A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported

Authors

Helmut Bölcskei

Reference

The Journal of Fourier Analysis and Applications, Vol. 5, No. 5, pp. 409-419, 1999.

Abstract

In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L2(R) for its minimal dual (Wexler-Raz dual) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L2(R) to have a band-limited minimal dual. As a consequence of these conditions we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactly supported (band-limited) minimal dual if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.

Keywords

Gabor expansions, Weyl-Heisenberg frames, compact support, Zak transform, unimodularity, polynomial matrices, tight frames, oversampling

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