Chirp-Induced Non-Separable Gabor Windows on $\mathbb{R}^d$

Abstract

We construct an explicit class of non-separable Gabor windows on $L^2(\R^d)$ by applying chirp deformations to tensor-product dual pairs on separable lattices. Starting from one-dimensional dual Gabor frames, we first obtain separable higher-dimensional dual pairs by tensorization. We then transport these systems through the unitary chirp operator $U_C f(x)=e^{\pi i x^T Cx}f(x)$ and the associated phase-space shear, obtaining Gabor systems on lower block-triangular lattices of the form $ \Lambda_{A,B,C}=\{(Ak,CAk+B\ell):k,\ell\in\Z^d\}. $ {The construction is governed by a single unitary conjugacy identity for the mixed Gabor reconstruction operator. As consequences, compact support, smoothness, frame bounds, exact duality, canonical duals, and approximate-duality errors are transported without loss.} {We also record a covariance-level diagnostic, depending on a chosen STFT reference window, which describes the second-order time--frequency tilt induced by the chirp. This diagnostic is separated from the unchanged frame stability constants and is not used here to claim improved sparsity, denoising, conditioning, or computational complexity.}