Spectral Efficiency vs Complexity in Downlink Algorithms for Reconfigurable Intelligent Surfaces

We consider a single user MISO communication system in which a transmitter equipped with $N_{\text{t}}$ antennas sends a data stream to a single-antenna receiver using both a direct LOS link and a RIS-aided NLOS link, the RIS being equipped with $N_{\text{RIS}}$ phase-shifters. The channel corresponding to the direct link between the transmitter and receiver is assumed to follow a Rayleigh channel model [13, 11, 12, 14, 15] and is modeled as ${\mathbf{h}}_{\text{d}}\in\mathbb{C}^{1\times N_{\text{t}}}$. The channel between the transmitter and the RIS surface is modeled as a Rayleigh matrix ${\mathbf{H}}_{1}\in\mathbb{C}^{N_{\text{RIS}}\times N_{\text{t}}}$, and the channel between the RIS and the receiver is modeled as ${\mathbf{h}}_{2}\in\mathbb{C}^{1\times N_{\text{RIS}}}$ using a Rayleigh channel model as well. The RIS behavior is modeled as $\bm{\Phi}\in\mathbb{C}^{N_{\text{RIS}}\times N_{\text{RIS}}}$, $\bm{\Phi}=\mathop{\mathrm{diag}}\{e^{{\mathsf{j}}\phi_{1}},\ldots,e^{{\mathsf{j}}\phi_{N_{\text{RIS}}}}\}$. We define an equivalent channel comprises of the LOS and NLOS paths between the BS and UE as follows

The received signal can then be expressed as

in which ${\mathbf{f}}\in\mathbb{C}^{N_{\text{t}}\times 1}$ is the transmit beamforming vector normalized so that $\|{\mathbf{f}}\|_{2}^{2}=1$, $s\in\mathbb{C}$ is the transmit symbol satisfying $\mathbb{E}\{|s|^{2}\}=1$, and $w\in\mathbb{C}$ is the receive circularly-symmetric additive white Gaussian noise sample $w\sim{\cal CN}(0,\sigma_{w}^{2})$. The SNR is defined as $\text{SNR}=\frac{1}{\sigma_{w}^{2}}$. In this paper, we focus on the joint problem of finding the beamformer ${\mathbf{f}}$ and RIS matrix $\bm{\Phi}$ that optimize the resulting spectral efficiency. Let ${\mathbf{h}}_{\text{eq}}\in\mathbb{C}^{1\times N_{\mathrm{t}}}$ be the equivalent channel including both the LOS and NLOS paths as $\mathbf{h}_{\text{eq}}=\left({\mathbf{h}}_{\text{d}}+{\mathbf{h}}_{2}\bm{\Phi}{\mathbf{H}}_{1}\right)$. The problem of finding ${\mathbf{f}}$ and $\bm{\Phi}$ can be formalized as

From (2), since the transmitted symbol is Gaussian, and the receive additive noise is also Gaussian, it follows that the spectral efficiency ${\cal R}({\mathbf{f}},\bm{\Phi})$ in (3) is given by

The spectral efficiency in (4) implicitly assumes a MISO communication link. To build intuition into the nature and solution to (3), in the next section we will first briefly review the SISO optimization, and then we will investigate the MISO optimization.

In [19], the simplest scenario of configuring the RIS elements to maximize spectral efficiency is considered for the single-user (SU) case. In the event that $N_{\text{RIS}}>1$, the resulting optimized RIS elements are a straightforward extension from the case of $N_{\text{RIS}}=1$. The spectral efficiency for the case $N_{\text{RIS}}>1$ is given by

in which $[\bm{\Phi}]_{i,i}=e^{{\mathsf{j}}\phi_{i}}$. From (5), it is clear that the optimum $\phi_{i}$, $1\leq i\leq N_{\text{RIS}}$, is the phase-shift that makes every complex element in the inner summation have the same phase as $h_{\text{d}}$. Let $\arg(z)$ denote the operation of extracting the phase of a complex number $z$. Then, the optimal elements of $\bm{\Phi}$ are provided as:

The $\phi_{i}^{*}$ are selected to coherently combine phases maximizing the squared magnitude of the equivalent channel $h_{\text{eq}}^{\text{SISO}}=h_{\text{d}}+{\mathbf{h}}_{2}\bm{\Phi}{\mathbf{h}}_{1}$, and in the case of $N_{\text{RIS}}=1$, the element $\phi_{i}$ is chosen to coherently combine the NLOS and LOS paths. The solution in (6) is very attractive due to its simplicity, but it has the drawback of not being easily extensible to the MISO scenario. As we’ll see next, the MISO cost function given below in (4) cannot be decoupled for the different $N_{\text{RIS}}$ phase elements $\phi_{i}$.

