Implementation of discretized Gabor frames and their duals

TL;DR

This paper explores efficient computation of dual windows for Gabor frames, analyzing iterative algorithms' stability and introducing direct methods for specific window types that converge rapidly to the canonical dual.

Contribution

It presents new direct algorithms for dual windows in Gabor frames with totally positive or exponential B-spline windows, and analyzes the stability of iterative approximation methods.

Findings

Iterative algorithms' stability analyzed.

Direct algorithms produce exact duals with compact support.

Dual windows converge exponentially fast to the canonical dual.

Abstract

The usefulness of Gabor frames depends on the easy computability of a suitable dual window. This question is addressed under several aspects: several versions of Schulz's iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability. For Gabor frames with totally positive windows or with exponential B-splines a direct algorithm yields a family of exact dual windows with compact support. It is shown that these dual windows converge exponentially fast to the canonical dual window.