RIS-assisted MMWave hybrid system under hardware damage: beamforming design and performance analysis (paper reading notes)

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foreword

This article is a reading note for the Hardware-Impaired RIS-assisted mmWave Hybrid
Systems: Beamforming Design and Performance
Analysis
paper

提示：以下是本篇文章正文内容，下面案例可供参考

1. Summary

Reconfigurable Smart Surface (RIS) is an innovative technology to assist millimeter wave (MMWave) communication. Due to the advantages of low hardware cost and low power consumption, the hybrid transceiver structure has also become an integral part of the MMWAVE system. However, due to the practical limitations of hardware devices, unavoidable hardware impairments (HWIs) usually occur in RIS-assisted MMWave communication. In this paper, under the practical discrete phase shift constraints, the MSE (sum) and the average rate of the RIS-assisted point topology MMWave MIMO system are minimized and the average rate is maximized by jointly optimizing the reflection coefficient of the hybrid transceiver and RIS, respectively. We first consider the single-antenna user case and propose an efficient Alternating Optimization (AO) algorithm to solve these two intractable problems. For the discrete optimization problem involved, a Binary-Oriented Exact Penalty (BEP) method is proposed, which can achieve a good trade-off between performance and complexity. The optimality of the AO algorithm under the condition of cascaded line-of-sight channels is analyzed, and the MSE floor effect and the average rate saturation effect under high SNR are revealed. The above research is then generalized to the general multi-antenna user case, and a low-complexity two-stage scheme is proposed, which aims to create favorable RIS cascaded channels in the first stage and improve system performance in the second stage. In the LOS scenario, this two-stage scheme has also been proven to achieve optimal performance. Numerical results validate our theoretical analysis and illustrate the superior performance of the proposed algorithm over various benchmark schemes.

2. Introduction

The proliferation of emerging data-hungry applications is driving future wireless networks toward the millimeter wave (MMWave) frequency band, which can provide abundant bandwidth to achieve gigabit-per-second throughput. However, due to the high carrier frequency, MMWave communication usually suffers from severe path loss, which greatly degrades the communication performance. To this end, massive multiple-input multiple-output (MIMO) technology with highly directional, high-gain beam capability is considered to be a promising solution. Typically, conventional all-digital beamforming architectures are impractical for large-scale arrays because it requires each antenna to be equipped with a dedicated radio frequency (RF) chain, which results in prohibitive hardware cost and power consumption [1]-[3]. As a remedy, hybrid beamforming architectures consisting of low-dimensional digital beamformers and high-dimensional analog beamformers have become an integral part of MMWave communications, where the number of expensive RF chains is greatly reduced, while the performance is only slightly degraded [2]. In addition, MMWave signals are also very susceptible to blocking, resulting in high penetration loss, such as concrete walls and buildings. A mature solution to this problem is to apply MIMO relay technology to establish multi-hop communication, but this requires complex signal processing and high energy consumption, so large-scale deployment is costly and impractical [3].

In recent years, Reconfigurable Smart Surface (RIS) technology has rapidly developed as a cost-effective and attractive technology that can help MMWave communication with severe propagation and penetration losses [4], [5]. RIS consists of an intelligent controller and a large number of passive reflective elements, each of which can dynamically adjust the amplitude and phase shift (collectively referred to as reflection coefficient) of the incident signal [4], [5]. Therefore, RIS is able to reshape the wireless propagation environment by adaptively tuning all reflection coefficients. Compared with conventional MIMO relays, RIS does not require radio frequency links and thus has lower energy consumption. In addition, due to its modularity and flexible deployment features, it is more suitable for large-scale upgrades of current wireless systems. Based on the above discussion, combining the three cutting-edge technologies of passive RIS, Massive MIMO and hybrid beamforming technology can greatly improve the coverage and performance of MMWave communication while maintaining low hardware cost and low energy overhead.

Extensive research has been done on the application of RISassisted mmWave communication in different configurations, such as single-user multiple-input multiple-output (SU-MIMO) [6]-[8] and multi-user multiple-input single-output [9]-[11] Research. For SU-MIMO configurations, the works of [6]–[8] are all for the joint design of RIS reflectance matrix and hybrid transceivers to maximize spectral efficiency. Specifically, the authors of [6] exploit the inherent structure of RIS cascaded mmWave MIMO channels to propose a two-stage scheme to optimize RIS and hybrid transceivers separately. This design method was generalized to broadband mmWave systems in [7]. Considering realistic low-resolution RIS phase shifts, the authors in [8] propose an efficient continuous refinement method for discrete optimization of RIS and analog precoder/synthesizer. For the MU-MISO structure, [9]-[11] studied the joint design of passive RIS and hybrid beamformer, which achieved the minimum mean square error (MSE), minimum transmit power [10] and maximum sum rate [11] respectively. . Therefore, various non-convex optimization techniques have been developed to solve these intractable joint designs, such as the gradient projection method [9], the penalty-based two-layer algorithm and manifold optimization [10], and the successive convex approximation (SCA) method [ 11]. Obviously, the above theoretical researches are all carried out under the assumption of perfect hardware. However, actual mmWave communication often has non-negligible hardware defects such as amplifier nonlinearity, phase noise, quantization error, in-phase/quadrature (I/Q) imbalance, etc., thus facing certain performance degradation [12]. Unfortunately, due to the stochastic and time-varying nature of hardware, HWIs cannot be fully mitigated even with effective compensation methods. Therefore, it has important practical significance to incorporate the signal distortion caused by hwi into the design of ris auxiliary system.