Considering a single RIS element in the SU-MISO setting yields a unique result. In this case, the beamformer ${\mathbf{f}}\in\mathbb{C}^{N_{\text{t}}\times 1}$ in (1) needs to be optimized as well, whose solution is given by the maximum ratio transmission (MRT) beamformer [20]. Similar to the SU-SISO scenario with both a single RIS element and multiple RIS elements in which the squared magnitude of the equivalent channel ${h_{\text{eq}}^{\text{SISO}}}$ was maximized, rate optimization in the MISO setting with a single RIS elements leads to optimization of the squared $\ell_{2}$-norm of the equivalent channel ${\mathbf{h}}_{\text{eq}}^{\text{MISO}}$, given by

Here, $\phi$ is the RIS element phase to be configured, $h_{2}\in\mathbb{C}^{\text{1x1}}$ is the channel scalar gain between the single RIS element and the UE, $\mathbf{h}_{d}\in\mathbb{C}^{1\times N_{\text{t}}}$ is the line of sight (LOS) path between the BS and UE, $\mathbf{h}_{1}\in\mathbb{C}^{1\times N_{\text{BS}}}$ is the MISO channel between the single element RIS and the BS. The above optimization yields the optimal $\phi^{\star}=-\arg(h_{2}{\mathbf{h}}_{1}{\mathbf{h}}_{\text{d}})$. This scenario is similar to that of a single element relay-assisted wireless communication system [8, 9], which is interesting for theoretical development, but less for practical purposes.

The scenario of more practical relevance is the SU-MISO scenario. Unlike the previously considered scenarios, this scenario does not allow for an optimal simple closed-form solution for the phases of the RIS elements. The optimization problem can be formally stated as follows:

Here, $\mathbf{h}_{eq}$ is defined in (1). The challenge posed by this scenario is handling the non-convex objective function in conjunction with the non-convex unit-magnitude constraints on each of the $N_{\text{RIS}}$ RIS elements. This constraint enforces the phase shifting operation of the RIS. Unlike relays, RIS do not amplify or decode-then-forward a received signal, as such RIS elements require unit magnitude [13, 14, 15, 16, 17, 18, 21, 22], which may be relaxed theoretically to achieve an approximate solution. Similar to the optimization in the prior scenarios, prior work optimizes the $||\mathbf{h}_{\text{eq}}||^{2}$, leading to suboptimal solutions. In the following section, we propose rate optimization schemes that either maximize $||\mathbf{h}_{\text{eq}}||^{2}$ or directly through the spectral efficiency.

The optimization problem is formulated as follows

The non-convex unit modal constraints in the optimization problem motivate the need for suboptimal tractable solutions to the rate optimization. The first approach to solve (LABEL:equation:optimization_magnitude) consists of an application of the power method. The power method is an iterative method which converges to the eigenvector corresponding to the strongest eigenvalue of a matrix. We consider the power method as a low complexity algorithm consisting of the repeated application of a diagonalizable matrix to a vector iteratively. Let $\mathbf{H}_{2}=\mathop{\mathrm{diag}}\left(\mathbf{h}_{2}\right)$, we consider the hermitian symmetric matrix $\mathbf{C}=\left(\mathbf{H}_{2}\mathbf{H}_{1}\right){\left(\mathbf{H}_{2}\mathbf{H}_{1}\right)}^{H}$ in Algorithm 1 listed below. We let $\mathbf{C}$ is contained in the expansion of the objective function $||\mathbf{h}_{\text{eq}}||_{2}^{2}$ where the diagonal $\mathbf{\Phi}$ is replaced with $\mathbf{H}_{2}$ and $\mathbf{h}_{2}$ . We also define a vector $\mathbf{b}$ as $\mathbf{H_{2}H_{1}h_{d}^{H}}$ and scalar $a$ as $\mathbf{h_{d}h_{d}^{H}}$.

In (10), $\mathbf{x}$ corresponds to the vectorization of $\mathbf{\Phi}^{c}$. The application of $\mathbf{C}$ over $k$ iterations leads to $\mathbf{x}_{k}$ being a linear combination of the eigenvectors of $\mathbf{C}$ with corresponding eigenvalues raised to the $k$th power. As $k$ tends to infinity, $\mathbf{x}$ tends to $\lambda_{0}^{k}x_{0}\mathbf{u_{0}}$; hence $\mathbf{x}_{k}$ points in the direction of the eigenvector $\mathbf{u}_{0}$ associated with the largest eigenvalue $\lambda_{0}$ of $\mathbf{C}$. The direction of $\mathbf{x}$ is of primary significance not the magnitude, as such we normalize $\mathbf{x}$ after each iteration to preserve unit length of $\mathbf{x}$ in order to avoid $\mathbf{x}$ from growing or shrinking without bound. Additionally we assume the maximum ratio transmisssion beamformer of form $\mathbf{f}=\mathbf{h}_{\text{eq}}^{H}/||\mathbf{h}_{\text{eq}}||_{2}$.

