Robust Beamforming Design and Antenna Selection for Dynamic HRIS-aided MISO Systems

As depicted in Fig. 1, a dynamic HRIS-aided uplink MISO communication system with a single-antenna user and an $N_{R}$-antenna BS is considered. The HRIS comprises $N_{a}$ active and $N_{p}$ passive elements, where $N_{a}+N_{p}=N$. To reduce the power consumption and hardware complexity, we assume the BS receives the data stream from $N_{R}$ antennas with $L<N_{R}$ dedicated RF chains. The antenna selection technique is employed to select $L$ out of $N_{R}$ antennas to explore the spatial diversity.

We denote ${\bf{h}}_{d}\in\mathbb{C}^{N_{R}\times 1}$ and ${\bf{G}}\in\mathbb{C}^{N\times N_{R}}$ as the full channel state information (CSI) associated with $N_{R}$ receive antennas from the BS to the user and the HRIS, respectively. The channel between the HRIS and the user is ${\bf{h}}_{r}\!\!\in\!\!\mathbb{C}^{N\times 1}$. Based on the receive antenna selection, $L$ out of $N_{R}$ or $N$ rows are selected to form the ${L\times 1}$ or ${L\times N}$ sub-channels, respectively. Denoting ${\bf{A}}\in\mathcal{A}$ as the index matrix for the $L$ selected antennas, where $\mathcal{A}\triangleq\{{\bf{A}}\in\mathbb{C}^{L\times N_{R}};A_{i,j}\in\{0,1\};\sum\nolimits_{j=1}^{N_{R}}A_{i,j}\!=\!1,\forall i;\sum\nolimits_{i=1}^{L}A_{i,j}\!\leq\!1,\forall j\}$, the sub-channels can be represented by ${\bf{A}}{\bf{h}}_{d}$ and ${\bf{A}}{\bf{G}}$. All channels are assumed to experience quasi-static flat fading and perfectly available using the recent advances in RIS and MISO channel estimation.¹¹1To delve into the unique effects of hardware impairments on the dynamic HRIS-assisted MISO communication system, we adopt the ideal CSI assumption, which is prevalent in contemporary studies [10, 11]. The development of a channel estimation method falls outside the scope of this work and is therefore reserved for future exploration. It is crucial to highlight that the optimization framework we propose is designed to be easily adaptable to scenarios characterized by imperfect CSI.

The dynamic HRIS, as shown in Fig. 1, is equipped with low-cost discrete phase shifters and RF switches. To avoid the bulky circuits in practical implementation, we adopt the “sub-connected architecture” for those active HRIS elements. This approach differs from the fully-connected one by employing a single power amplifier to serve all elements. The RF switches are used in conjunction with different phase-shift circuits, allowing HRIS elements to operate in either active or passive modes and independently control phase shifts while sharing a common amplification factor. The location and the number of active elements are optimized to enhance the system performance. Specifically, we denote ${\bm{\Phi}}\triangleq{\rm diag}(e^{j\phi_{1}},\dots,e^{j\phi_{N}})$ as the reflection phase matrix, where $e^{j\phi_{n}}$ denotes the discrete phase shift of the $n$-th element with $\phi_{n}\in\mathcal{X}\triangleq\{0,\frac{2\pi}{2^{B}},\dots,\frac{2\pi(2^{B}-1)}{2^{B}}\}$. $B$ is the number of quantization bits. The reflection amplitude matrix of HRIS is denoted as ${\bf{\Lambda}}\triangleq{\rm diag}(\omega_{1},\dots,\omega_{N})$, where $\omega_{n}$ represents the amplification factor of $n$-th element. For the active elements, we have $\omega_{n}>{\mu}_{\rm min}$ when $n\in\mathcal{N}_{a}$, where ${\mu}_{\rm min}\geq 1$ denotes the minimum amplification factor and $\mathcal{N}_{a}$ is the index set of all active elements. Otherwise, we have $\omega_{n}=1$ for those passive elements. With the same PA, all the active elements share one common amplification factor $\mu$. We introduce a binary diagonal selection matrix ${\bf{B}}$ to establish the relationship between the HRIS elements and the PA, i.e., ${\bf{B}}={\rm diag}({\bm{\gamma}})\in\mathcal{B}\triangleq\{{\bm{\gamma}}\in\mathbb{C}^{N\times 1};\gamma_{i}=1,\forall i\in\mathcal{N}_{a};\gamma_{i}=0,\forall i\notin\mathcal{N}_{a}\}$.

