RIS-enhanced Resilience in Cell-Free MIMO

The channel model considered in this paper assumes quasi-static block fading channels, where the channel coefficients remain constant within the coherence time, but may change independently among coherence blocks. We denote the direct channel link between AP $n$ and user $k$ as $\mathbi{h}_{n,k}\in\mathbb{C}^{L\times 1}$. The reflected-channel link provided by the RIS between AP $n$ and user $k$ is denoted by $\mathbi{G}_{n,k}=\mathbi{H}_{n}\mathsf{diag}(\mathbi{v})\mathbi{g}_{k}\in\mathbb{C}^{L\times 1}$, where $\mathbi{H}_{n}\in\mathbb{C}^{L\times M}$ denotes the link between AP $n$ and the RIS, $\mathbi{g}_{k}\in\mathbb{C}^{M\times 1}$ denotes the link between the RIS and user $k$ and $\mathsf{diag}(\mathbi{v})\in\mathbb{C}^{M\times M}$ denotes the reflection coefficient matrix with $\mathbi{v}=[v_{1},v_{2},\dots v_{M}]\in\mathbb{C}^{M\times 1}$. Here, $v_{m}=e^{j\theta_{m}}$ is the reconfigurable reflection coefficient at the $m$-th reflecting element, which is composed of a phase shift $\theta_{m}\in[0,2\pi]$. Further, we denote the aggregate direct channel vector of user $k$ as $\mathbi{h}_{k}=[\mathbi{h}_{1,k}^{T},\mathbi{h}_{2,k}^{T},\dots,\mathbi{h}_{N,k}^{T}]^{T}\in\mathbb{C}^{NL\times 1}$, the aggregate AP to RIS channel matrix as $\mathbi{H}=[\mathbi{H}_{1}^{T},\mathbi{H}_{2}^{T},\dots,\mathbi{H}_{N}^{T}]^{T}\in\mathbb{C}^{NL\times M}$ and the aggregate transmit signal vector as $\mathbi{x}=[\mathbi{x}_{1}^{T},\mathbi{x}_{2}^{T},\dots,\mathbi{x}_{N}^{T}]^{T}\in\mathbb{C}^{NL\times 1}$.

Utilizing the aggregate vectors, the received signal at user $k$ can be expressed as the sum of the direct and reflected channel vectors, namely

where $\mathbi{G}_{k}=\mathbi{H}\text{diag}(\mathbi{g}_{k})$ and $n_{k}\sim\mathcal{C}\mathcal{N}(0,\sigma_{k})$ is the AWGN (AWGN) sample.

Further, the symbols intended to be decoded by user $k$ are denoted by $s_{k}$. We assume that these messages form an i.i.d. (i.i.d.) Gaussian codebook. These symbols for user $k$ are transmitted by the $n$-th AP using the beamforming vector $\mathbi{w}_{n,k}\in\mathbb{C}^{L\times 1}$, which are both provided by the CP over an ideal fronthaul.

Hence, the overall transmit signal vector at the $n$-th AP is given as $\mathbi{x}_{n}=\sum_{k\in\mathcal{K}}\mathbi{w}_{n,k}s_{k}$, which is subject to the power constraint $\mathbb{E}\{\mathbi{x}_{n}^{H}\mathbi{x}_{n}\}\leq P^{\mathsf{Max}}_{n}$, or equivalently

The received signal (1) at user $k$ is then given by

where the first term is the desired signal at the user and the second term is the received interference from all other users. Thus, we formulate the SINR (SINR) of user $k$ decoding its message as

Using these definitions, the QoS demands for each user are satisfied, if the following conditions are met

where $B$ denotes the transmission bandwidth and $r_{k}$ denotes the rate of user $k$.

