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

We construct an explicit class of non-separable Gabor windows on $L^2({\mathbb{R}}^d)$ 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 ${\mathrm{\Lambda }}_{A,B,C}=\{(Ak,CAk+B\mathrm{\ell }):k,\mathrm{\ell }\in {\mathbb{Z}}^d\}\mathrm{.}$ {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.}