Due to the non-ideal phase shifters of HRIS, the phase error of the $n$-th element can be modeled as ${\bar{\phi}}_{n}\in\mathcal{U}[-\frac{\pi}{2^{B}},\frac{\pi}{2^{B}}]$, where $\mathcal{U}[\cdot]$ denotes the uniform distribution [12]. Therefore, the actual phase shift matrix with phase noise is ${\bm{\tilde{\Phi}}}={{\bm{\Phi}}}{\bm{\bar{\Phi}}}$, with ${\bm{\bar{\Phi}}}\triangleq{\rm diag}(e^{j{\bar{\phi}}_{1}},\dots,e^{j{\bar{\phi}}_{N}})$ denoting the phase noise matrix. To accurately model the realistic imperfect hardware, the additive HWIs are considered. Specifically, the aggregate residual HWIs for the transceiver can be modeled as the additive Gaussian distortion noise [13]. As such, the actual transmitted signal for the user is expressed as

where $s$ represents the transmitted symbol with $\mathbb{E}[ss^{H}]=1$. ${\bf{\kappa}}_{t}\!\sim\!\mathcal{CN}({{0}},{k_{t}^{2}}p)$ denotes the additive distortion noise whose variance is proportional to the transmit power $p$, where $k_{t}$ characterizes the distortion level.

Then, the uplink received signal at the BS is written as

where ${{\bf{y}}}$ denotes the un-distorted received signal and ${\bf{n}}_{b}$ stands for the additive white Gaussian noise (AWGN) whose elements are random variables with zero-mean and unit-variance, i.e., ${\bf{n}}_{b}\!\sim\!\mathcal{CN}({\bf{0}},{\sigma_{b}^{2}}{\bf{I}}_{L})$. ${\bf{G}}^{H}{\bf{B}}{\bf{\Lambda}}{\bm{\tilde{\Phi}}}{\bf{n}}_{a}$ in ${{\bf{y}}}$ represents the amplified noise associated with the active elements, where ${\bf{n}}_{a}\!\sim\!\mathcal{CN}({\bf{0}},{\sigma_{a}^{2}}{\bf{I}}_{N})$. ${\bm{\kappa}}_{r}$ stands for the receive distortion noise. We have ${\bm{\kappa}}_{r}\!\sim\!\mathcal{CN}(({\bf{0}},{k_{r}^{2}}{\rm\widetilde{diag}}(\mathbb{E}[{\bf{y}}{\bf{y}}^{H}]))$, whose variance is proportional to the average received power of the individual antennas at the BS. ${\rm diag}(\cdot)$ returns a square diagonal matrix whose diagonal elements are the same as the input. $k_{r}$ represents the received distortion level and $\mathbb{E}[{\bf{y}}{\bf{y}}^{H}]$ is derived as

where ${\bf{\Omega}}={\tilde{p}}\left({\bf{h}}{\bf{h}}^{H}\!+\!(1\!-\!\epsilon_{b}^{2}){\bf{G}}^{H}{\bf{\Lambda}}{\bm{\Phi}}{\rm\widetilde{diag}}({{\bf{h}}_{r}}{{\bf{h}}_{r}}^{H})({\bm{\Phi}}{\bf{\Lambda}})^{H}{\bf{G}}\right)+\sigma_{a}^{2}{\bf{G}}^{H}{\bf{B}}{\bf{\Lambda}}{\bf{\Lambda}}^{H}{\bf{B}}^{H}{\bf{G}}$ with ${\tilde{p}}=p(1+k_{t}^{2})$ and $\epsilon_{b}=\frac{\sin({\pi/2^{B}})}{{\pi/2^{B}}}$. ${\bf{h}}={\bf{h}}_{d}+\epsilon_{b}{\bf{G}}^{H}{\bf{\Lambda}}{\bm{\Phi}}{\bf{h}}_{r}$ denotes the average effective channel between the user and the BS. The equality $(a)$ holds since $\mathbb{E}[{\bm{\bar{\Phi}}}{\bf{A}}{\bm{\bar{\Phi}}}^{H}]=\epsilon_{b}^{2}{\bf{A}}+(1\!-\!\epsilon_{b}^{2}){\rm\widetilde{diag}}({\bf{A}})$ and $\mathbb{E}[{\bm{\bar{\Phi}}}{\bf{A}}]=\epsilon_{b}{\bf{A}}$.

