Effective Approach to Calculate Analysis Window in Infinite Discrete Gabor Transform

The long-periodic/infinite discrete Gabor transform (DGT) is more effective than the periodic/finite one in many applications. In this paper, a fast and effective approach is presented to efficiently compute the Gabor analysis window for arbitrary given synthesis window in DGT of long-periodic/infinite sequences, in which the new orthogonality constraint between analysis window and synthesis window in DGT for long-periodic/infinite sequences is derived and proved to be equivalent to the completeness condition of the long-periodic/infinite DGT. By using the property of delta function, the original orthogonality can be expressed as a certain number of linear equation sets in both the critical sampling case and the oversampling case, which can be fast and efficiently calculated by fast discrete Fourier transform (FFT). The computational complexity of the proposed approach is analyzed and compared with that of the existing canonical algorithms. The numerical results indicate that the proposed approach is efficient and fast for computing Gabor analysis window in both the critical sampling case and the oversampling case in comparison to existing algorithms.

[1]  Helmut Bi Icskei.  A Necessary and Sufficient Condition for Dual Weyl-Heisenberg Frames to be Compactly Supported , 1999 .

[2]  Liang Tao,et al.  Noise Reduction for NMR FID Signals via Oversampled Real-Valued Discrete Gabor Transform , 2005, IEICE Trans. Inf. Syst..

[3]  Dennis Gabor,et al.  Theory of communication , 1946 .

[4]  J. M. Morris,et al.  Noise reduction for NMR FID signals via Gabor expansion , 1997, IEEE Transactions on Biomedical Engineering.

[5]  Thomas Strohmer,et al.  Numerical algorithms for discrete Gabor expansions , 1998 .

[6]  Jason Wexler,et al.  Discrete Gabor expansions , 1990, Signal Process..

[7]  Kan Chen,et al.  Optimal biorthogonal functions for finite discrete-time Gabor expansion , 1992, Signal Process..

[8]  Liang Tao,et al.  Real discrete Gabor expansion for finite and infinite sequences , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[9]  Richard S. Laugesen,et al.  Gabor dual spline windows , 2009 .

[10]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[11]  Liang Tao,et al.  Block time-recursive discrete Gabor transform implemented by unified parallel lattice structures , 2005, 2005 IEEE International Symposium on Circuits and Systems.

[12]  Shidong Li,et al.  Sparse Dual Frames and Dual Gabor Functions of Minimal Time and Frequency Supports , 2013 .

[13]  Laura Bianco,et al.  Implementation of a Gabor Transform Data Quality-Control Algorithm for UHF Wind Profiling Radars , 2013 .

[14]  Nirmala Saini,et al.  Face and palmprint multimodal biometric systems using Gabor–Wigner transform as feature extraction , 2014, Pattern Analysis and Applications.

[15]  Maja Temerinac-Ott,et al.  Discrete Fourier-Invariant Signals: Design and Application , 2012, IEEE Transactions on Signal Processing.

[16]  Chao Lu,et al.  Parallel lattice structure of block time-recursive generalized Gabor transforms , 1997, Signal Process..

[17]  Gerhard Fettweis,et al.  Generalized Frequency Division Multiplexing in a Gabor Transform Setting , 2014, IEEE Communications Letters.

[18]  Yehoshua Y. Zeevi,et al.  Oversampling in the Gabor scheme , 1993, IEEE Trans. Signal Process..

[19]  Liang Tao,et al.  A Sparse Analysis Window for Discrete Gabor Transform , 2017, Circuits Syst. Signal Process..

[20]  S. Qian.  Introduction to Time-Frequency and Wavelet Transforms , 2001 .

[21]  Shie Qian,et al.  Discrete Gabor transform , 1993, IEEE Trans. Signal Process..

[22]  Peter L. Søndergaard,et al.  Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window , 2012 .

[23]  M. Bastiaans,et al.  Gabor's expansion of a signal into Gaussian elementary signals , 1980, Proceedings of the IEEE.

[24]  Stanley C. Ahalt,et al.  Computationally attractive real Gabor transforms , 1995, IEEE Trans. Signal Process..

[25]  S. Gazor,et al.  Local probability distribution of natural signals in sparse domains , 2014 .

[26]  Peter L. Søndergaard,et al.  Gabor frames by sampling and periodization , 2007, Adv. Comput. Math..

[27]  Benjamin Friedlander,et al.  Detection of transient signals by the Gabor representation , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[28]  Liang Tao,et al.  Multiwindow Real-Valued Discrete Gabor Transform and Its Fast Algorithms , 2015, IEEE Transactions on Signal Processing.

[29]  J. Zak.  FINITE TRANSLATIONS IN SOLID-STATE PHYSICS. , 1967 .

[30]  Ole Christensen,et al.  Pairs of Dual Gabor Frame Generators with Compact Support and Desired Frequency Localization , 2006 .