Reconfigurable Intelligent Surfaces Empowered Cooperative Rate Splitting with User Relaying

Consider a multi-user multiple-input single-output (MISO) transmission network, where a BS equipped with $N_{t}$ transmit antennas simultaneously serves two single-antenna users indexed by $U_{1}$ and $U_{2}$. Without loss of generality, $U_{1}$ is assumed to have a better channel to the BS than $U_{2}$, and it opportunistically acts as a half-duplex (HD) relay to forward the signal to $U_{2}$. There is one RIS with $N$ passive reflecting elements to assist the transmission from BS to the two users and from $U_{1}$ to $U_{2}$. The RIS elements are indexed by $\mathcal{N}=\{1,\ldots,N\}$. The channels between BS and RIS, BS and $U_{k}$, RIS and $U_{k}$, $U_{1}$ and $U_{2}$ are denoted by $\mathbf{G}\,\in\,\mathbb{C}^{N\times N_{t}},\,\mathbf{g}_{k}\,\in\,\mathbb{C}^{N_{t}\times 1}$, $\mathbf{h}_{k}\,\in\,\mathbb{C}^{N\times 1}$ and $h_{12}$, respectively. The system model is delineated in Fig. 1.

Two phases are involved in the proposed model, namely, the direct transmission phase and the cooperative transmission phase. In the direct transmission phase, also known as (a.k.a.) the first time slot, the BS transmits signals to two users based on the principle of 1-layer RSMA. In the meanwhile, the transmit signal is reflected by the RIS to the two users. In the cooperative transmission phase, a.k.a. the second time slot, the BS is silent and $U_{1}$ transmits the common stream of 1-layer RS to $U_{2}$ and the signal is reflected by the RIS to $U_{2}$. We assume $\beta\enspace(0<\beta\leq 1)$ as the fraction of time allocated to the first phase. The rest $(1-\beta)$ is allocated to the second phase.

The goal of this work is to maximize the user fairness. In particular, we jointly design the transmit beamforming $\mathbf{P}$, rate allocation $\mathbf{a}$, RIS phases $\bm{\Theta}=[\bm{\Theta}_{(1)},\bm{\Theta}_{(2)}]$ and time slot allocation $\beta$ with the aim of maximizing the minimum rate (max-min rate) of the users while satisfying the transmit power constraint. The max-min rate problem for the proposed RIS-aided CRS is formulated as

where constraint (9b) ensures that each user successfully decodes the common stream. Constraint (9e) is the range of the phase for each RIS element, and (9f) presents the transmit power constraint at the BS. Problem $\mathcal{P}_{1}$ is highly intractable and non-convex. To solve the problem, we develop an AO-based optimization algorithm in the next section to alternatively optimize the RIS phases $\mathbf{\Theta}$ and the remaining variables $\mathbf{P,a},\beta$.

With given $\bm{\Theta}$, the channels from BS to IRS and IRS to users are fixed. For notational simplicity, we denote the effective channel in the first time slot as $\mathbf{\widetilde{g}}_{k}=\mathbf{g}_{k}+\mathbf{G}^{H}\mathbf{\Theta}_{(1)}^{H}\mathbf{h}_{k}$. By introducing an auxiliary variable $t$ to denote the minimum rate of the two users, problem $\mathcal{P}_{1}$ is equivalently transformed to

$\mathcal{P}_{2}$ is still non-convex due to the rate expressions $r_{k}^{[1]}$ and $R_{c}$ in (10b) and (9b). To solve $\mathcal{P}_{2}$, we introduce slack variable vectors $\bm{\alpha}=[\alpha_{1},\alpha_{2}]$, $\bm{\alpha}_{c}=[\alpha_{c,1},\alpha_{c,2}]$, $\bm{\rho}=[\rho_{1},\rho_{2}]$ and $\bm{\rho}_{c}=[\rho_{c,1},\rho_{c,2}]$, where $\bm{\alpha}$ and $\bm{\alpha}_{c}$ denote the private and common stream rate vectors, respectively, $\bm{\rho}$ and $\bm{\rho}_{c}$ denote the signal-to-interference-plus-noise ratio (SINR) vectors for private and common streams, respectively. With the slack variables introduced above, $\mathcal{P}_{2}$ is equivalently transformed into

To handle the nonconvexity of constraints (11b)–(11d), (11g) and (11h), we next apply the SCA method. Constraint (11b) contains a bilinear function $\beta\alpha_{k}$, which can be rewritten as $\beta\alpha_{k}=\frac{1}{4}(\beta+\alpha_{k})^{2}-\frac{1}{4}(\beta-\alpha_{k})^{2}$. Therefore, $\beta\alpha_{k}$ can be approximated by the first-order Taylor approximation at the point $(\beta^{[l]},\alpha_{k}^{[l]})$, which is given as

With (12), (11b)–(11d) are approximated at iteration $l$ around the point $(\beta^{[l]},\bm{\alpha}^{[l]},\bm{\alpha}^{[l]}_{c})$ as

By further transforming (11g) and (11h) into the following difference-of-convex (DC) forms and using the first-order Taylor approximations to reconstruct the concave parts of the DC constraints, constraints (11g) and (11h) at iteration $l$ are approximated respectively at the point $(\mathbf{P}^{[l]},\bm{\rho}^{[l]},\bm{\rho}_{c}^{[l]})$ by

With the above approximations, problem $\mathcal{P}_{2}$ at iteration $l$ can be approximated by the following convex problem

Problem (15) is a convex problem, which can be solved using the SCA method. The detailed process of the SCA method to solve $\mathcal{P}_{2}$ is illustrated in Algorithm 1.

