TY - JOUR

T1 - A density theorem for time-continuous filter banks

AU - Janssen, A.J.E.M.

PY - 1998

Y1 - 1998

N2 - We show in this paper the following result. When a > 0 and gnm = gm(· - na), n, m e Z, is a frame for L2(), where each gm e L2() is localized in the frequency-domain around a point b m e , then ( = 1. Here is the asymptotic lower bound of the average number of bm in a symmetric interval around 0 as the interval length tends to 8. As a particular case it is found that a multi-window Gabor-type system gp(t - na) exp(2pimbt) with n, m e Z, p = 0, …, P - 1, can generate a frame for L2() only if P(ab)-1 = 1. The main result of this paperis based on the Ron-Shen criterion in the frequency-domain for duality of two shift- invariant systems of functions, combined with an amplification of Janssen's elementary proof of the fact that a Weyl-Heisenberg system g(t - na)exp(2pimbt) can generate a framefor L2() only if (ab)-1 = 1.

AB - We show in this paper the following result. When a > 0 and gnm = gm(· - na), n, m e Z, is a frame for L2(), where each gm e L2() is localized in the frequency-domain around a point b m e , then ( = 1. Here is the asymptotic lower bound of the average number of bm in a symmetric interval around 0 as the interval length tends to 8. As a particular case it is found that a multi-window Gabor-type system gp(t - na) exp(2pimbt) with n, m e Z, p = 0, …, P - 1, can generate a frame for L2() only if P(ab)-1 = 1. The main result of this paperis based on the Ron-Shen criterion in the frequency-domain for duality of two shift- invariant systems of functions, combined with an amplification of Janssen's elementary proof of the fact that a Weyl-Heisenberg system g(t - na)exp(2pimbt) can generate a framefor L2() only if (ab)-1 = 1.

U2 - 10.1016/S1874-608X(98)80020-4

DO - 10.1016/S1874-608X(98)80020-4

M3 - Article

SN - 1874-608X

VL - 7

SP - 513

EP - 523

JO - Wavelet Analysis and its Applications

JF - Wavelet Analysis and its Applications

ER -