Unimodular waveform design plays a crucial role in MIMO radar systems. Previous studies have mainly focused on continuous- and discrete-phase coding for single-pulse MIMO radar waveforms, as well as continuous-phase coding for pulse-Doppler MIMO radar waveforms. Although multi-pulse discrete-phase waveforms provide both high resolution and hardware simplicity, their design remains a challenging optimization problem. In this work, we go beyond prior approaches by investigating the design of pulse-Doppler MIMO waveforms under discrete phase constraints. We formulate the problem as optimizing the waveform phase matrix to minimize the weighted integrated sidelobe level (WISL) of the joint ambiguity function. The non-convexity of WISL and the discrete phase constraints make the problem particularly challenging. Noting that the Adam optimizer incorporates both adaptive learning rate and momentum mechanisms, making it suitable for solving non-convex optimization problems, and that nonlinear functions can be used to approximate quantization in a continuously differentiable form, we propose a soft quantization Adam optimization (SQAO) method to solve this problem. Simulations show that SQAO outperforms existing method.

Intelligent reflecting surface (IRS) and movable antenna (MA) technologies have been proposed to enhance wireless communications by creating favorable channel conditions. This paper investigates the joint beamforming and antenna position optimization for MA-enabled IRS (MA-IRS)-aided multi-user multiple-input single-output (MU-MISO) communication systems, where the MA-IRS is deployed to aid the communication between the MA-enabled base station (BS) and user equipment (UE). In contrast to conventional fixed position antenna (FPA)-enabled IRS (FPA-IRS), the positions of the reflecting elements of the MA-IRS can be controlled to enhances the wireless channel. To verify the system’s effectiveness and optimize its performance, we formulate a sum-rate maximization problem with a minimum rate threshold constraint for the MU-MISO communication. To tackle the non-convex problem, a product Riemannian manifold optimization (PRMO) method is proposed for the joint optimization of the beamforming and MA positions. Specifically, a product Riemannian manifold space (PRMS) is constructed and the corresponding Riemannian gradient is derived for updating the variables, and the Riemannian exact penalty (REP) method and a Riemannian Broyden-Fletcher-Goldfarb-Shanno (RBFGS) algorithm is exploited to obtain a feasible solution over the PRMS. Simulation results demonstrate that compared with the conventional FPA-IRS-aided communications, the reflecting elements of the MA-IRS can move to the positions with higher channel gain, thus enhancing the system performance. Furthermore, it is shown that optimizing the positions of the reflecting elements brings higher performance gain than controlling the phase shifts of the IRS, and integrating MA with IRS leads to higher performance gains compared to integrating MA with BS.

Driven by intelligent reflecting surface (IRS) and movable antenna (MA) technologies, movable IRS (MIRS) has been proposed to improve the adaptability and performance of conventional IRS, enabling flexible adjustment of the IRS reflecting element positions. This paper investigates MIRS-aided integrated sensing and communication (ISAC) systems. The objective is to minimize the power required for satisfying the quality-of-service (QoS) of sensing and communication by jointly optimizing the MIRS element positions, IRS reflection coefficients, transmit beamforming, and receive filters. To balance the performance-cost trade-off, we proposed two MIRS schemes: element-wise control and array-wise control, where the positions of individual reflecting elements and arrays consisting of multiple elements are controllable, respectively. To address the joint beamforming and position optimization, a product Riemannian manifold optimization (PRMO) method is proposed, where the variables are updated over a constructed product Riemannian manifold space (PRMS) in parallel via penalty-based transformation and Riemannian Broyden–Fletcher–Goldfarb–Shanno (RBFGS) algorithm. Simulation results demonstrate that the proposed MIRS outperforms conventional IRS in power minimization with both element-wise control and array-wise control. Specifically, with different system parameters, the minimum power is achieved by the MIRS with the element-wise control scheme, while suboptimal solution and higher computational efficiency are achieved by the MIRS with array-wise control scheme.