Assume that the receive beamforming at the BS is ${\bf{w}}$, then the estimated symbol can be denoted as

with ${\bf{\tilde{h}}}={\bf{h}}_{d}+{\bf{G}}^{H}{\bf{\Lambda}}{\bm{\tilde{\Phi}}}{\bf{h}}_{r}$. Therefore, the average MSE²²2 Due to the correlation between the minimum MSE and the minimum average bit error rate (BER), the MSE metric provide a elegant expression for measuring the quality of service compared to the BER, which is difficult or intractable due to their dependence on the integral $Q$ function [14]. can be derived as

with ${\bf{Q}}\!=\!{\bf{A}}{\bf{\Omega}}{\bf{A}}^{H}\!+\!k_{r}^{2}{\rm\widetilde{diag}}({\bf{A}}{\bf{\Omega}}{\bf{A}}^{H})\!+\!{\tilde{\sigma}}_{b}^{2}{\bf{I}}_{L}$ and ${\tilde{\sigma}}_{b}^{2}\!=\!\sigma_{b}^{2}(1+k_{r}^{2})$.

In this paper, we mainly focus on enhancing transmission reliability of the considered system in terms of MSE. Specifically, the average MSE is minimized under the total power budget $P_{\rm HRIS}$ of HRIS by jointly optimizing the receive antenna selection matrix ${\bf{A}}$, the mode selection matrix ${\bf{B}}$, the reflection phase matrix ${\bf{\Phi}}$, and the reflection amplitude matrix ${\bf{\Lambda}}$. The total power consumption at the HRIS is calculated as $P\!=\!\mathbb{E}[|{\bf{\Lambda}}{\bf{B}}{\bm{\tilde{\Phi}}}{\bf{h}}_{r}{\tilde{s}}|^{2}]\!+\!\mathbb{E}[|{\bf{B}}{\bf{\Lambda}}{\bm{\tilde{\Phi}}}{\bf{n}}_{a}|^{2}]={\tilde{p}}|{\bf{\Lambda}}{\bf{B}}{\bf{h}}_{r}|^{2}+\sigma_{a}^{2}|{\bf{B}}{\bf{\Lambda}}|^{2}$. Then, the average MSE minimization problem can be formulated as

where (6c) denotes the discrete phase coefficient constraint and (6d) limits the amplification factor of the active elements. Unfortunately, the problem (P1) is intractable and NP-hard due to the integer constraints and tightly coupled variables. In the sequel, we first transform these intractable constraints and then present a PEBCD algorithm to obtain a stationary solution.

Since the reflection amplitude matrix ${\bf{\Lambda}}\!=\!{\rm diag}({\bm{\omega}})$ and the mode selection matrix ${\bf{B}}\!=\!{\rm diag}({\bm{\gamma}})$ are tightly coupled, we first introduce the auxiliary variable ${\bm{\tilde{\omega}}}={\rm diag}({\bf{\Lambda}}{\bf{B}})$, representing the reflection amplification matrix related to those active HRIS elements, as follows

where $\mu$ denotes the common amplification factor for the active elements. Thus, we have ${\bm{\tilde{\omega}}}=\mu{\bm{\gamma}}$ and ${\bm{\omega}}=(\mu-1){\bm{\gamma}}+{\bm{1}}$. The power constraint in (6b) can be transformed into $\mu^{2}{\bm{\gamma}}^{H}({\tilde{p}}{\rm\widetilde{diag}}({\bf{h}}_{r}{\bf{h}}_{r}^{H})+\sigma_{a}^{2}{\bf{I}}){\bm{\gamma}}\leq P_{\rm HRIS}$. Therefore, we equivalently optimize the common amplification factor $\mu$ and the mode selection vector ${\bm{\gamma}}$ in the following.

