In this paper, we consider the problem of joint waveform and passive beamforming design for MIMO integrated sensing and communication (ISAC) systems, where a reconfigurable intelligent surface (RIS) is deployed to assist the downlink communication. The objective is to maximize the signal-to-interference-and-noise-ratio (SINR) for radar and meanwhile minimizing the multi-user interference for communication. To address this problem, a block coordinate descent (BCD) method is proposed. Specifically, given the fixed waveform, an Element-wise Closed-Form (ECF) technique is derived for the reflection coefficients optimization, where the high-dimensional problem is divided into multiple one-dimensional problems with closed-form solution. Then, given fixed reflection coefficients, a dinkelbatchs-ECF algorithm is developed to optimize the waveform. Simulation results show that the proposed method achieves joint sensing and communication improvement and presents a clear performance advantage over existing methods.

Waveform design with constant modulus constraint (CMC) is of great importance in multiple-input multiple-output (MIMO) radar systems. Both the relaxations in model-based waveform design methods and data limitation in data driven Deep Neural Networks (DNNs) methods could result in performance degradation. Nevertheless, these two types of methods have their unique advantages. This motivates us to develop a model-based Learned Complex Circle Manifold Network (LCCM-Net) method without relaxation, by exploiting the advantages of both gradient descent models over CCM and DNNs with limited data for learning. More concretely, we propose to formulate the waveform design problem as an Unconstrained Quadratic Fractional Problem (UQFP) on the CCM. To solve the resultant problem, the gradient descent algorithm is unfolded as the network layer over the CCM, and the step sizes are adaptively learned. Furthermore, for discrete phases adopted in practical systems, we develop a Quantized LCCM-Net (QLCCM-Net), where a low-resolution nonuniform quantizer is designed to quantize the phase. This quantizer relies on a uniquely designed soft staircase function, which incorporates learnable parameters that allow it to adaptively fine-tune the decision region. The performance superiority of the proposed method is evidenced by comparing with the existing methods.

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 trade-off 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 usually computationally unaffordable. To solve the problem efficiently, the Parallel Product Complex Circle Manifold (P2C2M) framework is devised using the natural constant modulus 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 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.

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 non-convex 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 obtain 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 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 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.