We observed that a random initialization of $\mathbf{x}$ was sufficient. We investigated initializing $\mathbf{x}$ with the solution to the linear system of equations $\mathbf{Cx}=\mathbf{b}$, and did not observe significant performance difference between that of $\mathbf{x}$ being randomly initialized.
To optimize $||\mathbf{h}_{\text{eq}}||_{2}^{2}$, we also consider two gradient-based approaches in Algorithms 2 and3. In Algorithm 3, we seek to optimize over the phases of $\mathbf{x}$ directly. In order to define the necessary gradient, we define $\mathbf{A}$ as $\mathop{\mathrm{diag}}{e^{-j\angle{\mathbf{x}}}}$, where $\angle{\mathbf{b}}$ is this initial set of RIS phases considered. The gradient of (11) with respect to the phases of $\mathbf{x}$ is

In Algorithm 2 we perform gradient ascent on $\mathbf{x}$, rather than on $\angle{\mathbf{x}}$ and define the corresponding gradient below:

We enforce the unit magnitude constraint on each RIS elements in $\mathbf{x}$ by projecting the solution at each iteration onto the feasible set by considering only the phases of the updated $\mathbf{x}$ with unit magnitude.

All iterative methods use a constant learning rate $\mu=.01$; a decaying learning rate can be considered in future work in efforts to enhance the gradient-based approach performances. Additionally, we consider a convergence threshold $\epsilon=10^{-10}$ for both methods.

Majority of prior work tries to optimize (10). Important contributions of this paper are twofold: i) to optimize the problem in (3) directly, and ii) to determine if there is a performance difference between optimizing over ${\cal R}({\mathbf{f}},\bm{\Phi})$ directly versus optimizing over $||\mathbf{h}_{\text{eq}}||_{2}^{2}$ in (10). This is an important analysis because in previous application cases an optimal closed-form solution for the elements of $\mathbf{\Phi}$ could be obtained; however in the SU-MISO with multiple RIS elements case– in which an optimal solution is not available– such an analysis is pertinent particular for future extensions into the MIMO case and beyond.

The proposed algorithm is a projected gradient ascent over ${\cal R}({\mathbf{f}},\bm{\Phi})$. We derive $\nabla_{\mathbf{\Phi}}{\cal R}({\mathbf{f}},\bm{\Phi})$, the gradient of ${\cal R}({\mathbf{f}},\bm{\Phi})$ with respect to $\mathbf{\Phi}$ below:

Similar to prior work [14, 15] and the other proposed algorithms, we employ a MRT beamformer at each iteration to configure ${\mathbf{f}}$. Unlike Algorithms 3 and 2, a decaying $\mu$ is utilized in order to facilitate convergence. We initialize $\mu$ to $0.5$ We also investigated initializing Algorithm 4 with the power method solution in comparison to random initialization and observed a performance difference which is shown in the following section. Additionally, we observed reinitializing the learning rate after the learning rate had decayed sufficiently– in order to escape a potential local optimal– did not lead to substantial performance difference between that of before reinitialization. The proposed projected gradient ascent method is detailed in Algorithm 4.

In our analysis we make a comparison with two algorithms from prior work which aim to optimize the spectral efficiency in a MISO RIS-assisted communication system [14, 15]. These particular algorithms were selected as appeared to be most relevant to the single user scenario considered. Furthermore, they are regarded as state-of-the-art solutions according to in [5].

From Fig. 1 and 2 we can observe how the spectral efficiency performance of proposed algorithms and comparison algorithms evolve as $N_{\text{t}}$ and SNR increase. We observe that Algorithm 3 is the only Algorithm which does not outperform prior work. We observe a similar behavior for different values of $\mathop{\mathrm{SNR}}$ or $N_{\text{t}}$. The Algorithms 2 and 3 have similar spectral efficiency performance when $N_{\text{t}}$ and $N_{\text{RIS}}$ are small, but as $N_{\text{t}}$ and $N_{\text{RIS}}$ increase, the performance of Algorithm 3 decays to the performance level of the SDR algorithm from [15]; while the Algorithm 2 maintains higher performance than both comparison algorithms. As such, we determine it is a better design choice to conduct gradient ascent over ${\mathbf{x}}$ rather than $\angle{\mathbf{x}}$ when optimizing (10). Furthermore, in Figs. 1 and 2 we observe that optimization directly over the spectral efficiency cost function yields the highest performance gains over the other proposed algorithms. Using the solution from Algorithm 1 as the initialization to algorithm 1 unlocks further spectral efficiency gains over the base Algorithm 4.

Another critical criterion for evaluating algorithms includes the computational complexity [21, 23]. We quantify the runtime complexity as the number of Floating Point Operations (FLOPs), and obtain the average spectral efficiency and average number of FLOPs in a Pareto scatter plot in Fig. 3; furthermore Fig. 3 allows understanding the Pareto frontier of the algorithms investigated within this paper. We aim to generate an algorithm which is characterized by high spectral efficiency performance and low complexity. We observe that, for the given experimental scenario, Algorithm 4 provides relatively high achievable spectral efficiency and relatively low computational complexity. Algorithm 4 using the power method initialization yields slightly higher spectral efficiency at the cost of multiplying the complexity of the algorithm by a factor of $4$ approximately. Algorithms 1, 3, and 2 are a cluster of algorithms classified by similar middle range performance in terms of spectral efficiency and complexity, however; amongst these middle range algorithms, Algorithm 1 provides the highest spectral efficiency and lowest complexity.