Constant Modulus Constraint Research Articles

Hybrid Nonlinear Transceiver Optimization for the RIS-Aided MIMO Downlink

The hybrid nonlinear transceiver optimization problem of reconfigurable intelligent surface (RIS)-aided multi-user multiple-input multiple-output (MU-MIMO) downlink is investigated. Specifically, the Tomlinson-Harashima precoder (THP) and the hybrid transmit precoder (TPC) of the base station are jointly optimized with the linear digital receivers of mobile users. The triangular feedback matrix of the THP is optimized and the optimal solution is derived in closed form based on a matrix inequality. Moreover, in order to tackle the nonconvexity of the constant-modulus constraints imposed on the analog TPC, the Majorization-Minimization (MM) based reconfigurable optimization framework is proposed, which strikes a trade-off between the implementation complexity and system performance in a reconfigurable manner. Explicitly, our MM-based reconfigurable optimization framework is capable of optimizing the analog TPC in a dynamically reconfigurable manner on an element-by-element, column-by-column, row-by-row or block-by-block basis. Moreover, an MM-based reconfigurable algorithm is proposed for the optimization of the phase shifting matrix at RIS, which also suffers from constant-modulus constraints. In the proposed MM-based reconfigurable algorithm, the RIS can be partitioned into a series of subarrays for striking different performance vs. complexity tradeoffs. Finally, our numerical results demonstrate the performance advantages of the proposed nonlinear hybrid transceiver optimization techniques.

A Low Complexity Algorithm for Achievable Rate Maximization in mmWave Systems Aided by IRS

This letter investigates the weighted sum-rate maximization in a multi-user millimeter wave (mmWave) wideband downlink system aided by double intelligent reflecting surfaces (IRS). We formulate a joint optimization problem of digital beamforming (BF) matrix, analog BF matrix, and IRS phase shifts matrix, which is non-convex. To solve this problem, we propose an alternating optimization (AO) algorithm based on the projection gradient method (PGM). PGM is used to deal with non-convex element-wise constant-modulus constraints for analog BF matrix and IRS phase shifts. Furthermore, the performance loss caused by using discrete phase shift is studied by the proposed algorithm and quantization method. Simulation results show that the proposed AO algorithm based on PGM achieves the same achievable rate with a lower complexity compared to existing methods.

Hybrid beamforming for millimeter-wave massive MIMO heterogeneous networks

In this paper, we investigate the hybrid beamforming designs in millimeter wave (mmWave) massive MIMO relaying heterogeneous networks (HetNet), where small cells are overlaid onto the service area of a macrocell. The deployment of HetNet and utilization of the adequate bandwidth in the mmWave band are capable of addressing the explosive growth of data rates in next-generation wireless networks. However, the severe path loss and attenuation in the mmWave band and complex interference in HetNet should be addressed. For the macrocell, we propose a minimum mean squared error (MMSE)-based alternative iterative hybrid analog/digital beamforming algorithm for the downlink multi-user mmWave relaying system. With the constant modulus constraints and the higher-order polynomial function, we reformulate the optimization problem as three convex quadratic subproblems, and the coupling approximate optimal precoders and combiners are jointly obtained by the proposed alternative iterative algorithm. For the picocells, we propose a hybrid beamforming algorithm for downlink mmWave massive MIMO system. An alternative iterative optimization algorithm based on manifold optimization is developed to approximate the optimal beamforming designs. The baseband digital beamforming scheme is investigated by leveraging the mean square error (MSE) criteria for macrocell users and picocell users to manage cross-tier and inter-user interference. Simulation results demonstrate the obvious advantages of the proposed hybrid beamforming algorithms in terms of achievable sum-rate for various setting scenarios.

