Abstract
For a finite abelian group A and g ∈ L(A), we define the concept of a translate of g by an element in A × A* and form systems of functions in L(A) by translates of g over subgroups Δ of A × A*. Such systems are called Weyl-Heisenberg (W-H) systems and expansions of signals over W-H systems are called W-H expansions. W-H expansions naturally embed into many time-frequency constructions and distributions resulting in significant simplification in the form of basic formulas and in the interpretation of computations.
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References
The fundamental work of Gabor [19] along with extensions of Bastiaans [4] form the basis of the critical sampling case. During the last five years the over-sampling case has been extensively studied by Zibulski-Zeevi [66–71], Janssen [24], Morris-Lu [31], Wexler-Raz [61], Redding-Newsan [41–43], Qian-Chen [36], Qian-Chen-Li [37], and Brodzik-An-Gertner-Tolimieri [7]. The introduction of biorthogonals has greatly inspired much of this work. In these works, the Zak transform and sometimes frame theory play important roles. A somewhat different approach to the study of over-sampled W-H systems using frame operators can be found in the works of Bölckei-Feichtinger-Hlawatch [5], Qui-Feichtinger [38, 39], and Qui-Feichtinger-Strohmer [40].
The relationship of W-H systems to other time-frequency representations can be found in the work of Wexler-Raz [62]. Many of these papers contain applications to imaging and computer vision.
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© 1998 Birkhäuser Boston
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Tolimieri, R., An, M. (1998). Weyl-Heisenberg systems. In: Time-Frequency Representations. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4152-2_6
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DOI: https://doi.org/10.1007/978-1-4612-4152-2_6
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8676-9
Online ISBN: 978-1-4612-4152-2
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