This paper considers a specific target throughput, e.g., determined by the QoS requirements by the network with $r_{k}^{\mathsf{des}}$ being user $k$’s desired QoS requirement. Note that the QoS demands are assumed to remain constant within the observation interval. In contrast, the network’s sum throughput is dependent on the allocated resources at time $t$, and thus captured in $\sum_{k=1}^{K}r_{k}(t)$, where $r_{k}(t)$ is the allocated data rate for user $k$ at time $t$. Regarding the time axis, there are two major cornerstones on the resilience behavior, namely $t_{0}$, the initial time at which the degradation manifests and $t_{n}$, the time of recovery. With those aspects at hand, and considering the resilience metric proposed in [resiliencemetric, RobertRes], we define the networks absorption, adaption, and time-to-recovery metrics as

respectively, where $T_{0}$ is the network operator’s desired recovery time, i.e., the time for which a functionality degradation is tolerable. A linear combination of the equations (6)-(7) yields the considered resilience metric

with fixed weights $\lambda_{i}$, $i\in\{1,2,3\}$, denoting the network operator’s needs, e.g., emphasizing the robustness, the quality of adaption, or the recovery time. We also let $\sum_{i=1}^{3}\lambda_{i}=1$, thus, the best-case value for the resilience is $r=1$.

Note that the proposed resilience metric (8) can be temporally divided in anticipatory actions, i.e., $r_{\text{abs}}$, that take place before the outage occurs and reactionary actions , i.e., $r_{\text{ada}}$ and $r_{\text{rec}}$, which occur after the outage. Regarding the anticipatory actions, various work are concerned with designing wireless communication networks to be robust against adverse conditions, e.g., [scalableRS]. However, the literature on designing resilient wireless communication systems from the physical layer resource management perspective with a focus on adaption and time-to-recovery remains limited in breadth and depth. Thus, this paper mainly focuses the reactionary actions as they are sufficient to demonstrate the trade-off between the quality of a solution and the time necessary to obtain it. Consequently, we assume $\lambda_{1}=0$ throughout this work.

As a result of the alternating optimization approach, the phase shifters are assumed to be fixed for the duration of the beamforming design. Thus, problem (P1) can be rewritten as

where the introduction of the slack variables $\mathbi{q}=[q_{1},\dots,q_{K}]$ convexifies the rate expressions and $\mathbi{q}\geq 0$ signifies an element-wise inequality. However, the constraints in (12) are still non-convex but can be convexified using the SCA approach. To this end, we rewrite (12) as

and apply the first-order Taylor approximation around the point $(\tilde{\mathbi{w}},\tilde{\mathbi{q}})$ on the fractional term. Consequently, the following convex approximation of (14) can be derived, see also [SynBenefits],

Thus, the approximation of problem (P2) can be written as

Problem (P2.1) is convex and can be solved iteratively using the SCA method. More precisely, we define $\mathbi{\Lambda}_{z}^{w}=[\mathbi{w}_{z}^{T},\mathbi{\kappa}_{z}^{T}]^{T}$ as a vector stacking the optimization variables of the beamforming design problem at iteration $z$, where $\mathbi{\kappa}_{z}=[\mathbi{r}_{z}^{T},\mathbi{q}_{z}^{T}]^{T}$. Similarly $\hat{\mathbi{\Lambda}}_{z}^{w}=[\hat{\mathbi{w}}_{z}^{T},\hat{\mathbi{\kappa}}_{z}^{T}]^{T}$ and $\tilde{\mathbi{\Lambda}}_{z}^{w}=[\tilde{\mathbi{w}}_{z}^{T},\tilde{\mathbi{\kappa}}_{z}^{T}]^{T}$ denote the optimal solutions and the point, around which the approximations are computed, respectively. Thus, with a given point $\tilde{\mathbi{\Lambda}}_{z}^{w}$, an optimal solution $\hat{\mathbi{\Lambda}}_{z}^{w}$ can be obtained by solving problem (P2.1).

During the design process of the phase shifters at the RIS, the beamformers are assumed to be fixed due to the application of the alternating optimization approach. With the intent of utilizing a similar problem structure as in (P2), we denote $|(\mathbi{h}_{i}+\mathbi{G}_{i}\mathbi{v})^{H}\mathbi{w}_{k}|^{2}$ $=$ $\tilde{h}_{i,k}+\tilde{\mathbi{G}}_{i,k}\mathbi{v}$, where $\tilde{h}_{i,k}=\mathbi{w}_{k}^{H}\mathbi{h}_{i}$ and $\tilde{\mathbi{G}}_{i,k}=\mathbi{w}_{k}^{H}\mathbi{G}_{i}$. With the above definitions, the SINR constraints can be written similar to (14) as