Training Beam Design for Channel Estimation in Hybrid mmWave MIMO Systems

Training beam design for channel estimation with infinite-resolution and low-resolution phase shifters (PSs) in hybrid analog-digital milimeter wave (mmWave) massive multiple-input multiple-output (MIMO) systems is considered in this paper. By exploiting the sparsity of mmWave channels, the optimization of the sensing matrices (corresponding to training beams) is formulated according to the compressive sensing (CS) theory. Under the condition of infinite-resolution PSs, we propose relevant algorithms to construct the sensing matrix, where the theory of convex optimization and the gradient descent in Riemannian manifold is used to design the digital and analog part, respectively. Furthermore, a block-wise alternating hybrid analog-digital algorithm is proposed to tackle the design of training beams with low-resolution PSs, where the performance degeneration caused by non-convex constant modulus and discrete phase constraints is effectively compensated to some extent thanks to the iterations among blocks. Finally, the orthogonal matching pursuit (OMP) based estimator is adopted for achieving an effective recovery of the sparse mmWave channel. Simulation results demonstrate the performance advantages of proposed algorithms compared with some existing schemes.

Autofocus algorithms with phase error correction for synthetic aperture radar imagery

Due to the unknown platform motion and/or signal propagation delays, phase errors are introduced to synthetic aperture radar (SAR) imagery. With the fact that the pixel values in the low-return region of the focused image are close to zero, we devise two autofocus SAR imaging methods with phase error correction in this paper. One is a fractional model to maximize the pixel values in the remaining region, termed high-return region, while minimizing those in the low-return region. And another is a model with the high-return region pixel value maximization and the controlled peak low-return pixel value. The resultant optimization problems are difficult to solve due to the coupled numerator and denominator of the quadratic fractional programming problem formulation with the constant modulus constraints and the controlled optimization problem with the nonconvex inequality constraints. To tackle them, we derive two iterative solutions via introducing auxiliary variables to decouple the numerator and denominator and the inequality constraints. Alternating direction method of multipliers is utilized in the algorithm development so that their convergence is guaranteed. Numerical examples are provided to show the effectiveness of the developed methods.

Constant modulus shaped beam synthesis via linear fractional semidefinite relaxation

Excitation amplitude control is a critical issue regarding the shaped beam synthesis of array antennas. Ideally, excitations can have the same amplitude, and the desired beam can be achieved only by changing the phases. This method is called constant modulus synthesis, which can reduce the complexity of array antennas and is more conducive to the feeding, but it is really difficult to realize. In this paper, the authors derive a novel constant modulus beam synthesis algorithm, which uses the linear fractional semidefinite relaxation to relax the constant modulus constraints. Simulation results show that this method is stable and can maintain the excitation modulus while realizing the shaped beam.

RIS-Aided Multiple User Interference Mitigation via Fast Successive Upper-Bound Minimization Method

Reconfigurable intelligent surfaces (RIS) aided multiple user interference mitigation with constant modulus constraint (CMC) is a key issue in the future 6G communication systems. Usually, the existing methods solve this problem either by relaxing the CMC with performance degrading or by relaxing the cost function with huge computational cost. To achieve a good trade-off between the performance and the complexity, a Fast Successive Upper-bound Minimization (FSUM) method is proposed. In the proposed method, the problem is first reformulated as a penalty quadratic function with CMC. Then, by efficiently finding the surrogate function, a FSUM algorithm is derived to obtain the solution. Compared with the existing methods, the proposed method can achieve 0.4 bits/Hz/s average sum-achievable rate gain enhancement.

RMOCG: A Riemannian Manifold Optimization-Based Conjugate Gradient Method for Phase-Only Beamforming Synthesis

The phase-only null beamforming synthesis is widely used in communication, radar, and sonar. Due to the phase-only constraint (constant modulus constraint), the problem is nonconvex and NP-hard. The existing methods mainly solve the problem in an indirect way, by relaxing the problem to a more tractable form. Nevertheless, these methods either need huge computational cost or degrade the performance. To break the tradeoffs between the performance and computational complexity, a direct method is proposed without relaxation. In the proposed method, the problem is first transformed into an unconstrained problem on a complex circle manifold. Then, by deriving the gradient descent direction and the step size, a Riemannian manifold optimization-based conjugate gradient algorithm is derived to obtain the direct solution for the transformed problem. Compared with the existing methods, the proposed method can achieve 4 dB null gain reduction and 1 magnitude complexity decreasing.

