Uncertainty in Finite Frames with application to Quantization

We begin by motivating a characterization of localization/uncertainty in the finite setting using some recent work in Sigma Delta quantization. This work exploits an alternate dual, ie. a non-canonical dual, to reduce the error of the quantization process. This alternate dual is found by minimizing the $\ell^2$ norms of the dual frame vectors after one applies a finite difference matrix. Using a finite difference Matrix (D) and the Discrete Fourier Transform (F) leads to a natural representation of the Heisenberg product , $\|D v \| \|DF v \|$, in the finite setting. A number of authors have used these matrices to develop finite versions of the Gauss and Hermite functions as eigenvectors of the Discrete Fourier Transform, as well as some finite versions of the classical uncertainty principle. Inspired by the Balian-Low theorem, we present some initial findings for both general finite frames and finite Gabor systems. (Joint work with Blum/Powell/Yilmaz and Fickus/Powell)

Uncertainty in Finite Frames with application to Quantization Presentation