A Riemannian Distance Approach to Probing Signal Design for MIMO Radar

Abstract

We consider the problem of designing probing signals for a Multi-Input Multi-Output (MIMO) radar. The goal is to design a signal vector having a desired covariance while ensuring the sidelobes 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 measured 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 a Riemannian distance (RD). An optimization problem for the design of the probing signal is formulated for each distance metric, with the waveform being represented as a linear combination of a set of orthonormal signals. In both cases, the optimization problem is quartic in the coefficients. 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 are used in our design examples, including the Walsh functions, the cosine functions and a set of functions designed for optimal time-frequency concentration. When the time-frequency constraints are tight, the selection of the orthonormal set plays a significant role in the design, with the functions with optimal time-frequency concentration.