For the receive antenna selection matrix ${\bf{A}}$, we define ${\bf{a}}\triangleq{\rm vec}({\bf{A}}^{T})$ with ${\bf{a}}=\left[{\bf{a}}_{1}^{T},{\bf{a}}_{2}^{T},\dots,{\bf{a}}_{L}^{T}\right]^{T}$ and ${\bf{a}}_{i}\neq{\bf{a}}_{j},\forall i\neq j$. ${\bf{a}}_{i}\in\mathcal{S}_{a}\triangleq\{{\bf{a}}\in\mathbb{R}^{N_{r}}|{\bf{1}}^{T}{\bf{a}}=1,[{\bf{a}}]_{m}\in\{0,1\},\forall m\}$ denotes each ${\bf{a}}_{i}$ belongs to a special ordered set of type 1 (SOS1). On the other hand, to efficiently deal with the discrete phase coefficients, we adopt the binary representation for the discrete variable ${\bm{\theta}}\triangleq{\rm diag}({\bm{\Phi}})$. Defining ${\bm{\theta}}_{s}$ as ${\bm{\theta}}_{s}=[1,e^{j\frac{2\pi}{2^{B}}},\dots,\frac{2\pi(2^{B}-1)}{2^{B}}]^{T}$, we can rewrite ${\bm{\theta}}$ as ${\bm{\theta}}={\bf{Z}}{\bm{\theta}}_{s}$, where ${\bf{Z}}=[{\bf{z}}_{1},{\bf{z}}_{2},\dots,{\bf{z}}_{N}]\in\mathbb{C}^{N\times 2^{B}}$. Each ${\bf{z}}_{n}$ belongs to the SOS1, i.e., ${\bf{z}}_{n}\in\mathcal{S}_{z}\triangleq\{{\bf{z}}\in\mathbb{R}^{2^{B}}|{\bf{1}}^{T}{\bf{z}}=1,[{\bf{z}}]_{m}\in\{0,1\},\forall m\}$. Similar with the receive antenna selection matrix ${\bf{A}}$, we further define ${\bf{z}}\triangleq{\rm vec}({\bf{Z}}^{T})$ with ${\bf{z}}=\left[{\bf{z}}_{1}^{T},{\bf{z}}_{2}^{T},\dots,{\bf{z}}_{L}^{T}\right]^{T}$.

The binary variables $\{{{\bm{\gamma}},{\bf{a}},{\bf{z}}}\}$ still render the problem intractable and NP-hard. Though the optimal solution can be obtained by the state-of-the-art Gurobi solver embedded with the branch-and-bound (BB) method, the worst-case complexity increases exponentially with the scale of the problem. To balance the complexity and optimality, we propose a variational reformulation of the binary constraints. Specifically, these constraints can be equivalently transformed into the $l_{2}$ box non-separable constraints with the following lemma.

Based on Lemma 1 and introducing the auxiliary variables $\{{\bf{\tilde{u}}},{\bf{\tilde{v}}},{\bf{\tilde{q}}}\}$, the problem (P1) equivalently turns into the following mix-integer optimization problem:

where ${\bm{\tilde{\gamma}}}\triangleq 2{\bm{\gamma}}-{\bf{1}}$ and $\{{\bf{\tilde{a}}},{\bf{\tilde{z}}},{\bf{\tilde{v}}},{\bf{\tilde{u}}},{\bf{\tilde{q}}}\}$ are similarly defined w.r.t. $\{{\bf{a}},{\bf{z}},{\bf{v}},{\bf{u}},{\bf{q}}\}$.

To efficiently tackle the equality constraints in (10e), we penalize the complementary error directly by a penalty function. The resultant objective function is defined as

where the penalty term $\mathcal{J}_{\rho}={\rho}[(N\!-\!{\bm{\tilde{\gamma}}}^{T}{\bf{\tilde{u}}})\!+\!(LN_{r}\!-\!{\bf{\tilde{a}}}^{T}{\bf{\tilde{v}}})\!+\!(2^{B}N\!-\!{\bf{\tilde{z}}}^{T}{\bf{\tilde{q}}})]$. Thus, the problem (P3) turns into

In the sequel, we alternatively optimize these variables with the remaining ones fixed.

