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Vector-valued Gabor frames associated with periodic subsets of the real line

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  • Li, Yun-Zhang
  • Zhang, Yan

Abstract

The notion of vector-valued frame (also called superframe) was first introduced by Balan in the context of multiplexing. It has significant applications in mobile communication, satellite communication, and computer area network. For vector-valued Gabor analysis, existent literatures mostly focus on L2(R,CL) instead of its subspace. Let a>0, and S be an aZ-periodic measurable set in R (i.e. S+aZ=S). This paper addresses Gabor frames in L2(S,CL) with rational time–frequency product. They can model vector-valued signals to appear periodically but intermittently. And the projections of Gabor frames in L2(R,CL) onto L2(S,CL) cannot cover all Gabor frames in L2(S,CL) if S≠R. By introducing a suitable Zak transform matrix, we characterize completeness and frame condition of Gabor systems, obtain a necessary and sufficient condition on Gabor duals of type I (resp. II) for a general Gabor frame, and establish a parametrization expression of Gabor duals of type I (resp. II). All our conclusions are closely related to corresponding Zak transform matrices. This allows us to easily realize these conclusions by designing the corresponding matrix-valued functions. An example theorem is also presented to illustrate the efficiency of our method.

Suggested Citation

  • Li, Yun-Zhang & Zhang, Yan, 2015. "Vector-valued Gabor frames associated with periodic subsets of the real line," Applied Mathematics and Computation, Elsevier, vol. 253(C), pages 102-115.
    • Handle: RePEc:eee:apmaco:v:253:y:2015:i:c:p:102-115
    DOI: 10.1016/j.amc.2014.12.046
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    Cited by:

    1. Qiaofang Lian & Zhao Wang, 2018. "Generalized super Gabor duals with bounded invertible operators," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(3), pages 413-430, September.
    2. Perraudin, Nathanaël & Holighaus, Nicki & Søndergaard, Peter L. & Balazs, Peter, 2018. "Designing Gabor windows using convex optimization," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 266-287.

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