HWIs in RIS-assisted wireless systems can basically be divided into two categories, aggregated additive transceiver HWIs and multiplicative RIS HWIs generated by finite-resolution phase shifts (also known as RIS phase noise) [13]. [14] The author made a preliminary attempt on the design of HWI perceptual beamforming for RIS-assisted MISO system. In the RIS-assisted MISO system, only the transceiver HWIS is considered, and the ideal phase shift of RIS is infinite resolution. Some recently published works [15]–[20] also consider the additional HWI of RIS. For example, the authors in [15] modeled RIS HWI as random uniformly distributed phase noise, and derived a closed expression for the average rate of RIS-assisted SISO systems on this basis. Under the same system HWIS assumption, the papers [16] and [18] analyzed the spectral efficiency and energy efficiency of the Ris-assisted MISO system. In contrast, the authors of [17] modeled RIS phase noise with a von Mises distribution and studied the MSE minimization problem for RIS-assisted MIMO systems. In addition, considering that it is difficult to obtain ideal channel state information (CSI) in practical applications, the joint robust beamforming design of HWIS-based RIS-assisted MISO system is studied based on statistical CSI [19] and estimated CSI [20], respectively.

In fact, there are still several research gaps in [15]–[20] that need to be filled. First, the algorithms proposed in these works cannot be directly generalized to RIS-assisted MIMO systems because HWIS leads to more complex objective functions. Second, passive RIS designs in [18], [19] do not take into account the randomness of unknown phase noise, and thus generally perform poorly against HWIS. Third, these algorithms are also not directly applicable to the case of hybrid beamforming, where the inherent unit-mode constraint imposed on the analog precoder/synthesizer may make the joint design more difficult. To the best of the authors' knowledge, there is no published literature on joint hybrid transceiver and RIS designs for RIS-aided MMWAVE systems with HWIS.

In this paper, we consider a hardware-impaired RisAssisted MMWave system where HWIs are present on both hybrid transceivers and RIS. Our goal is to jointly design a hybrid precoder, RIS reflection coefficient, and hybrid combiner to minimize the mean squared error sum and maximize the average rate, respectively, under practical discrete phase shift constraints. These two problems are often challenging because their objective functions are complex or even non-analytic, contain hardware distortion noise and RIS phase noise, and the discrete unit mode constraints involved are non-convex. To address the above issues, we explore the intrinsic structure of these two problems and propose efficient algorithms for different system configurations. In addition, the influence of HWIS in the transceiver and RIS on the average rate performance of the system is analyzed theoretically. The main contributions of our work are outlined below:

• First, for single-antenna users whose optimization variables simplify more, the fractional average MSE minimization (average rate maximization) problem is transformed into an equivalent problem in the form of parameter subtraction, and an efficient alternate optimization (AO) algorithm is proposed to Find a local optimum. Specifically, by exploiting binary representations of discrete variables and equivalent continuous implication of binary constraints, we propose a Binary-Oriented Exact Penalty (BEP) method for solving all involved discrete optimization subproblems, which is able to Strike a balance between performance and complexity.
• Second, for the general multi-antenna user case, we propose a low-complexity two-stage scheme to address these two issues, avoiding AO among high-dimensional variables. In the first stage, we jointly design a discrete analog precoder/synthesizer and RIS phase shift to construct favorable propagation channels. More specifically, RIS is designed to be the maximum effective channel gain for which the proposed BEP method is still applicable. Inspired by MIMO transmission strategies based on optimal feature space alignment, we propose a unitary matching scheme for analog precoder/synthesizer design. In the second stage, a low-dimensional AO process and a digital precoder design based on maximizing the signal-to-distortion-to-noise ratio (SDNR) are proposed, which minimize the average and MSE and maximize the average rate, respectively.
• Then, the optimality of the AO algorithm and the two-phase scheme are analyzed for single-antenna and multi-antenna users under the cascaded line-of-sight (BS-RIS) user channel setting, respectively. Furthermore, both the irreducible MSE floor effect and the average rate saturation effect induced by HWIS in hybrid transceivers and RIS are revealed under high SNR conditions. In particular, we establish the equivalence between average MSE minimization and average rate maximization for an ideal RIS phase shift. Numerical results verify the correctness of the theoretical analysis and show that the proposed algorithm outperforms existing benchmark schemes.

The rest of this paper is structured as follows. Section II presents the system model and problem formulation. Section III studies the joint design of RIS and hybrid transceivers for the single-antenna user case. This study is then extended to the general multi-antenna user case in Section IV. Sections 5 and 6 analyze the complexity of the proposed algorithm and conduct numerical experiments for performance evaluation. Finally, Section VII concludes the paper.

