Probing signal design for MIMO radar: A Riemannian distance approach

We consider the problem of designing probing signals for a Multi-Input Multi-Output (MIMO) radar. The goal is to design a signal vector with a desired covariance while ensuring the side-lobes of the ambiguity functions are small. We will also consider cases in which a bandwidth constraint is placed on the signal. Since covariance matrices are structurally constrained, they form a manifold in the signal space. Hence, we argue that the difference between these matrices should not be in terms of the conventional Euclidean distance (ED), rather, the distance should be measured along the surface of the manifold, i.e., in terms of Riemannian distance (RD). In either case, the optimization problem is quartic in the design variables. An efficient algorithm based on iterative convex quadratic optimization is developed and is effective in producing good solutions. In addition, we show that by optimizing over the manifold, the number of iterations can be significantly reduced in comparison to optimizing in the Euclidean space. Several orthonormal signal sets, including the Walsh functions, the cosine functions and the WLJ functions, are employed to design the transmission signal vector, giving us different feasible regions in the optimization. It is observed that the WLJ othornormal basis yields feasible solutions under tight time-frequency constraints.