On the structures of Grassmannian frames

A common criterion in the design of finite Hilbert space frames is minimal coherence, as this leads to error reduction in various signal processing applications. Frames that achieve minimal coherence relative to all unit-norm frames are called Grassmannian frames, a class which includes the well-known equiangular tight frames. However, the notion of “coherence minimization” varies according to the constraints of the ambient optimization problem, so there are other types of “minimally coherent” frames one can speak of. In addition to Grassmannian frames, we consider the class of frames which minimize coherence over the space of frames which are both unit-norm and tight, which we call 1-Grassmannian frames. We observe that these two types of frames coincide in many settings, but not all; accordingly, we investigate some of the differences between the resulting theories. For example, one noteworthy advantage of 1-Grassmannian frames is that their optimality properties are preserved under the Naimark complement.