CHARACTERIZATION OF RATIONAL TIME-FREQUENCY MULTI-WINDOW GABOR FRAMES AND THEIR DUALS

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θa transform in , and using Θa‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.

This paper addresses multiwindow Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. We give some necessary and/or sufficient conditions for multiwindow Gabor systems to foe frames on discrete periodic sets, and characterize two multiwindow Gabor Bessel sequences to foe dual frames on discrete periodic sets. For a given multiwindow Gabor frame, we derive all its Gabor duals, among which we obtain an explicit expression of the canonical Gabor dual. In addition, we generalize multiwindow Gabor systems to the case of a different sampling rate for each window, and investigate multiwindow Gabor frames and dual frames in this case. We also show the properties of the multiwindow Gabor systems are essentially not changed by replacing the exponential kernel with other kernels.

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

Time-frequency partitions and characterizations of modulation spaces with localization operators

We establish a necessary density criterion for the identifiability of time-frequency structured classes of Hilbert-Schmidt operators. The density condition is based on the density criterion for Gabor frames and Riesz bases in the space of square integrable functions. We complement our findings with examples of identifiable operator classes.

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.