Time-frequency shift invariance of Gabor spaces generated by integer lattices

Time-frequency shift invariance and the Amalgan Balian Low Theorem

We consider smoothness properties of the generator of a principal Gabor space on the real line which is invariant under some additional translation-modulation pair. We prove that if a Gabor system on a lattice with rational density is a Riesz basis for its closed linear span, and if the closed linear span, a Gabor space, has any additional translation-modulation invariance, then its generator cannot decay well in time and in frequency simultaneously.

Gabor 框架界优化

Gabor frame bound optimizations

本文研究了矩形格上不同窗和整数过采样率下 Gabor 系统的尖锐框架界，发现某些情况下正方形格最优，其他情况下矩形格最优，且有时最优格不存在而退化系统最优。

Non-Invertible Gabor Transforms

Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. The redundancy of such representations is often directly related to the computational load of any algorithm operating in the transform domain. To reduce complexity, it may be desirable to increase the time and frequency sampling intervals beyond the point where the transform is invertible, at the cost of an inevitable recovery error. In this paper we initiate the study of recovery procedures for non-invertible Gabor representations. We propose using fixed analysis and synthesis…

Asymptotic boundary forms for tight Gabor frames and lattice localization domains

We consider Gabor localization operators $G_{ϕ,Ω}$ defined by two parameters, the generating function $ϕ$ of a tight Gabor frame $\{ϕ_λ\}_{λ\in Λ}$, parametrized by the elements of a given lattice $Λ\subset \Bbb{R}^2$, i.e. a discrete cocompact subgroup of $\Bbb{R}^2$, and a lattice localization domain $Ω\subset \Bbb{R}^2$ with its boundary consisting of line segments connecting points of $Λ$. We find an explicit formula for the boundary form $BF(ϕ,Ω)=\text{A}_Λ\lim_{R\rightarrow \infty}\frac{PF(G_{ϕ,RΩ})}{R}$, the normalized limit of the projection functional $PF(G_{ϕ,Ω})=\sum_{i=0}^{\infty}λ…

Non-separable lattices, Gabor orthonormal bases and Tilings

Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $Λ=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(Λ)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}χ_K, Λ)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block trian…

A quantitative Balian-Low theorem

We study functions generating Gabor Riesz bases on the integer lattice. The classical Balian-Low theorem restricts the simultaneous time and frequency localization of such functions. We obtain a quantitative estimate that extends both this result and other related theorems.