Movable antennas (MAs) represent a novel approach that enables flexible adjustments to antenna positions, effectively altering the channel environment and thereby enhancing the performance of wireless communication systems. However, conventional MA implementations often adopt fully digital beamforming (FDB), which requires a dedicated RF chain for each antenna. This requirement significantly increase hardware costs, making such systems impractical for multi-antenna deployments. To address this, hardware-efficient analog beamforming (AB) offers a cost-effective alternative. This paper investigates the physical layer security (PLS) in an MA-enabled multiple-input single-output (MISO) communication system with an emphasis on AB. In this scenario, an MA-enabled transmitter with AB broadcasts common confidential information to a group of legitimate receivers, while a number of eavesdroppers overhear the transmission and attempt to intercept the information. Our objective is to maximize the multicast secrecy rate (MSR) by jointly optimizing the phase shifts of the AB and the positions of the MAs, subject to constraints on the movement area of the MAs and the constant modulus (CM) property of the analog phase shifters. This MSR maximization problem is highly challenging, as we have formally proven it to be NP-hard. To solve it efficiently, we propose a penalty constrained product manifold (PCPM) framework. Specifically, we first reformulate the position constraints as a penalty function, enabling unconstrained optimization on a product manifold space (PMS), and then propose a parallel conjugate gradient descent algorithm to efficiently update the variables. Simulation results demonstrate that MA-enabled systems with AB can achieve a well-balanced performance in terms of MSR and hardware costs.

Reconfigurable Intelligent Surface (RIS) is a key technology for radar and communication systems. This paper focuses on designing RIS-aided communication-compatible MIMO radar unimodular waveform design for radar and communication coexistence. The goal is to minimize the RIS-aided spatial Integrated Sidelobe Level Ratio (ISLR) under spectral constraint and unimodular constraints on both the waveform and RIS phase shifts. This is a challenging non-convex problem that existing methods cannot solve directly. We observe that the spectral constraint can be rewritten as a smooth non-negative function, and the Product Complex Circle Manifold (PCCM) naturally satisfies the unimodular constraints. Based on these insights, we propose an Inequality Constrained Product Manifold Optimization (ICPMO) framework. The spectral constraint is handled using a smooth penalty function, reformulating the problem as an unconstrained optimization on the PCCM. We then develop a Parallel Conjugate Gradient Descent (PCGD) algorithm without relaxing the objective. Simulations show our method reduces beam sidelobes by about 10 dB and improves energy distribution nulling compared to non-RIS methods.

Movable antennas (MA) allow flexible positioning within a specified region, enhancing wireless communication performance. This paper explores leveraging MA to improve physical layer security in analog beamforming (AB) systems. Specifically, we aim to maximize the secrecy rate by jointly optimizing the AB and MA positions under constant modulus (CM) and position constraints. To solve the resulting non-convex problem, we propose a penalty product manifold (PPM) method, which converts MA position constraints into a penalty function, reformulating the problem as unconstrained optimization on the product manifold space (PMS). We then derive a parallel conjugate gradient descent (PCGD) algorithm to efficiently update both AB and MA positions, providing analytical solutions at each step and ensuring convergence to a KKT point. Simulation results confirm that the MA system achieves a higher secrecy rate than systems with fixed antenna positions.

Polarization adds an additional dimension to the radar signals, contributing to waveform diversity. Codesign of unimodular waveforms and filters with polarimetric power allocation for maximizing the signal-to-interference-plus-noise ratio (SINR) plays a key role in the polarimetric radar system. The problem is challenging to solve due to the nonconvex nature of the objective function and constraints, coupled with the interdependence of multiple variables. Existing methods mainly solve this problem by fixing the power allocation or relaxing the objective function and obtaining the receive filters with matrix inversion. We directly address this problem without matrix inversion by using the proposed adaptive unified manifold optimization (AUMO) framework. Specifically, a unified manifold space (UMS) is constructed to satisfy the constraints of unimodular waveform, filters, and power, transforming the problem to an unconstrained optimization problem over the manifold. To solve this problem, a parallel conjugate gradient (PCG) algorithm is derived. This algorithm can adaptively change the step size by exploring the local features of the manifold space. The experimental results based on the measured data show that the proposed method outperforms existing methods in terms of SINR gain and execution time.

