Nonstationary Gabor frames for

Abstract

Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.

Approximately dual frames of nonstationary Gabor frames for l2(Z) and reconstruction errors

“…Let (g, M) and (h, M) be two NSG systems in l 2 (Z). From the proof of Theorem 2.3 in Lian, 23 we have…”

“…It is known from Christensen and Laugesen 24 that if two Bessel sequences are approximately dual frames, then each of them is a frame. So we can reformulate the existence theorem of NSG frames given in Lian 23 in the context of approximately dual frames. ) −1 g n for n ∈ Z.…”

“…A special class of NSG systems are collections of compactly supported windows. They were studied in Theorem 2.3 in Lian, 23 where the frame property under the conditions given in the following definition is derived. A system (h, M) is called a painless NSG frame for l 2 (Z) if h n is compactly supported with card(supp(h n )) ≤ M n for n ∈ Z, and there exist M) is a painless NSG frame for l 2 (Z), and (g, M) is a Bessel sequence in l 2 (Z) with g n (j) = h n (j) for j ∈ supph n .…”

“…Recently, the first author of this paper provided existence results for NSG frames and constructed NSG frames with non-compactly supported window functions in l 2 (Z). 23 Frames have important applications in signal reconstruction, in which dual frames play an essential role. It is well known that there exists at least one dual frame for a given frame.…”

“…n∈Z such that for M n ≥ M 0 n , n ∈ Z, (g, M) and (h, M) are approximately dual frames, where h n = ( G g, g 0) −1 g n for n ∈ Z.Proof. By Theorem 2.2 in Lian,23 there exists a sequence { M 0 n } n∈Z such that for M n ≥ M 0 n , n ∈ Z, (g, M) is a Bessel sequence in l 2 (Z), and R g, g < inf ∈Z G g, g 0 ( ). Let h n = ( G g, g 0…”