For any given $\{{\mu},{\bf{v}},{\bf{u}},{\bf{q}},{\bm{\gamma}},{\bf{a}},{\bf{z}}\}$, the linear MMSE detector is the optimal receive beamforming. Thus, the optimal receive beamforming ${\bf{w}}^{\star}$ is written as

In this subsection, we optimize the common amplification factor for the active elements of the HRIS with the other variables fixed. Since the penalty term $\mathcal{J}_{\rho}$ is irrelevant to the amplification factor $\mu$, we first rewrite the objective $f_{\rm MSE}$ as $f_{\rm MSE}\!=\!f_{{\bm{\omega}},{\bm{\tilde{\omega}}}}\!+\!{\tilde{p}}{\bf{h}}_{d}^{H}{\bf{X}}{\bf{h}}_{d}\!+\!{\tilde{\sigma}}_{b}^{2}{\bf{w}}^{H}{\bf{w}}\!-\!2\sqrt{p}\Re\{{\bf{h}}_{d}^{H}{\bf{A}}^{H}{\bf{w}}\}\!+\!1$ utilizing the matrix identity of ${\rm Tr}({\rm{diag}}({\bf{a}})^{H}{\bf{B}})\!=\!{\bf{a}}^{H}{\rm{diag}}({\bf{B}})$ and ${\rm Tr}({\bf{A}}{\rm{diag}}({\bf{b}}){\bf{C}}{\rm{diag}}({\bf{b}})^{H})\!=\!{\bf{b}}^{H}({\bf{C}}^{T}\odot{\bf{A}}){\bf{b}}$. And $f_{{\bm{\omega}},{\bm{\tilde{\omega}}}}$ is defined as

with the auxiliary terms ${\bf{k}}={\tilde{p}}\epsilon_{b}{\rm diag}({\bf{\Phi}}^{H}{\bf{G}}{\bf{X}}{\bf{h}}_{d}{\bf{h}}_{r}^{H})-\sqrt{p}\epsilon_{b}{\rm diag}({\bf{\Phi}}^{H}{\bf{G}}{\bf{A}}^{H}{\bf{w}}{\bf{h}}_{r}^{H})$ and ${\bf{K}}=({\bf{\Phi}}^{H}{\bf{G}}{\bf{X}}{\bf{G}}^{H}{\bf{\Phi}})\odot({\tilde{p}}\epsilon_{b}^{2}{{\bf{h}}_{r}}{{\bf{h}}_{r}}^{H}+{\tilde{p}}(1-\epsilon_{b}^{2}){\rm\widetilde{diag}}({{\bf{h}}_{r}}{{\bf{h}}_{r}}^{H}))^{T}$. ${\bf{X}}$ is defined as ${\bf{X}}={\bf{A}}^{H}({\bf{w}}{\bf{w}}^{H}+k_{r}^{2}{\rm\widetilde{diag}}({\bf{w}}{\bf{w}}^{H})){\bf{A}}$.

Substituting ${\bm{\tilde{\omega}}}=\mu{\bm{\gamma}}$ and ${\bm{\omega}}=(\mu\!-\!1){\bm{\gamma}}\!+\!{\bm{1}}$ into $f_{{\bm{\omega}},{\bm{\tilde{\omega}}}}$, it can be further transformed to $f_{{\bm{\gamma}},{\mu}}$ as follows:

With the fixed ${\bm{\gamma}}$, the problem for optimizing $\mu$ at $(t\!+\!1)$-th iteration can be formulated as

where the parameters $a,b$ and $\mu_{\rm ref}$ are computed as

The $\mu_{\rm ref}$ represents the maximum amplification factor that is constrained by the transmit power allocated to the active elements at the current iteration. Then we have the following proposition concerning the amplification factor selection:

Then, we consider the non-trivial case with $P_{\rm HRIS}\geq P_{\rm min}$, where the number of active elements $N_{\rm act}>0$. The optimal amplification factor for the unconstrained minimization in problem (P4) can be derived as

Thus, the optimal $\mu^{\star}$ for problem (P4) can be obtained by comparing $\tilde{\mu}$ and $\mu_{\rm ref}$ as follows:

For the auxiliary variables ${{\bf{u}},{\bf{v}},{\bf{q}}}$, the problem turns into the following individual convex optimization problems:

The optimal closed-form solutions ${{\bf{u}}^{\star},{\bf{v}}^{\star},{\bf{q}}}^{\star}$ are readily derived as

In this subsection, we alternatively optimize the dynamic mode selection vector ${\bm{\gamma}}$, the receive antenna selection vector ${\bf{a}}$ and the reflection phase selection vector ${\bf{z}}$ using the same optimization oracles.