In this paper, we focus on the joint beamforming for intelligent reflecting surface (IRS) aided integrated sensing and communication (ISAC) systems, where a multi-antenna base station (BS) performs multi-user multi-input single-output (MU-MISO) communication and radar sensing simultaneously. Specifically, the direction-of-arrival (DoA) estimation is considered as the task of radar sensing, and we aim to optimize the MU-MISO communication while enhancing the estimation accuracy by ensuring a Cramér-Rao bound (CRB) lower bound. First, for the CRB-constrained sum rate maximization problem, we propose a product Riemannian manifold optimization (PRMO) framework to solve the problems without relaxing the objective functions. Specifically, a product Riemannian manifold space (PRMS) is constructed to satisfy the constraints of the precoding matrix and IRS phase shifts, and the constraint of the CRB threshold is tackled by a Riemannian exact penalty (REP) method. A parallel Riemannian Broyden-Fletcher–Goldfarb-Shanno (P-RBFGS) algorithm is derived to update the parameters over the PRMS. Then, considering the fairness of the MU-MISO communication, the PRMO is further extended to tackle the CRB-constrained max-min optimization by maximizing the minimum rate among all users. Simulation results demonstrate that with the same CRB constraint, the PRMO outperforms the existing method in sum rate maximization with lower computational complexity, and the extended PRMO enables the users to obtain approximately equal rates, thus guaranteeing the fairness of the system.

Joint design of waveforms and filters with adaptive power alloction has important applications in improving the target detection performance of polarization radar. The resulting problem is maximizing polarimetric radar target detection performance through unimodular waveform and filter design with power allocation, which is a non-convex and NP-hard problem. Existing methods mainly solve this problem by relaxing the objective function and obtaining filters with matrix inversion, which may introduce relaxation errors and high complexity. We construct a method based on unified product manifold to directly address this problem without matrix inversion. Specifically, a product manifold space is constructed to satisfy the constraints of unimodular waveform, filters and power. Then, an unconstrained problem is obtained by projecting the problem onto the unified product manifold. To solve this problem, a parallel conjugate gradient algorithm is derived. This algorithm can adaptively change the step size and fully explore the manifold space. Simulation results show that the proposed method can achieve better performance in shorter time than existing methods.

Resource allocation for enhancing the target capacity of multiple target tracking (MTT) with desired accuracies for given transmit power is the key issue in radar networks. Most existing methods solve this problem with heuristic evolutionary methods or convex relaxation methods with high computational cost, which lack the real-time adaptability for dynamic threat scenarios. To overcome this issue, we propose a real-time Adaptive Manifold Optimization (AMO) framework. This is achieved by utilizing the inherent real oblique characteristic of the power matrix constraints. Specifically, we construct a real oblique manifold that satisfies the constraints, enabling the problem to be rephrased as an unconstrained problem over the manifold space. Then, we derive a conjugate gradient algorithm for direct optimization of the problem. Simulation results demonstrate that the proposed method outperforms existing approaches in terms of target capacity, MTT accuracy and computational cost.

Polyphase waveform design for the minimization of Integrated Sidelobe Level Ratio (ISLR) is the key technology in Multiple-Input Multiple-Output (MIMO) radar systems. Due to the discrete phase constraint, the problem is non-convex and challenging to solve. Existing methods often rely on experience-based hard quantization with fixed thresholds, leading to limited performance due to the hard quantization. To address this issue, we propose the Optimal Loss Function-Based Soft Quantization (OLF-SQ) method with adjustable quantization thresholds. Firstly, the Complex Circle Manifold (CCM) space is constructed to satisfy the constant modulus constraint, and then the Gradient Descent (GD) model-driven network layer based on the CCM is devised to obtain the continuous waveform. Secondly, the soft quantization function is derived with adjustable quantization thresholds, and then the soft quantization network layer is devised to obtain the discrete waveform, where the quantization thresholds are learned by the unsupervised learning network. Compared with the existing methods, the proposed method has better performance in terms of ISLR and beampattern shaping.

