Discrete Subspace Multiwindow Gabor Frames and Their Duals

Yun‐Zhang Li ,Yan Zhang

doi:10.1155/2013/357242

Abstract

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

Highlights

Let H be a separable Hilbert space
A frame for H is said to be a Riesz basis if it ceases to be a frame for H whenever an arbitrary element is removed
This paper addresses Gabor systems G(g, N, M) of the form

Summary

Introduction

Let H be a separable Hilbert space. An at most countable sequence {hi}i∈I in H is called a frame for H if there exist 0 < A ≤ B < ∞ such that. We always denote by N the least common multiple of Nl with 1 ≤ l ≤ L, by λl the positive integer satisfying N = λlNl for each 1 ≤ l ≤ L, by p and q two relatively prime positive integers satisfying N/M = p/q, and by Q the number Q = q ∑Ll=1 λl They are all uniquely determined by Assumptions 1 and 2. Example 38 and the arguments before it in the last section show that, for Gabor duals, multi-window Gabor frames behave differently from single-window ones. Motivated by the above works, we in this paper study subspace Gabor frame of the form (2) under Assumptions 1 and 2. We characterize the uniqueness of Gabor duals of type I (type II) and obtain a class of examples of subspace multiwindow Gabor frames (Riesz bases, orthonormal bases) and their Gabor duals of type I (type II) (see Theorems 36 and 37)

Zak Transforms and Zak Transform Matrices

Zak transform

Frame Characterization

Gabor Dual Characterization