Designing Unimodular Waveforms for MIMO Radar Based on Manifold Optimization Method

TL;DR

This paper introduces a novel Riemannian manifold optimization framework for designing unimodular waveforms in MIMO radar, featuring new algorithms with reduced complexity and proven convergence guarantees.

Contribution

It develops a Riemannian manifold-based optimization approach and algorithms for unimodular waveform design, improving efficiency and convergence over existing methods.

Findings

The proposed UM-SVRG algorithm reduces computational complexity significantly.

Numerical results confirm the effectiveness of the algorithms.

Theoretical convergence guarantees are established for both algorithms.

Abstract

In this paper, we design unimodular waveforms with good correlation properties for multi-input multi-output (MIMO) radar systems. Specifically, first, we analyze the geometric properties of the unimodular constraint in the fourth-order polynomial minimization problem using Riemannian geometry theory. By embedding it into the search space, we transform the original non-convex optimization problem into an unconstrained problem on a Riemannian manifold. Then, we construct the manifold corresponding to the search space and the operators required for the customized optimization algorithm. Second, we develop a customized low-complexity unimodular manifold gradient descent (UM-GD) algorithm on the constructed manifold to solve the optimization problem in the normal-scale case, and propose its acceleration version unimodular manifold accelerated gradient descent (UM-AGD) algorithm, to speed up the convergence. In the large-scale case, we transform the objective function into the form of a summation of a large but finite number of loss functions and develop a customized unimodular manifold stochastic variance reduced gradient (UM-SVRG) algorithm to solve this problem. Compared to the existing bechmark method, which has a computational complexity of roughly $\mathcal{O}(M^4+3M^2N+3|\mathcal{D}| |\hat{\Theta}|^2MN)$, UM-SVRG algorithm effectively reduces the computational complexity of each iteration to roughly $\mathcal{O}(|\mathcal{D}||\hat{\Theta}|^2MN)$. Thirdly, we provide theoretical guarantees of convergence for both the UM-GD and UM-SVRG algorithms through appropriate parameter selection, and prove that the proposed algorithms can converge to a stationary point. Finally, numerical examples demonstrate the effectiveness of the proposed UM-GD and UM-SVRG algorithms.