Given transmit beamforming vector $\mathbf{P}$, rate allocation $\mathbf{a}$, and time slot allocation $\beta$, we also introduce an auxiliary variable $t$ and reformulate problem $\mathcal{P}_{1}$ as

As the channel between $U_{1}$ and $U_{2}$ is known, the RIS phase in the cooperative transmission time slot can be designed as $\theta^{(2)}_{n}=\arg(h_{12})-\arg([\mathbf{h}_{1,r}]_{n}[\mathbf{h}_{r,2}]^{\ast}_{n})$ [15]. Hence, we only need to optimize the RIS phase matrix $\mathbf{\Theta}_{(1)}$. We denote $\nu_{n}=e^{j\theta_{n}^{(1)}},\forall n\in\mathcal{N}$, $\bm{\nu}=[\nu_{1},\ldots,\nu_{N}]^{T}$, and $R_{0}=2^{\frac{a_{1}+a_{2}}{\beta}}-1$. With the assistance of $\bm{\nu}$, we obtain $\mathbf{h}_{k}^{H}\mathbf{\Theta}_{(1)}\mathbf{G}\mathbf{p}_{i}=\mathbf{d}_{k,i}^{H}\bm{\nu}$, where $\mathbf{d}_{k,i}=(\textrm{diag}(\mathbf{h}_{k}^{H})\mathbf{G}\mathbf{p}_{i})^{\ast}\in\mathbb{C}^{N}$. To ease notations, we denote $g_{k,i}=\mathbf{g}_{k}^{H}\mathbf{p}_{i}$. We further introduce slack variables $\bm{\eta}=[\eta_{1},\eta_{2}]$, $\bm{\delta}=[\delta_{1},\delta_{2}]$, $\eta_{c,2}$, and $\delta_{c,2}$. $\bm{\eta}$ and $\bm{\delta}$ denote the SINR and rate vectors of private streams, respectively. $\eta_{c,2}$ and $\delta_{c,2}$ denote the SINR and rate of the common stream of $U_{2}$ in the first time slot. Then, problem $\mathcal{P}_{3}$ is reformulated as

To handle the non-convexity of constraint (17f), we adopt the penalty method to transform $\mathcal{P}_{3.1}$ into

where $C$ is a large positive constant. The objective function can be approximated around the point $(\bm{\nu}^{[l]})$ at iteration $l$ by the first-order Taylor approximation of $C\sum_{n=1}^{N}(|\nu_{n}|^{2}-1)$, which is given by

To handle the nonconvexity of constraints (17g) and (17h), we introduce variables $\bm{\kappa}=[\kappa_{1},\kappa_{2}]$ and $\kappa_{c,2}$, and constraints (17g) and (17h) are equivalent to

and

(20) and (21) are still non-convex, we therefore adopt the first-order Taylor approximation at point $(\bm{\nu}^{[l]},\bm{\eta}_{k}^{[l]},\bm{\kappa}^{[l]},\eta_{c,2}^{[l]},\kappa_{c,2}^{[l]})$. (20) and (21) become

Similarly, constraint (17i) can be approximated by

Based on the above approximation methods, problem $\mathcal{P}_{3}$ is approximated by the following convex problem at iteration $l$.

The detailed SCA method to solve $\mathcal{P}_{3}$ is illustrated in Algorithm 2.

The proposed AO algorithm to jointly optimize the RIS phases, transmit beamforming, common rate and time slot allocation is shown in Algorithm 3. Starting from a feasible beamforming matrix $\mathbf{P}^{[0]}$, time slot allocation $\beta^{[0]}$, rate allocation $\mathbf{a}^{[0]}$, a RIS phase matrix $\mathbf{\Theta}^{[0]}_{(1)}$, in $l$-th iteration, we first update $(\mathbf{P}^{[l-1]},\beta^{[l-1]},\mathbf{a}^{[l-1]})$ with a fixed $\mathbf{\Theta}^{[l-1]}_{(1)}$ by Algorithm 1. For a given $(\mathbf{P}^{[l]},\beta^{[l]},\mathbf{a}^{[l]})$, the RIS phase matrix $\mathbf{\Theta}^{[l]}_{(1)}$ is then updated based on the SCA method with $\mathbf{\Theta}^{[l-1]}_{(1)}$ as the initialization point. The max-min rate $t$ is then calculated based on the updated $(\mathbf{P}^{[l]},\beta^{[l]},\mathbf{a}^{[l]})$ and $\mathbf{\Theta}^{[l]}$. The process is repeated until convergence.

Convergence Analysis: For the precoder optimization problem $\mathcal{P}_{2}$ , the proposed SCA algorithm ensures the monotonic increase of the objective function. This is due to the fact that the solution of problem (15) at iteration $l-1$ is a feasible point of problem (15) at iteration $l$. Due to the transmit power constraint (9f), the solution sequence $\{t^{[l]}\}^{l=\infty}_{l=1}$ is bounded above, which implies that the convergence of Algorithm 1 is guaranteed. Similarly, for the RIS phase problem $\mathcal{P}_{3}$, the solution of problem (24) at iteration $l-1$ is also a feasible point for the problem (24) at iteration $l$. Due to the modulus constraint (9e), the solution sequence $\{t^{[l]}\}^{l=\infty}_{l=1}$ is bounded above, which guarantees the convergence of Algorithm 2. The monotonic increase of the objective functions for $\mathcal{P}_{2}$ and $\mathcal{P}_{3}$ implies that the solution of $\mathcal{P}_{1}$ at iteration $l-1$ is also a feasible point for $\mathcal{P}_{1}$ at iteration $l$. Hence, the convergence of AO algorithm is guaranteed.