On optimum multi‐input multi‐output radar signal design: Ambiguity function, manifold structure and duration‐bandwidth

Abstract

Abstract A design technique is developed for the probing signals of a Multi‐Input Multi‐Output (MIMO) radar. The concentration of the energy of the signal in its essential duration and essential bandwidth is achieved through the use of a class of time‐frequency concentrated functions called the WLJ functions as the synthesizing signal set. The goal is to design a signal vector having a pre‐specified desired covariance (CoV) matrix while ensuring that the side‐lobes of the ambiguity functions are small. Since CoV 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, that is, in terms of a Riemannian distance (RD). In either case, the signal optimisation problem is non‐convex in the design variables, involving, respectively, a quartic and a square‐root objective function. An efficient algorithm based on successive convex approximation is developed in which the original non‐convex problems are transformed so that they can be approximated by a convex quadratically constrained quadratic problem at each stage, resulting in good approximate solutions. Comparing the designs using ED and RD, we find that the convergence of the algorithm can be significantly faster when optimising over the manifold (RD) than when optimising over the whole space (ED). More importantly, for tight constraints, the use of RD yields solutions which satisfy the constraints far better than the use of ED.

Keywords

• approximation theory
• concave programming
• convex programming
• covariance matrices
• iterative methods
• matrix algebra