Constructing Tight Gabor Frames Using CAZAC Sequences

TL;DR

This paper investigates conditions under which CAZAC sequences can generate tight Gabor frames, providing theoretical criteria and practical methods for verifying tightness in applications like signal processing.

Contribution

It establishes necessary and sufficient conditions for tight Gabor frames using CAZAC sequences via Janssen's representation and ambiguity functions, and introduces an alternative Gram matrix method.

Findings

Necessary and sufficient conditions for tightness using zeros of the ambiguity function

Construction of examples based on CAZAC sequences

An alternative Gram matrix-based method for verifying tightness

Abstract

The construction of finite tight Gabor frames plays an important role in many applications. These applications include significant ones in signal and image processing. We explore when constant amplitude zero autocorrelation (CAZAC) sequences can be used to generate tight Gabor frames. The main theorem uses Janssen's representation and the zeros of the discrete periodic ambiguity function to give necessary and sufficient conditions for determining whether any Gabor frame is tight. The relevance of the theorem depends significantly on the construction of examples. These examples are necessarily intricate, and to a large extent, depend on CAZAC sequences. Finally, we present an alternative method for determining whether a Gabor system yields a tight frame. This alternative method does not prove tightness using the main theorem, but instead uses the Gram matrix of the Gabor system.