The joint design of waveforms and filters has key applications in polarimetric radar target detection. This letter studies the joint design of waveforms and filters to maximize the signal-to-interference-to-noise ratio (SINR) of polarimetric radar, which is a nonconvex and NP-hard problem. Most existing works solve it based on matrix inversion and problem relaxation, which inevitably introduce high complexity and relaxation errors. We notice that a unified manifold space naturally satisfies the constant modulus constraint (CMC) and the norm constraint. Based on this characteristic, we proposed a parallel manifold joint optimization (PMJO) method to solve it without relaxing the objective function. Specifically, the unified product manifold is constructed to satisfy both waveform and filter constraints. Subsequently, the problem is transformed into an unconstrained one by projecting it onto the product manifold space. Finally, a parallel conjugate gradient method is proposed to simultaneously optimize waveforms and filters, which can adaptively adjust the step size and fully explore the product manifold space. Simulation results show that our method can obtain a 2-dB performance advantage compared with the existing methods, while having a half-order of magnitude advantage in time complexity.

Multiple Input Multiple Output (MIMO) radar unimodular waveform design with range-ISL optimization is a key technology in remote sensing. Due to the non-convex quartic objective function and constant modulus constraint (CMC), the problem is NP-hard and non-convex. Existing methods mainly include relaxation methods or non-relaxation methods with huge computational cost. We notice that complex circle manifold (CCM) naturally satisfies the CMC. By projecting onto the CCM, the problem is transformed into an unconstrained minimization problem that can be addressed using the Riemannian gradient descent (RGD) algorithm. Furthermore, we notice that the RGD algorithm can be unfolded into a deep learning model. Hence, a computationally efficient method without relaxation, Iterative Deep Unfolding Network (IDUN), is proposed. First, this problem is converted into an unconstrained fourth-order polynomial minimization problem on the CCM. Then, by unfolding RGD algorithm as the network layer, IDUN is developed with adaptively learning the step sizes. Compared with existing methods, the proposed method has superior performance and less computational cost.

In this letter, we investigate the per-user MUI energy-controllable waveform design problem in Dual-Function Radar-Communication (DFRC) systems. Due to the unimodular constraint and per-user MUI energy constraint, the problem is non-convex and difficult to solve. To address it, the Inequality Constrained Manifold Optimization (ICMO) method is proposed. First, we transform the per-user MUI energy constraint into a penalty function added to the objective function through the exact penalty technique, resulting in a transformed problem containing only the unimodular constraint. Then, we note that the Complex Circle Manifold (CCM) naturally satisfies the unimodular constraint, the problem can be further converted into an unconstrained problem over CCM, and we derive a conjugate gradient descent (CGD) algorithm to solve it. Compared with existing methods, the proposed method exhibits advantages in terms of per-user communication performance, signal-to-interferenceand-noise ratio (SINR), and beampattern performance. Besides, our method has lower computational costs.

Intelligent reflecting surface (IRS) is a key technique for enhancing the performance of wireless communications. In this letter, we focus on the sum-path-gain maximization (SPGM) problem in an IRS-aided MIMO communication system, which is non-convex due to the constant modulus constraint. The existing works mainly include the relaxation method with relaxation error and the non-relaxation methods with high complexity. Different from the existing methods, we notice that constant modulus constraint can naturally satisfy the Riemannian manifold, and the deep learning method has strong non-convex learning ability. By exploiting these characteristics, the Riemannian gradient descent network (RGD-Net) is proposed. In the proposed method, we first project the non-convex SPGM problem to the Riemannian manifold. Then, the Riemannian gradient descent iterations are unfolded as the network layers. Finally, the step sizes of each layer are learned in unsupervised manner to ensure converged performance. Compared with the existing methods, the proposed method achieves higher spectral efficiency with lower computational cost.

