On the size of the DOA manifold

We discuss the size of the set of all vectors that can be described as a linear combination of K steering vectors relative to the size of the M-dimensional ambient space. This set is an infinite union of K-dimensional subspaces or a Kdimensional manifold in C^M. If this set was a finite union of subspaces, e.g., because only steering vectors corresponding to an angular grid of, say, N grid points are allowed, then K log N < M would be an adequate measure in the context of compressive sensing. We discuss why this is a good measure and how a generalization to the grid-less case can be obtained.