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3. System model

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Additive hardware impairments on transceivers

Gaussian distribution

RIS phase noise

We also consider random phase noise in RIS caused by discrete phase shifts [16], [19]. Since only B bits are used to quantize the RIS phase shift, the $The\; phase\; noise\; at\; m$ RIS units can be modeled as${\bar{i}}_m\in \mathcal{U}\left[-\frac{}{2^B},\frac{}{2^B}\right]$,其中$U\left[\cdot \right]$ meansuniform distribution. Therefore, the actual RIS reflection matrix with phase noise is expressed as$\widetilde{Phi}=Phi\overline{Phi}$,index $\overline{Phi}=\mathrm{diag}\left\{e^{j{\bar{i}}_1},\cdots ,e^{j{\bar{i}}_M}\right\}$ denotes the diagonal phase noise matrix at RIS.

Received signal model with hardware impairments

$$\mathbf{y}={\mathbf{H}}_{\widetilde{Phi}}\mathbf{x}+\mathbf{n}+{the}_{\mathrm{r}}=\underset{\tilde{y}}{\underbrace{H_{\widetilde{Phi}}\left({\mathbf{F}}_{\mathrm{RF}}{\mathbf{F}}_{\mathrm{BB}}\mathbf{S}+{the}_{\mathrm{t}}\right)+\mathbf{n}}}+{the}_r$$

Signal model after equalization at the receiver

$$\begin{array}{rl}\hat{\mathbf{s}}=& {\mathbf{W}}_{\mathrm{BB}}^H{W}_{\mathrm{RF}}^H\mathbf{y}\\ =& W_{BB}^H{W}_{RF}^H{\mathbf{H}}_{\widetilde{Phi}}\left({\mathbf{F}}_{\mathrm{RF}}{\mathbf{F}}_{\mathrm{BB}}s+{the}_{\mathrm{t}}\right)+W_{BB}^H{W}_{RF}^H{the}_{\mathrm{r}}\\ & +W_{BB}^H{W}_{RF}^H\mathbf{n}.\end{array}$$

binary-oriented exact penalty (BEP)

Binary programming, an augmented biconvex optimization problem with bilinear equality constraints.

Binary optimization is a central problem in mathematical optimization, and its application is very wide. To address this problem, we propose a new class of continuous optimization techniques based on mathematical programming with balance constraints. We first transform binary programming into an equivalent augmented biconvex optimization problem with bilinear equality constraints , and then propose an exact penalty function method to solve it. The resulting algorithm seeks an ideal solution to the original problem by solving a series of linear programming convex relaxation subproblems . Furthermore, we demonstrate that the penalty function obtained by adding complementary constraints to the objective is exact , i.e. , it has the same local and global minima as the original binary program when the penalty parameter exceeds a certain threshold . The convergence of this algorithm is guaranteed because it is essentially block coordinate descent in the literature . Finally, we demonstrate the effectiveness of our method on the problem of dense subgraph discovery . Extensive experiments demonstrate that our method outperforms existing techniques such as iterative hard thresholding and linear programming relaxation.

Discussion for Single Antenna User Case (MISO)

In this section, we aim to address problem (9) in the Single Antenna user case, where only a single data stream is required. Therefore, the RIS-user channel $G$ is reduced to${\mathbf{g}}^H\in {\mathbb{C}}^{1\times M}$ , the optimization variable in problem (9) degenerates into{ W , Φ , f RF , f BB } \left\{\mathrm{W}, \mathbf{\Phi}, \mathbf{f}_{\ mathrm{RF}}, f_{\mathrm{BB}}\right\}{ W,F ,fRF​,fBB​}。

three special cases

Such as line-of-sight (BS-RIS) channel, high SNR transmission and ideal RIS phase shift to illustrate the optimality of our proposed AO algorithm for MSE minimization and average rate maximization.

1) LOS channel:

By assuming that BS and RIS are properly deployed at high altitudes such that there is a deterministic LOS link between them, we give insight into the equivalent MSE minimization problem (12) and the average rate maximization problem (22) in the following propositions the optimal solution of .
$$\begin{array}{rl}{i}^{\mathrm{opt}}& =\underset{i_m\in \mathcal{B},\forall m}{\mathrm{argmax}}{\left|i^H\mathrm{diag}\left({\mathbf{g}}^H\right){\mathbf{a}}_r\left(f_r^r,i_r^r\right)\right|}^2,\\ {\mathbf{f}}_{\mathrm{RF}}^{\mathrm{opt}}& =\underset{{\mathbf{f}}_{\mathrm{RF},\mathrm{i}}\in \mathcal{B},\forall i}{\mathrm{argmax}}{\left|a_t^H\left(f_r^t,i_r^t\right)f_{\mathrm{RF}}\right|}^2.\end{array}$$

2)High-SNR Regime：

At high SNR, that is, $P_t\to \infty$ , the MSE expression in (10) is reduced to

Analysis of some simulation results