The unimodular waveform design for simultaneous sensing and communication plays an important role in the integrated sensing and communication (ISAC) systems. The existing studies mainly include the tradeoff waveform design without Reconfigurable Intelligent Surface (RIS); or the RIS aided-waveform design with optimal performance in a certain aspect, which usually degrade the comprehensive performance. To address these issues, the comprehensive waveform design with RIS is proposed, in which the waveform and RIS are coupled. The existing decoupled methods are mainly Alternating Optimization (AO), which are computationally unaffordable. To solve the problem efficiently, the Parallel Product Complex Circle Manifold (P2C2M) framework is devised using the natural constant mod-ulus characteristic of both the waveform and RIS. Concretely, the problem is converted to the Unconstrained Coupling Quartic Problem (UCQP) over the P2C2M. Based on the P2C2M, the Parallel Conjugate Gradient algorithm is derived to optimize the waveform and RIS in parallel. Compared with the existing methods, the proposed method achieves better comprehensive performance with less computational cost.

The Multiple-Input Multiple-Output (MIMO) radar transmit beampattern matching under the Constant Modulus Constraint (CMC) is a key technology. Most existing approaches address this problem by relaxation, which result in performance degradation. Different from these methods, we notice that the CMC is the product of complex circles. Based on this characterisic, a Riemannian Complex Circle Manifold (RCCM) method without relaxation is developed. More precisely, the aforementioned problem is firstly reformulated as an unconstraint quartic function over the RCCM. After that, an efficient Riemannian conjugate gradient algorithm is developed to solve it. Compared with the existing methods, the proposed method obtains better performance with lower computational cost.

The phase-only beamforming synthesis is widely applied in millimeter wave communication, radar and sonar. Due to the CMC, the problem is non-convex. The most current methods solve the problem by designing the phase, which either degrades the performance or needs huge complexity. To address this issue, a low-complexity Riemannian Manifold Optimization based Conjugate Gradient (RMOCG) method is proposed. First, the original problem is transformed into an unconstrained prob-lem on a complex circle manifold. Then, a RMOCG algorithm is derived, by deriving the gradient descent direction and the step size for ensuring the cost function non-increasing. Comparing with the existing methods, the proposed method has the following advantages: 1) the null depth is respectively 8 dB deeper than [6] and 3 dB deeper than [12]. 2) The computational cost is 2 magnitude lower than [6] and 1 magnitude lower than [12].

The signal-to-interference plus noise ratio (SINR) maximization with constant modulus (CM) constraint is a key issue in Multiple-Input-Multiple-Ouput (MIMO) radar system. This problem is hard to solve, due to the SINR function and CM constraint both are nonconvex. Usually, the existing methods indirectly optimize the problem by relaxing SINR function or CM constraint to a more tractable form. These methods usually degrade the performance due to relaxation. To address this issue, the deep learning (DL) based method is proposed, by using the strong and robust nonlinear fitting capabilities of the DL. Firstly, the CM constraint problem was converted into an unconstrained phase optimization problem. Then, we construct an optimization training network (OTN) directly sloving this nonconvex problem without relaxation. Simulation results show that our proposed method acheived better performance compared with the existing methods.

The constant modulus (CM) waveform design for integrated sensing and communication (ISAC) systems is a key technology. The joint design of maximizing the Signal-to-Interference-and-Noise-Ratio (SINR) for radar and minimizing the Multiple User Interference (MUI) for communication is studied. The problem is nonconvex and NP-hard, due to the fractional expression and the CM constraint. To address this issue, a low-complexity Accelerated Coordinate Descent (ACD) method is proposed. First, the problem is simplified to a quadratic function with CMC by dinkelbatchs method. Then, a CD method is derived, by transforming the problem into a decomposable problem with multiple one-dimensional subproblems. Finally, the ACD algorithm is derived to accelerate the convergence by using the square iterative technique. Simulation results show that the proposed method obtains favorable trade-off performance between SINR and MUI.