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Time-Frequency Spectra with the S-transform

Lalu Mansinha - Watch Now

Most interesting data series consist of signal and noise series that are usually non-stationary, i.e. the properties fluctuate with time.Fourier analysis (FT) of the whole time series provides the spectrum of the whole time series, but is not capable of showing the time variation of the spectrum.The S-transform, also known as the Stockwell Transform (ST), uses a scalable, translating Gaussian window to determine the local spectrum at every point on the time series.The local spectrum supplements the local temporal information in the time series, and aids in the detection of onsets and cessations of events.For a N point time series, the output of the FT is a N point complex spectrum; the output of ST is a N x N, 2D time-frequency matrix, giving a N point spectrum at every point on the time series.Since the original time series contains only N points, the additional N2 - N points is a measure of the non-independent, redundant information computed for the ST.The redundancy is useful in presenting the similarities in the neighbouring local spectra and contributes to the visual continuity and smoothness to the 2-D time-frequency spectrum. The computation and storage of N2 - N additional points is a major drag on the usage of ST. Several approaches to reducing the computational burden are presented.Local spectra aids in analysis of 1D data. In images, the 2D local spectra aids in definition of texture and image segmentation.For 3 and 4 colour images, Trinion ST and Quaternion ST have been defined.In use since 1997, ST has found applications in numerous disciplines, including medical data series and images; power quality disturbance; atmospheric physics; exploration geophysics etc.The ST has been implemented in ARM and Raspberry Pi processors by several researchers.

This presentation is introductory, for the interested practitioner.Mathematical content will be at the absolute minimum.

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HardRealTime
Score: 0 | 3 weeks ago | 1 reply

Thank you! I especially appreciated the resources and backgrounds of Gabor and Hamilton.

LaluSpeaker
Score: 0 | 2 weeks ago | no reply

Happy that you liked the presentation. Appreciate the compliment.

fred_h
Score: 0 | 4 weeks ago | 1 reply

Boulem Boashash did generalized gabor transforms... any relationship?

LaluSpeaker
Score: 0 | 3 weeks ago | no reply

Boulem Boashash was the Associate Editor of IEEE Trans Signal Processing who handled our original S-Transform paper in 1993. So I am assuming that the ST at that time (1993-1996) did not overlap with anything he had worked on. Boushash has worked extensively on time-frequency methods. It is possible that he has extended the fixed length Gabor Transform (which uses a Gaussian) by scaling. I did a cursory Google search 'Boulem Boashash Generalised Gabor Transform' but did not find a specific reference. It maybe in his book.

woodpecker
Score: 0 | 4 weeks ago | 1 reply

Hi Lalu,
Thanks for your interesting talk. I'm wondering if the S-Transform has applications in signal source separation ? - I'm thinking of audio/music.

LaluSpeaker
Score: 0 | 3 weeks ago | no reply

Thanks for listening to my presentation. I have not addressed this problem before. Off the cuff, the ST may be able to help in source separation, if the sound from a source arrives at a specific time, and has a specific spectral characteristic. Can be an interesting aspplication.

SlightlyChaotic
Score: 0 | 4 weeks ago | 2 replies

Hello, thank you for the presentation. The S-transform pictures starting at time 14:20 of the presentation look like a matlab spectrogram plot. As I listen to your presentation, I am thinking that the matlab spectrogram must be a type of time-frequency analysis. Is it similar to the S-transform? Do you know, would the only difference be the windowing function?

LaluSpeaker
Score: 0 | 3 weeks ago | no reply

I would like to add to my earlier comment. Yes, a spectrogram is the very earliest type of time-frequency analysis, using a STFT (Short Term Fourier Transform). It is fast, but does not resolve the lowest frequencies atr all.

LaluSpeaker
Score: 0 | 4 weeks ago | no reply

You are right. It is a Matlab spectogram, with a Hann window. Sorry the video stalled. I was just a few slides from showing the S-transform plot.

Steve_Hageman
Score: 0 | 3 weeks ago | 1 reply

Wonderful, thank you for the presentation.

LaluSpeaker
Score: 0 | 3 weeks ago | no reply

Thanks for your comments. Hope you managed to see the entire video.

Stephane.Boucher
Score: 0 | 4 weeks ago | 1 reply

This video is broken at the end (cut). There are about 15 minutes missing. Will try to fix this asap.

Stephane.Boucher
Score: 0 | 4 weeks ago | no reply

Fixed now. The video should play through the end.

fred_h
Score: 0 | 4 weeks ago | 1 reply

This is similar to proportional bandwidth spectrum analysis. BW is proportional to center frequency... Called constant Q filter banks... Optimal matched filter for detecting and estimating linear FM sweeps is BW proportion to sqrt center frequency. Once built 10 octave spectrum analyzer with 1000 proportional bands per octave. Ear is constant Q filter bank.

LaluSpeaker
Score: 0 | 4 weeks ago | no reply

You are right Fred. Viewing this problem as a filter banks concept also leads to very similar results. Wavelet Transform can (and is) treated as filter banks.

Arthur
Score: 0 | 4 weeks ago | no reply

S Transform of the Brain Fever Bird call was missing at the end

fred_h
Score: 0 | 4 weeks ago | no reply

the ppt contains the missing segment...

Leonard
Score: 0 | 4 weeks ago | no reply

Stephane, can you open the zoom early?

fred_h
Score: 0 | 4 weeks ago | no reply

missed the last few minutes.... darn

Leonard
Score: 0 | 4 weeks ago | no reply

seems like the video stopped

LaluSpeaker
Score: 0 | 4 weeks ago | no reply

Something has happened to the video

Rafa_Zamora
Score: 0 | 4 weeks ago | no reply

I think we missed the last part of the presentation. :(

Brewster
Score: 0 | 4 weeks ago | 1 reply

Is it just me or does it end too soon? For me it stopped at 46:43 kind of in the middle of a sentence. I tried reloading but got the same thing.

AlexeyS
Score: 0 | 4 weeks ago | no reply

The same for me. Seems the video was cut off during upload or something.

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