Construction of Multivariate Gaussian Weyl--Heisenberg Frames, (I)

TL;DR

This paper demonstrates that for any generalized Gaussian, a suitable symplectic transformation exists making the associated Weyl--Heisenberg system a frame, advancing the understanding of multivariate Gaussian frames in phase space.

Contribution

It establishes the existence of a symplectic matrix transforming a rectangular lattice into a frame for multivariate Gaussian states.

Findings

Existence of a positive definite symplectic matrix for Gaussian frames

Construction of multivariate Gaussian Weyl--Heisenberg frames

Foundation for a converse result in future work

Abstract

Let {\phi} be an arbitrary generalized Gaussian (squeezed coherent state), {\Lambda}_{{\alpha}{\beta}}=({\alpha}_1 Z \times\cdot\cdot\cdot\times \alpha_{n}\mathbb{Z)\times}(\beta_{1}\mathbb{Z}\times\cdot\cdot\cdot \times\beta_{n}\mathbb{Z)}$ a rectangular lattice. We show that there exists a positive definite symplectic matrix M (depending on {\phi}) such that the multivariate Weyl--Heisenberg system G({\phi},M{\Lambda}_{{\alpha}{\beta}}) is a frame. In a forthcoming Note we will prove a converse to this result.