Joint Transmit Waveform and Passive Beamforming Design for RIS-Aided DFRC Systems

Reconfigurable intelligent surface (RIS) is a promising technology for 6 G networks owing to its superior ability to enhance the capacity and coverage of wireless communications by smartly creating a favorable propagation environment. In this paper, we investigate the potential of employing RIS in dual-functional radar-communication (DFRC) systems for improving both radar sensing and communication functionalities. In particular, we consider a RIS-assisted DFRC system in which the multi-antenna base station (BS) simultaneously performs both multi-input multi-output (MIMO) radar sensing and multi-user multi-input single-output (MU-MISO) communications using the same hardware platform. We aim to jointly design the dual-functional transmit waveform and the passive beamforming of RIS to maximize the radar output signal-to-interference-plus-noise ratio (SINR) achieved by space-time adaptive processing (STAP), while satisfying the communication quality-of-service (QoS) requirement under one of three metrics, the constant-modulus constraint on the transmit waveform, and the unit-modulus constraint of RIS reflecting coefficients. An efficient algorithm framework based on the alternative direction method of multipliers (ADMM) and majorization-minimization (MM) methods is developed to solve the complicated non-convex optimization problem. Simulation results verify the advancement of the proposed RIS-assisted DRFC scheme and the effectiveness of the developed ADMM-MM-based joint transmit waveform and passive beamforming design algorithm.

Passive Beamforming for 3-D Coverage in IRS-Assisted Communications

In this letter, we study the passive beamforming design for optimizing the three-dimensional (3-D) coverage performance in an intelligent reflecting surface (IRS)-assisted communication system. In particular, we maximize the minimum (i.e., max-min) beamforming gain over all locations in the target area by jointly optimizing the passive reflection of all IRS elements. The resulting problem has a non-convex and semi-infinite beamforming gain constraint and a non-convex constant-modulus constraint. Both of them are reformulated to be finite and convex in the lifted matrix space with an additional rank-one constraint, and a difference-of-convex technique is developed to solve the reformulated problem efficiently. Finally, we demonstrate that the proposed passive beamforming solution outperforms the existing methods and can approach the upper bound on the max-min beamforming gain.

Hybrid Beamforming for MISO System via Convolutional Neural Network

Hybrid beamforming (HBF) is a promising approach to obtain a better balance between hardware complexity and system performance in massive MIMO communication systems. However, the HBF optimization problem is a challenging task due to its nonconvex property in terms of design complexity and spectral efficiency (SE) performance. In this work, a low-complexity convolutional neural network (CNN)-based HBF algorithm is proposed to solve the SE maximization problem under the constant modulus constraint and transmit power constraint in a multiple-input single-output (MISO) system. The proposed CNN framework uses multiple convolutional blocks to extract more channel features. Considering that the solutions for the HBF are hard to obtain, we derive an unsupervised learning mechanism to avoid any labeled data when training the constructed CNN. We discuss the performance of the proposed algorithm in terms of both the generalization ability for multiple CSIs and the specific solving ability for an individual CSI, respectively. Simulations show its advantages in both SE and complexity over other related algorithms.

An anti-jamming method against interrupted sampling repeater jamming based on designing transmitted waveforms and receiving filters for simultaneous polarimetric radar

The realistic false targets caused by the interrupted sampling repeater jamming can obscure the real target echo, resulting in radar target detection failure. In this paper, based on reducing the energy of the interference signal after the pulse compression operation, a method is first proposed to suppress the interrupted sampling repeater jamming by jointly designing the transmitted waveforms and receiving filters for the simultaneous polarimetric radar. The joint design optimization problem is formulated by minimizing the energy of the sidelobes of the pulse compression output signal with the constant modulus constraint. An alternately optimized iterative algorithm is proposed to optimize the transmitted waveforms and receiving filters, respectively. Numerical and indoor experimental results are presented to show the effectiveness of the proposed method to suppress the interrupted sampling repeater jamming.

