Tight Gabor Sets on Discrete Periodic Sets

Abstract

This paper addresses Gabor analysis on a discrete periodic set. Such a scenario can potentially find its applications in signal processing where signals may present on a union of disconnected discrete index sets. We focus on the Gabor systems generated by characteristic functions. A sufficient and necessary condition for a set to be a tight Gabor set in discrete periodic sets is obtained; discrete periodic sets admitting a tight Gabor set are also characterized; the perturbation of tight Gabor sets is investigated; an algorithm to determine whether a set is a tight Gabor set is presented. Furthermore, we prove that an arbitrary Gabor frame set can be represented as the union of a tight Gabor set and a Gabor Bessel set.

Multi-window Gabor frames and oblique Gabor duals on discrete periodic sets

“…Also observing that S h, g = I on l 2 (S), by Lemma 2.7, we have ( i.e., card(S ∩ N N ) = LM by Lemma 2 in [20]. The proof is completed.…”

“…(i) When L = 1, the theorem is an immediate consequence of Theorem 4 in [20]. Next we turn to the case L > 1.…”

“…So, by Theorem 4 in [20], for each l ∈ N L , we can construct a set E l ⊂ S l + N Z such that {E m M T nN χ E l : m ∈ N M , n ∈ Z} forms a tight frame for l 2 (S l + N Z) with frame bound M . Put g = {χ E l : l ∈ N L }.…”

Super Gabor frames on discrete periodic sets

“…(ii) Lcard(S ∩ N N ) = M. (iii) L k∈N p χ S ( j + kM) = q for j ∈ N M q .Proof By Lemma 2 in[24], we havecard(S ∩ N N ) = j∈N M q card(B j ) = j∈N M q k∈N p χ S ( j + kM). S ( j + kM) = M.…”

“…We refer to [1,7,8,20,21,[26][27][28][29] and the references therein for Gabor analysis on l 2 (Z), and to [23][24][25] for Gabor analysis on general l 2 (S). The framework of Gabor analysis on l 2 (S) can model a signal to appear periodically but intermittently.…”