bf{H}}_{\lambda}}{\rm diag}\left({{{\bf{h}}_{\lambda,k}}}\right)$ denotes the channel from the BS to ${\rm{RIS\lambda}}$ to the $k$-th user via ${\rm{RIS\lambda}}$, where $\lambda\in\left\{{1,2}\right\}$. The joint channel can be rewritten as

Let ${{\bf{Q}}_{n_{2},k}}\!=\!{{\bm{{\rm H}}}_{1}}{\rm diag}(\!{{{\bf{\tilde{d}}}}_{n_{2},k}}\!)$ represent the channel from the BS to the $k$-th user via RIS1 and RIS2. Both amplitude coefficients and the phase shifts are not considered in the single-hop and multi-hop channels mentioned above. Accordingly, signal-to-interference-plus-noise ratio (SINR) of the received information at the $k$-th user is formulated as

where $S$ denotes the received signal power, which is given by

and $I$ denotes the interference power, which is given by

Assuming the RISs adopt a linear energy harvesting model [9]. Hence, harvested energy at RIS1 and RIS2 are given by

where $\eta$ is the energy harvesting efficiency.

We aim to minimize the transmission power at the BS. The optimization is subject to QoS constraints for each user and energy consumption of the RISs. Thus, the problem is stated as follows

where ${\overline{\Gamma}_{k}}$, $\mu$, and ${P_{\max}}$ denote the desired QoS for each user, energy consumption of each RIS element, and transmission power budget at the BS, respectively. Constraint (10i) is QoS constraints, which means that the SINR of each user should satisfy the SINR target. Constraints (10i) and (10i) are energy consumption constraints, which means that energy consumption of each RIS is no more than harvesting energy of each RIS. Constraints (10i)$-$(10i) represent transmission power budget and the limits of amplitudes and the phase shifts at the RISs, respectively.

Given ${{\bf{\Theta}}_{1}}$ and ${{\bf{\Theta}}_{2}}$, we can write the subproblem as

problem (P2) can not be solved directly because constraints (10i), (10i), (10i) are non-convex. In this section, we introduce a precoding algorithm utilizing SCA to approximate the problem of minimizing transmission power as a convex optimization problem. We phase shifting the constraint (10i) and then perform a first-order Taylor expansion at the given point with $n$-th iteration ${\bf{w}}_{k}^{\left(n\right)}$, the constraint (10i) is equivalent to

similarly, constraints (10i) and (10i) are equivalent to

then problem (P2) can be reformulated as

which can be solved directly.

Given ${\bf{w}}_{k}$, the problem is modeled as

obviously, due to the coupling between ${{\bf{\Theta}}_{1}}$ and ${{\bf{\Theta}}_{2}}$, problem (P3) remains non-convex. We optimize ${{\bf{\Theta}}_{1}}$ when fixing ${{\bf{\Theta}}_{2}}$, the problem can be modeled as

where ${{\bf{q}}_{k,i}}={\left({\sum\nolimits_{{n_{2}}=1}^{{N_{2}}}{{{\bf{Q}}_{n_{2},k}}{\beta_{1}}{\beta_{2}}{\theta_{2,{n_{2}}}}}+{{\bf{R}}_{1,k}}{\beta_{1}}}\right)^{H}}{{\bf{w}}_{i}}$, ${{{{\bar{q}}}}_{k,i}}={\left({{\beta_{2}}{{\bf{R}}_{2,k}}{{\bm{\theta}}_{2}}}\right)^{H}}{{\bf{w}}_{i}}$. Due to constraint (19c), problem (P3.1) remains non-convex. Constraint (19c) is equivalent to

where

with ${{\bf{\Psi}}_{1}}\succeq 0$ and $\rm rank\left({{{\bf{\Psi}}_{1}}}\right)=1$, problem (P3.1) can transform into

Previous researches have often employed Gaussian randomization to handle the rank-one constraint and devise approximate solutions when addressing similar problems [9]. However, this approach does not yield satisfactory results in the alternating optimization algorithm. Hence, we adopt a penalty-based method as an alternative [10]. To be specific, the objective function is rewritten as

To solve the concave function (27), we rewrite (27) by employing the first-order Taylor expansion [10]

where

in which $\tau$ is penalty factor. ${{{\bm{\varphi}}_{1\max}}}$ denotes the eigenvector associated with the maximum eigenvalue of a matrix ${{{{\bf{\hat{\Psi}}}}_{1}}}$, which can be solved directly.

Next, the solution methods for ${{\bf{\Theta}}_{1}}$ and ${{\bf{\Theta}}_{2}}$ are symmetrical. By optimizing ${{\bf{\Theta}}_{2}}$ in the same manner, we can rewrite problem (P3) as

where ${{\bf{p}}_{k,i}}={\left({\left[{{{\bf{Q}}_{1,k}}{{\bm{\theta}}_{1}},\cdots,{{\bf{Q}}_{{N_{2}},k}}{{\bm{\theta}}_{1}}}\right]{\beta_{1}}{\beta_{2}}+{{\bf{R}}_{2,k}}{\beta_{2}}}\right)^{H}}{{\bf{w}}_{i}}$, ${{\bar{p}}_{k,i}}={\left({{\beta_{1}}{{\bf{R}}_{1,k}}{{\bm{\theta}}_{1}}}\right)^{H}}{{\bf{w}}_{i}}$. We can solve (P3.4) by following the procedure outlined in (P3.1) to (P3.3).

Firstly, for problem (P1), we have obtained optimal solutions for two subproblems, and the objective function monotonically decreases as the number of iterations increases. Secondly, to ensure normal operation of the system, there exists a lower bound for transmission power.

Calculating the complexity of two subproblems separately. In the subproblem of optimizing ${\bf{w}}_{k}$ based on SCA, the complexity is ${\cal O}\left\{\left({{M^{3}}{K^{3.5}}}\right){S_{1}}\right\}$, where ${S_{1}}$ is the number of iterations. In the subproblem of optimizing ${\bf{\Theta}}$ based on alternating optimization, the complexity is ${\cal O}\left\{\left(\sqrt{K+N}\left({{M^{3}}{K^{3}}}\right)\right){S_{2}}\right\}$, where ${S_{2}}$ is the number of iterations. In conclusion, overall complexity of the algorithm is ${\cal O}\left\{\left({\left({{M^{3}}{K^{3.5}}}\right){S_{1}}+\sqrt{K+N}\left({{M^{3}}{K^{3}}}\right){S_{2}}}\right)Z\right\}$, where $Z$ is the number of iterations in the outer loop.