Designing Unimodular Waveforms for MIMO Radar Based on Manifold Optimization Method

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.