Weighted multiwindow discrete Gabor transform

Abstract According to the Heisenberg uncertainty principle, an analysis window with a high time resolution in time domain will result in a low frequency resolution in frequency domain, and vice versa. To obtain the Gabor spectrum with high time-frequency resolution and concentration, weighted multiwindow discrete Gabor transform (M-DGT) using weights and the biorthogonal analysis method for analyzing long (or infinite) sequences is proposed in this paper, in which the combined Gabor coefficients constructed by a combination of M-DGT coefficients can be adaptively changed according to the time-frequency distributions of an analyzed signal containing multiple time-varying frequencies. To obtain the weights of the weighted M-DGT, the M-DGT is converted into a sparse problem with l 1 - l 2 regularization, then an efficient iterative algorithm for solving the weights in terms of real-valued matrix and real-valued vector is derived. The convergence of the iterative algorithm is proved by optimization theory. The experimental results demonstrate that the proposed method is an effective and efficient tool for nonstationary time-frequency analysis of signals.

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

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

[3]  Liang Tao,et al.  Gabor time-frequency representation for transient signals using multiwindow discrete Gabor transform , 2017, Int. J. Wavelets Multiresolution Inf. Process..

[4]  Yehoshua Y. Zeevi,et al.  Existence Conditions for Discrete Noncanonical Multiwindow Gabor Schemes , 2007, IEEE Transactions on Signal Processing.

[5]  Zahra Moravej,et al.  Gabor transform-based fault location method for multi-terminal transmission lines , 2018, Measurement.

[6]  Qiao-Fang Lian,et al.  Time-domain characterization of multiwindow Gabor systems on discrete periodic sets , 2013 .

[7]  S. Qian,et al.  Joint time-frequency analysis , 1999, IEEE Signal Process. Mag..

[8]  Liang Tao,et al.  Multi-window real-valued discrete Gabor transform for long and infinite sequences , 2015, 2015 IEEE International Symposium on Circuits and Systems (ISCAS).

[9]  K. Gröchenig,et al.  Sharp Results on Sampling with Derivatives in Shift-Invariant Spaces and Multi-Window Gabor Frames , 2017, Constructive Approximation.

[10]  Liang Tao,et al.  Fast Parallel Approach for 2-D DHT-Based Real-Valued Discrete Gabor Transform , 2009, IEEE Transactions on Image Processing.

[11]  Jinghuai Gao,et al.  Seismic Time–Frequency Analysis via STFT-Based Concentration of Frequency and Time , 2017, IEEE Geoscience and Remote Sensing Letters.

[12]  Yonina C. Eldar,et al.  Blind Phaseless Short-Time Fourier Transform Recovery , 2018, IEEE Transactions on Information Theory.

[13]  Liang Tao,et al.  Novel DCT-Based Real-Valued Discrete Gabor Transform and Its Fast Algorithms , 2009, IEEE Transactions on Signal Processing.

[14]  Tomasz P. Zielinski.  Joint time-frequency resolution of signal analysis using Gabor transform , 2001, IEEE Trans. Instrum. Meas..

[15]  Adel Belouchrani,et al.  Performance analysis and computational cost evaluation of high-resolution time-frequency distributions derived from compact support time-lag kernels , 2018, Digit. Signal Process..

[16]  Liang Tao,et al.  Multirate-based fast parallel algorithms for DCT-kernel-based real-valued discrete Gabor transform , 2012, Signal Process..

[17]  Yehoshua Y. Zeevi,et al.  Discrete multiwindow Gabor-type transforms , 1997, IEEE Trans. Signal Process..

[18]  Rui Li,et al.  Parallel Lattice Structure for Dual Windows Computation in Multiwindow Gabor Transform , 2019, IEEE Access.

[19]  Chen Lin,et al.  Parallel-computing-based implementation of fast algorithms for discrete Gabor transform , 2015, IET Signal Process..

[20]  Jian Zhou,et al.  Sparse Gabor Time–Frequency Representation Based on $$\ell _{1/2}$$ℓ1/2–$$\ell _2$$ℓ2 Regularization , 2019, Circuits Syst. Signal Process..

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

[22]  Hui Ji,et al.  Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames , 2016, Journal of Fourier Analysis and Applications.

[23]  Ali Gholami,et al.  Sparse Time–Frequency Decomposition and Some Applications , 2013, IEEE Transactions on Geoscience and Remote Sensing.

[24]  Shidong Li.  Discrete multi-Gabor expansions , 1999, IEEE Trans. Inf. Theory.

[25]  Mads S. Jakobsen,et al.  Gabor duality theory for Morita equivalent C∗-algebras , 2019, International Journal of Mathematics.

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

[27]  Liang Tao,et al.  Parallel lattice structures of block time-recursive discrete Gabor transform and its inverse transform , 2008, Signal Process..

[28]  Nathanael Perraudin,et al.  Designing Gabor windows using convex optimization , 2014, Appl. Math. Comput..

[29]  James M. Murphy,et al.  Discrete directional Gabor frames , 2016, Applied and Computational Harmonic Analysis.

[30]  Liang Tao,et al.  Block Time-Recursive Real-Valued Discrete Gabor Transform Implemented by Unified Parallel Lattice Structures , 2005, IEICE Trans. Inf. Syst..

[31]  Miguel Arevalillo-Herráez,et al.  Efficient Analysis and Synthesis Using a New Factorization of the Gabor Frame Matrix , 2018, IEEE Transactions on Signal Processing.

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

[33]  Emil Solsbæk Ottosen,et al.  A Characterization of Sparse Nonstationary Gabor Expansions , 2016, 1606.08647.

[34]  Yehoshua Y. Zeevi,et al.  Macromolecular sequence analysis using multiwindow Gabor representations , 2008, Signal Process..

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

[36]  Deyun Wei,et al.  Time-frequency analysis method based on affine Fourier transform and Gabor transform , 2017, IET Signal Process..

[37]  Ahmad Salemnia,et al.  Method for classification of PQ events based on discrete Gabor transform with FIR window and T2FK-based SVM and its experimental verification , 2017 .

[38]  Wei Zhang,et al.  Time–frequency analysis of nonstationary vibration signals for deployable structures by using the constant-Q nonstationary gabor transform , 2016 .

[39]  Feng Zhou,et al.  Discrete Gabor transforms with complexity O (NlogN) , 1999, Signal Process..

[40]  Xuefeng Chen,et al.  Ridge-Aware Weighted Sparse Time-Frequency Representation , 2021, IEEE Trans. Signal Process..

[41]  Karlheinz Gröchenig,et al.  Implementation of Discretized Gabor Frames and Their Duals , 2015, IEEE Transactions on Information Theory.

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

[43]  Irena Orovic,et al.  Compressive sensing meets time-frequency: An overview of recent advances in time-frequency processing of sparse signals , 2017, Digit. Signal Process..

[44]  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).

[45]  Hon Keung Kwan,et al.  Multiwindow discrete Gabor transform using parallel lattice structures , 2020, IET Signal Process..

[46]  Liang Tao,et al.  Efficient discrete Gabor transform with weighted linear combination of analysis windows , 2016 .

[47]  Raj Kumar,et al.  Recognition of Underlying Causes of Power Quality Disturbances Using Stockwell Transform , 2020, IEEE Transactions on Instrumentation and Measurement.

[48]  Yue Wang,et al.  Clustered fractional Gabor transform , 2020, Signal Process..

[49]  Liang Tao,et al.  Multirate-Based Fast Parallel Algorithms for 2-D DHT-Based Real-Valued Discrete Gabor Transform , 2012, IEEE Transactions on Image Processing.