Construction of k-angle tight frames

TL;DR

This paper presents a new, explicit construction method for k-angle tight frames, generalizing equiangular tight frames, with applications in signal processing and connections to graph theory.

Contribution

It introduces a simple construction for k-angle tight frames that generalize ETFs and relates them to regular graphs and association schemes.

Findings

Explicit construction of ETFs of d+1 vectors in d-dimensional space.

Generalization of ETFs to k-angle tight frames with at most k distinct angles.

Connection of these frames to regular graphs and association schemes.

Abstract

Frames have become standard tools in signal processing due to their robustness to transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d+1 vectors in a d dimensional space always exists. This paper gives an explicit construction of ETFs of d+1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do so is to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This paper presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes.