Reassignment and synchrosqueezing for general time-frequency filter banks, subsampling and processing

In this contribution, we extend the reassignment method (RM) and synchrosqueezing transform (SST) to arbitrary time-frequency localized filters and, in the first case, arbitrary decimation factors. A sufficient condition for the invertibility of the SST is provided. RM and SST are techniques for deconvolution of short-time Fourier and wavelet representations. In both methods, the partial phase derivatives of a complex-valued time-frequency representation are used to determine the instantaneous frequency and group delay associated to the individual representation coefficients. Subsequently, the coefficient energy is reassigned to the determined position. Combining RM and frame theory, we propose a processing scheme that benefits both from the improved localization of the reassigned representation and the frame properties of the underlying complex-valued representation. This scheme is particularly interesting for applications that require low redundancy not achievable by an invertible SST.

[1]  Liguo Han,et al.  Spectral decomposition and de‐noising via time‐frequency and space‐wavenumber reassignment , 2014 .

[2]  Christina Gloeckner.  Foundations Of Time Frequency Analysis , 2016 .

[3]  Thibaud Necciari,et al.  The ERBlet transform: An auditory-based time-frequency representation with perfect reconstruction , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[4]  I. Daubechies,et al.  Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool , 2011 .

[5]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[6]  Nicki Holighaus,et al.  Theory, implementation and applications of nonstationary Gabor frames , 2011, J. Comput. Appl. Math..

[7]  Anssi Klapuri,et al.  A Matlab Toolbox for Efficient Perfect Reconstruction Time-Frequency Transforms with Log-Frequency Resolution , 2014, Semantic Audio.

[8]  Prashant Parikh.  A Theory of Communication , 2010 .

[9]  Kannan Ramchandran,et al.  Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms , 1993, IEEE Trans. Signal Process..

[10]  M. Vetterli.  Filter banks allowing perfect reconstruction , 1986 .

[11]  Thomas Grill,et al.  A Framework for Invertible, Real-Time Constant-Q Transforms , 2012, IEEE Transactions on Audio, Speech, and Language Processing.

[12]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[13]  Axel Röbel,et al.  Sound Analysis and Synthesis Adaptive in Time and Two Frequency Bands , 2011, ArXiv.

[14]  O. Christensen.  Frames and Bases: An Introductory Course , 2008 .

[15]  Ole Christensen,et al.  Frames and Bases , 2008 .

[16]  Hau-Tieng Wu,et al.  The Synchrosqueezing algorithm for time-varying spectral analysis: Robustness properties and new paleoclimate applications , 2011, Signal Process..

[17]  Chuan Li,et al.  A generalized synchrosqueezing transform for enhancing signal time-frequency representation , 2012, Signal Process..

[18]  Patrick Flandrin,et al.  Wigner-Ville spectral analysis of nonstationary processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[19]  Monika Dörfler,et al.  Quilted Gabor frames - A new concept for adaptive time-frequency representation , 2009, Adv. Appl. Math..

[20]  Kelly Fitz,et al.  Correction to: 'On the Use of Time/Frequency Reassignment in Additive Sound Modeling' , 2002 .

[21]  Patrick Flandrin,et al.  Improving the readability of time-frequency and time-scale representations by the reassignment method , 1995, IEEE Trans. Signal Process..

[22]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[23]  K. Kodera,et al.  A new method for the numerical analysis of nonstationary signals , 1976 .

[24]  Judith C. Brown.  Calculation of a constant Q spectral transform , 1991 .

[25]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[26]  Sony Akkarakaran,et al.  10 - Nonuniform Filter Banks: New Results and open Problems , 2003 .

[27]  Thibaud Necciari,et al.  A Perceptually Motivated Filter Bank with Perfect Reconstruction for Audio Signal Processing , 2016, ArXiv.

[28]  Sylvain Meignen,et al.  On the mode synthesis in the synchrosqueezing method , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[29]  P. Flandrin,et al.  Differential reassignment , 1997, IEEE Signal Processing Letters.

[30]  Jun Xiao,et al.  Multitaper Time-Frequency Reassignment for Nonstationary Spectrum Estimation and Chirp Enhancement , 2007, IEEE Transactions on Signal Processing.

[31]  Leon Cohen,et al.  Time Frequency Analysis: Theory and Applications , 1994 .

[32]  Jelena Kovacevic,et al.  Wavelets and Subband Coding , 2013, Prentice Hall Signal Processing Series.

[33]  Ingrid Daubechies,et al.  A Nonlinear Squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models , 2017 .

[34]  Gaigai Cai,et al.  Nonlinear squeezing time-frequency transform for weak signal detection , 2015, Signal Process..

[35]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[36]  Nicki Holighaus,et al.  The Large Time-Frequency Analysis Toolbox 2.0 , 2013, CMMR.

[37]  Ljubisa Stankovic,et al.  Synchrosqueezing-based time-frequency analysis of multivariate data , 2015, Signal Process..

[38]  Sylvain Meignen,et al.  Time-Frequency Reassignment and Synchrosqueezing: An Overview , 2013, IEEE Signal Processing Magazine.

[39]  Yi Wang,et al.  ConceFT: concentration of frequency and time via a multitapered synchrosqueezed transform , 2015, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[40]  Sylvain Meignen,et al.  Second-Order Synchrosqueezing Transform or Invertible Reassignment? Towards Ideal Time-Frequency Representations , 2015, IEEE Transactions on Signal Processing.

[41]  Brian R Glasberg,et al.  Derivation of auditory filter shapes from notched-noise data , 1990, Hearing Research.

[42]  A. Ron.  Review of An introduction to Frames and Riesz bases, applied and numerical Harmonic analysis by Ole Christensen Birkhäuser, Basel, 2003 , 2005 .