# On the structures of Grassmannian frames

## TL;DR

This paper explores the structures of Grassmannian and 1-Grassmannian frames, analyzing their coherence properties, differences, and the preservation of optimality under Naimark complement, with implications for signal processing.

## Contribution

It introduces and compares Grassmannian and 1-Grassmannian frames, highlighting their properties, differences, and the impact of Naimark complement on their optimality.

## Key findings

- Grassmannian and 1-Grassmannian frames often coincide but differ in some cases.
- 1-Grassmannian frames maintain optimality under Naimark complement.
- The paper clarifies the theoretical distinctions between different minimal coherence frames.

## Abstract

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 enjoyed by 1-Grassmannian frames is that their optimality properties are preserved under the Naimark complement.