Properties of certain multi-angle tight frames in [formula omitted

Abstract

Frames are now standard tools in signal processing, and have applications ranging from compressed sensing, to communication systems and quantum sensing. Designing frames with some special structure such as equiangularity and tightness is useful in frame theory and its applications. In practice, constructing such frames with a given size in a specific dimension can be difficult or impossible in some cases. This leads one to consider the construction of frames with few distinct angles among pairs of frame vectors. We look at a specific construction done previously, which, for a given dimension d and integer 1<k≤d, gives a unit norm tight frame such that the number of distinct angles among the vectors is bounded above by k. Here we give several properties of this multi-angle tight frame. In particular, we provide a precise value of the number of angles which depends on a relationship between d and k, and give a formula for the multiplicity of each angle. We also show how one can strategically choose subsets of such a multi-angle tight frame that will be equiangular or orthogonal. This property is meaningful in the context of erasures when one has to deal with subsets of a given frame that may or may not be a frame. Finally, we show a connection between certain unit norm tight frames with three angles and adjacency matrices of regular graphs.