Abstract
In this paper, we introduce the reassigned spectrogram of Stockwell transform by re-mapping the surface of the spectrogram of Stockwell transform with the aim to improve its readability. We first define the channelized instantaneous frequency and the local group delay for the Stockwell transform. At any given point in the time-frequency domain, the associated local group delay and channelized instantaneous frequency provide a re-estimation of the time arrival and instantaneous frequency of the signal component observed at that point. The reassigned spectrogram of Stockwell transform therefore has signal energy highly concentrated at the instantaneous frequency/group delay curves and greatly increases the resolution and readability of the time-frequency structure of the underlying signal. The instantaneous frequencies of signal components can then be extracted by detecting the local energy peaks in the reassigned spectrogram of Stockwell transform. The improvement of the reassigned spectrogram of Stockwell transform over the conventional spectrogram of Stockwell transform is illustrated using both synthetic and real signals.
This research is supported by NSERC grant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.G. Stockwell, L. Mansinha, R.P. Lowe, Localization of the complex spectrum: the S-transform. IEEE Trans. Signal Proc. 44 (4), 998–1001 (1996)
K. Gröchenig, Foundations of Time-frequency Analysis (Birkhäuser, Boston, 2001)
R.G. Stockwell, Why use the S-transform? in Pseudo-differential Operators: Partial Differential Equations and Time-Frequency Analysis. Fields Institute Communications Series, vol. 52 (American Mathematical Society, Providence, 2007), pp. 279–309
P.C. Gibson, M.P. Lamoureux, G.F. Margrave, Stockwell and wavelet transform. J. Fourier Anal. Appl. 12, 713–721 (2006)
M.W. Wong, H. Zhu, A characterization of the Stockwell transform, in Modern Trends in Pseudo-differential Operators. Operator Theory: Advances and Applications, vol. 172 (Birkhäuser, Basel/Boston, 2006), pp. 251–257
J. Du, M.W. Wong, H. Zhu, Continuous and discrete formulas for the Stockwell transform. Integral Transform. Spec. Funct. 8, 537–543 (2007)
H. Zhu, B.G. Goodyear, M.L. Lauzon, R.A. Brown, G.S. Mayer, A.G. Law, L. Mansinha, J.R. Mitchell, A new local multiscale Fourier analysis for MRI. Med. Phys. 30 (6), 1134–1141 (2003)
C.R. Pinnegar, Polalization analysis and polarization filtering of three component signals with the time-frequency S-transform. Geophys. J. Int. 165 (2), 596–606 (2006)
C. Liu, W. Gaetz, H. Zhu, The Stockwell transform in studying the dynamics of brain functions, in Proceedings of the International Workshop Pseudo-differential Operators: Complex Analysis and Partial Differential Equations, vol. 205 (Birkhäuser, Basel/Boston, 2008) pp. 277–291
B. Boashash, Estimating and interpreting the instantaneous frequency of a signal-part 1: fundamentals. Proc. IEEE 80, 520–538 (1992)
L. Cohen, Time-Frequency Analysis (Prentice Hall, Englewood Cliffs, 1995)
K. Kodera, R. Gendrin, C.D. Villedary, Analysis of time-varying signals with small BT values. IEEE Trans. Acoust. Speech, Signal Process. 26, 64–76 (1978)
F. Auger, P. Flandrin, Improving the readability of time frequency and time scale representations by the reassignment method. IEEE Trans. Signal Proc. 43, 1068–1089 (1995)
S.A. Fulop, K. Fitz, Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram with applications. J. Acoust. Soc. Am. 119, 360–371 (2006)
D.J. Nelson, Cross-spectral methods for processing speech. J. Acoust. Soc. Am. 110 (5), 2575–2592 (2001)
K. Fitz, L. Haken, S. Lefvert, C. Champion, M. O’Donnell, Cell-utes and flutter-tongued cats: sound morphing using Loris and the reassigned bandwidth-enhanced model. Comput. Music J. 27, 44–65 (2003)
T.J. Gardner, M.O. Magnasco, Instantaneous frequency decomposition: an application to spectrally sparse sounds with fast frequency modulations. J. Acoust. Soc. Am. 117, 2896–2903 (2005)
C. Liu, A new adaptive multi-resolution time-frequency representation with applications in studying brain functions. Unpublished Ph.D. thesis, York University (2011)
Q. Guo, S. Molahajloo, M.W. Wong, Phase of modified Stockwell transforms and instantaneous frequencies. J. Math. Phys. 51, 052101 (2010)
P. Flandrin, On detection-estimation procedures in the time-frequency plane, in International Conference on Acoustics, Speech and Signal Processing, Tokyo, vol. 43, no. 5 (1986), pp. 1–4
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Liu, C., Zhu, H. (2017). The Reassigned Spectrogram of the Stockwell Transform. In: Wong, M., Zhu, H. (eds) Pseudo-Differential Operators: Groups, Geometry and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47512-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-47512-7_11
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-47511-0
Online ISBN: 978-3-319-47512-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)