At this point, the overall optimization problem for the phase shift design can be formulated as

where the penalty method [PenaltyMethod] for the phase shift constraints (10) is adopted and $C$ is a large positive constant. Similar to (14), (16) can be approximated by calculating the first-order Taylor approximation of (16) on $\mathbi{v}$ around the point $(\tilde{\mathbi{v}},\tilde{\mathbi{q}})$, denoted by $(\widetilde{\ref{SINR_rewritten_v}})$. Further, the objective function can be approximated around the point $\tilde{\mathbi{v}}$ by first-order Taylor approximation of the penalty term $C\sum_{m=1}^{M}(|v_{m}|^{2}-1)$, which is given by $C\sum_{m=1}^{M}\text{Re}\{2\tilde{v}_{m}^{*}v_{m}-|\tilde{v}_{m}|^{2}\}$. Based on the above approximation methods, problem (P3) is approximated by the following convex problem:

Due to the similarity of the problem formulation and the utilization of the same SCA framework, problem (P3.1) can be solved by defining $\mathbi{\Lambda}_{z}^{v}=[\mathbi{v}_{z}^{T},\mathbi{\kappa}_{z}^{T}]^{T}$ and following the same iterative procedure as for solving problem (P2.1).

In this section, we outline an alternating optimization procedure, that is suitable to be utilized for resilience applications. Usually, when considering an SCA approach in the literature, the goal is to retrieve the highest-quality solution for a given problem [SynBenefits, PenaltyMethod, SynBenefitsConf]. Hence, these works do not take the duration of obtaining these solutions into consideration and employ time-intensive outer and inner loops, until some convergence criteria are met.

In the context of resilience, however, we are not necessarily interested in the highest-quality results. Instead we aim to obtain a solution, which satisfies a specific quality-time trade-off specified by the weights $\lambda_{i}$ in (8). Consequently, the proposed resilience-aware alternating optimization works towards minimizing the network wide gap $\Uppsi$ and stops as soon as it lies below a certain threshold $\tau$. Further, it dispenses from the convergence criteria of the inner loops when solving the sub-problems. Instead, a fixed amount of iterations $T_{o},\forall o\in\{w,v\}$ of solving the problems (P2.1) and (P3.1) are introduced. This is possible due to the utilization of the same framework for both sub-problems because it makes $\tilde{\mathbi{\kappa}}$ a feasible point for both problems without additional adaption. The advantage of using fixed iterations lies in the fact that the intermediate solutions $\hat{\mathbi{w}}_{z}$ and $\hat{\mathbi{v}}_{z}$ of these sub-problems are also suitable to be evaluated in order to improve the resilience metric. This enables the algorithm to react to different values of $\lambda_{3}$, i.e., the weight of the recovery-time, efficiently. Note, that for $\lambda_{3}\ll 1$ and large $T_{o}$, the proposed algorithm reduces to behaving like the conventional convergence-based algorithms, thus representing a generalization to them. The detailed steps of the resilience-aware alternating optimization are illustrated in Algorithm 1.

First we study the effect of the amount of fixed iterations $T_{o}$ employed in Algorithm 1. To this end, we define two approaches: 1) an alternating approach (alt), in which we set $T_{o}=1,\forall o\in\{w,v\}$ and 2) a convergence-based approach (conv), in which each-subproblem is optimized until convergence. At this point, it also becomes important to decide which sub-problem is solved first after an outage occurs. Thus, we also compare the performance of both approaches above when initializing with either the beamforming problem $o=w$ (alt-BF, conv-BF) or the phase shifting problem $o=v$ (alt-PS, conv-PS) after an outage has occurred. Fig. LABEL:altVSconv depicts an outage, occuring at $t_{0}=1000$ms, and the changes in $r_{ada}$ of the different approaches over the time required to solve the respective sub-problems. Note that the slope between any two points of a curve represents the quality-time trade-off between those points.

It becomes apparent that both red curves, representing the approaches starting with the beamforming sub-problem, perform better right after the outage has occurred. The rationale behind this behaviour lies in the fact that the beamformer of any AP-link that is affected by an outage should be redirected to the RIS first, before reflecting any incoming signals at the RIS to the users. Fig. LABEL:altVSconv also shows that the alternating approaches are performing better with regards to the quality-time ratio between each iteration than the convergence-based ones. In addition, the alternating approach starting with the beamforming sub-problem (alt-BF) not only performs the best considering the quality-ti