Gabor's signal expansion and a modified Zak transform for a quincunx-type sampling geometry

2003 ◽  
Author(s):  
M.J. Bastiaans ◽  
A.J. van Leest
Keyword(s):  
Zak Transform

High-concentration time–frequency analysis for multi-component nonstationary signals based on combined multi-window Gabor transform

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Qiang Wang ◽  
Chen Meng ◽  
Cheng Wang
Keyword(s):  
Optimal Solution ◽  
Iteration Process ◽  
Perfect Reconstruction ◽  
Gabor Transform ◽  
Zak Transform ◽  
High Concentration ◽  
Content Type ◽  
Nonstationary Signals ◽  
Time Frequency ◽  
Optimization Function

PurposeThis study aims to reveal the essential characteristics of nonstationary signals and explore the high-concentration representation in the joint time–frequency (TF) plane.Design/methodology/approachIn this paper, the authors consider the effective TF analysis for nonstationary signals consisting of multiple components.FindingsTo make it, the authors propose the combined multi-window Gabor transform (CMGT) under the scheme of multi-window Gabor transform by introducing the combination operator. The authors establish the completeness utilizing the discrete piecewise Zak transform and provide the perfect-reconstruction conditions with respect to combined TF coefficients. The high-concentration is achieved by optimization. The authors establish the optimization function with considerations of TF concentration and computational complexity. Based on Bergman formulation, the iteration process is further analyzed to obtain the optimal solution.Originality/valueWith numerical experiments, it is verified that the proposed CMGT performs better in TF analysis for multi-component nonstationary signals.

scholarly journals The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Joachim Toft
Keyword(s):  
Operator Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Modulation Spaces ◽  
Zak Transform ◽  
Periodic Functions ◽  
Distribution Spaces ◽  
Necessary And Sufficient

AbstractWe characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

Gabor’s signal expansion and the Zak transform

1994 ◽  
Vol 33 (23) ◽  
pp. 5241 ◽  
Author(s):  
Martin J. Bastiaans
Keyword(s):  
Zak Transform

Zak transform

1998 ◽  
pp. 57-75
Author(s):  
Richard Tolimieri ◽  
Myoung An
Keyword(s):  
Zak Transform

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

2005 ◽  
Vol 03 (02) ◽  
pp. 257-281 ◽  
Author(s):  
J. A. HOGAN ◽  
J. D. LAKEY
Keyword(s):  
Summation Formula ◽  
Sampling Theorem ◽  
Poisson Summation Formula ◽  
Zak Transform ◽  
Self Similarity ◽  
Critical Rate ◽  
Sampling Schemes ◽  
Bandlimited Signals ◽  
Shift Invariant Spaces ◽  
Invariant Spaces

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

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