The phase only beamforming synthesis via an accelerated coordinate descent method

Phase-only (constant modulus) beamforming synthesis is a key issue in radar, communication and sonar systems. Due to the constant modulus constraint (CMC), the problem is non-convex and NP-hard. Most existing methods either indirectly solve the problem by relaxing with performance degrading, or by directly solve the problem in the high dimensional space with huge computational complexity. To address this problem, the low-complexity Accelerated Coordinate Descent (ACD) based method is proposed by directly solving the problem in the low dimensional space. First, the Coordinate Descent (CD) based method is proposed, where the high-dimensional problem is divided into multiple one-dimensional problems, to which the optimal solutions are found. Then, the ACD method is derived by the square iterative technique, which needs less computational complexity due to it only uses linear-complexity storage. Both the continuous and discrete phases are designed in the case of the uniform and random linear arrays. Compared with the existing methods, the proposed method achieves 4 dB beampattern sidelobe reduction and 1.5 magnitude complexity decreasing.

Deep Learning-Based Overhead Minimizing Hybrid Beamforming for Wideband mmWave Systems

Hybrid beamforming (HBF) design for wideband millimeter-wave (mmWave) systems has two challenges: 1) design complexity due to frequency selectivity; 2) large pilot overhead and feedback overhead due to heavy dependence on the channel information. This letter proposes a deep learning-based HBF method for wideband mmWave systems. First, we simplify the complex HBF design problem into a network optimization problem by designing the loss function with spectral efficiency as an optimization objective. Then, we replace the instantaneous channel state information with channel statistics information of multiple time slots to achieve low system overhead. Specifically, we develop an HBF network (HBFNet) based on convolutional neural network with the channel covariance matrix (CCM) as input and the hybrid beamformer as output. Meanwhile, we design a constraint layer to satisfy the constant modulus constraint and power constraint in HBF design. In addition, we adopt the imperfect CCM for training to make the proposed method more robust.

A Precoding Approach for Dual-Functional Radar-Communication System With One-Bit DACs

In this paper, we investigate the precoder design for multiple-input multiple-output (MIMO) dual-functional radar-communication (DFRC) system with one-bit digital-to-analog converters (DACs). In order to form the dual-functional beam-pattern, we formulate the precoding problem as a weighted optimization problem with the constant modulus constraint, which aims at minimizing the average error power and guaranteeing radar waveform similarity. The problem is divided into three sub-problems corresponding to the multiple variables, i.e., the precoding factor, transmit signal matrix, and radar waveform matrix. Due to the discrete and non-convex properties of the optimization problem, we propose a multi-variable alternating minimization (MVAM) framework to achieve the near-optimal solutions. The precoding factor and radar waveform can be solved in closed-forms. For the transmit signal matrix, we devise a binary particle swarm optimization-simulated annealing (BPSO-SA) algorithm to obtain it under the MVAM framework. Extensive simulations validate the effectiveness of the proposed approach under various scenarios, including the case without perfect channel state information. The simulation results show that, compared with existing non-linear precoders, the proposed approach achieves 7dB SNR gain at the bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−4</sup> in the 8-antenna system, and the gain of SNR is 0.2dB in the massive MIMO system with 128 antennas.

MIMO Radar Beampattern Design Based on Manifold Optimization Method

The transmit beampattern design with constant modulus (CM) constraint is a key technical issue in Multiple-Input Multiple-Output (MIMO) radar system. The problem is non-convex and direct solution is impossible in general, due to the CM constraint. Usually, most existing methods indirectly solve this problem by relaxing it to a more tractable form. These methods usually either degrade the performance due to relaxation or consume expensive computational cost. Different from the relaxation methods, a direct method is proposed to solve the non-convex problem without relaxation. In the proposed method, the original problem is first converted to an Unconstrained Quadratic Lagrange Multiplier Problem (UQLMP) over the non-convex complex circle manifold. Then, by efficiently finding the descent direction and step size to ensure the objective function to be non-increasing, a novel Manifold Optimization-based Conjugate Gradient (MOCG) approach is derived to directly solve UQLMP. Compared with current methods, the proposed method has better performance in terms of nulling depth.

