Grassmannian Frames with Applications to Coding and Communication

TL;DR

This paper studies Grassmannian frames, which minimize maximal correlation among uniform frames, explores their bounds, constructions, and applications in coding and wireless communication, extending previous results and introducing infinite-dimensional cases.

Contribution

It provides bounds, explicit constructions, and applications of Grassmannian frames, including infinite-dimensional cases and connections to coding and communication.

Findings

Derived bounds on minimal correlation for Grassmannian frames

Constructed explicit Grassmannian frames using graph theory and coding theory

Applied Grassmannian frames to wireless communication and multiple description coding

Abstract

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.