Training Signal Design for Sparse Channel Estimation in Intelligent Reflecting Surface-Assisted Millimeter-Wave Communication

In this paper, the problem of training signal design for intelligent reflecting surface (IRS)-assisted millimeter-wave (mmWave) communication under a sparse channel model is considered. The problem is approached based on the Cramér-Rao lower bound (CRB) on the mean-square error (MSE) of channel estimation. By exploiting the sparse structure of mmWave channels, the CRB for the channel parameter composed of path gains and path angles is derived in closed form under Bayesian and hybrid parameter assumptions. Based on the derivation and analysis, an IRS reflection pattern design method is proposed by minimizing the CRB as a function of design variables under constant modulus constraint on reflection coefficients. Extensions of the proposed design to a multi-antenna transceiver, a uniform planar array (UPA)-based IRS, and multi-user case are discussed. Numerical results validate the effectiveness of the proposed design method for sparse mmWave channel estimation.

Weighted Sum Rate Maximization of the mmWave Cell-Free MIMO Downlink Relying on Hybrid Precoding

The cell-free MIMO concept relying on hybrid precoding constitutes an innovative technique capable of dramatically increasing the network capacity of millimeter-wave (mmWave) communication systems. It dispenses with the cell boundary of conventional multi-cell MIMO systems, while drastically reducing the power consumption by limiting the number of radio frequency (RF) chains at the access points (APs). In this paper, we aim for maximizing the weighted sum rate (WSR) of mmWave cell-free MIMO systems by conceiving a low-complexity hybrid precoding algorithm. We formulate the WSR optimization problem subject to the transmit power constraint for each AP and the constant-modulus constraint for the phase shifters of the analog precoders. A block coordinate descent (BCD) algorithm is proposed for iteratively solving the problem. In each iteration, the classic Lagrangian multiplier method and the penalty dual decomposition (PDD) method are combined for obtaining near-optimal hybrid analog/digital precoding matrices. Furthermore, we extend our proposed algorithm for deriving closed-form expressions for the precoders of fully digital cell-free MIMO systems. Moreover, we present the convergency analysis and complexity analysis of our proposed method. Finally, our simulation results demonstrate the superiority of the algorithms proposed for both fully digital and hybrid precoding matrices.

Learning a compressive sensing matrix with structural constraints via maximum mean discrepancy optimization

We introduce a learning-based algorithm to obtain a measurement matrix for compressive sensing related recovery problems. The focus lies on matrices with a constant modulus constraint which, e.g., represent a network of analog phase shifters in hybrid precoding/combining architectures. We interpret a matrix with restricted isometry property as a mapping from a high- to a low-dimensional hypersphere. We argue that points on the low-dimensional hypersphere should be uniformly distributed to combat measurement noise. This notion is formalized as an optimization problem which uses a maximum mean discrepancy metric as objective function. Recent success of such metrics in neural network related topics motivate a solution of the optimization problem based on machine learning. Numerical experiments show a better performance than random matrices that are typical for compressive sensing. Further, we adapt a method from the literature to the constant modulus constraint. This method can also compete with random matrices and harmonizes well with the proposed algorithm if it is used as an initialization. Lastly, we describe how other structural matrix constraints, e.g., a Toeplitz constraint, can be taken into account as well.

Direct transmit waveform design to match a desired beampattern under the constant modulus constraint

In this paper, we study waveform optimization methods for transmit beampattern matching with a multiple-input-multiple-output (MIMO) radar. The aim is to minimize the squared error between the desired beampattern and the actual one. We propose two algorithms, which are based on cyclic optimization and majorization-minimization (MM), to tackle the highly nonlinear and non-convex optimization problem. The proposed algorithms have simple closed-form solutions at every iteration, which reduce the computational complexity significantly. Moreover, the convergence of the proposed methods can be guaranteed, due to the descent property of MM algorithms and cyclic optimization. In addition, we extend the proposed algorithm to design waveforms for dual-function radar-communication systems. Finally, simulation results demonstrate the superiority of the proposed algorithms compared with state-of-the art methods.