Time-Frequency Analysis
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Personal research web page for Kelly Fitz, Ph.D.
Time-frequency representations are used to analyze or characterize signals whose energy distribution varies in time and frequency. They map the one-dimensional time-domain signal into a two-dimensional function of time and frequency. A time-frequency representation describes the variation of spectral energy over time, much as a musical score describes the variation of musical pitch over time.
One of the best-known time-frequency representations is the spectrogram, defined as the squared magnitude of the short-time Fourier transform.
This figure shows the spectrogram of an acoustic bass tone. A very long analysis window is needed to resolve the harmonic components in the decay of the bass tone, which are spaced at approximately 73.4 Hz. Consequently, the onset of the tone is smeared considerably.
Spectrogram of the onset of a plucked acoustic bass tone at approximately 73 Hz.
The Method of Reassignment
As a time-frequency representation, the spectrogram has relatively poor resolution. Time and frequency resolution are constrained such that greater concentration in one domain is accompanied by greater smearing in the other.
The acoustic bass sound is difficult to analyze using the spectrogram because it requires high temporal resolution to capture the abrupt attack without smearing, and high frequency resolution to capture the individual harmonics. In fact, in order to capture the transient behavior of the attack, a window shorter than 3 ms, much shorter than a single period of the waveform (approximately 13.6 ms) is needed. Any window that achieves the desired temporal resolution will fail to resolve the harmonic components.
The method of time-frequency reassignment is used to correct the nominal time and frequency coordinates of the spectral data, and map it back onto the true regions of support of the analyzed signal.
The figure below shows the reassigned spectrogram for the same bass tone, computed using the same analysis window as the smeared spectrogram, above. The reassigned spectral data captures both the harmonic components in the decay and the transient behavior of the abrupt attack.
Reassigned spectrogram of the onset of a plucked acoustic bass tone at approximately 73 Hz.
This research is conducted in collaboration with Sean Fulop at the California State University at Fresno, and Lippold Haken from the Department of Electrical and Computer Engineering at the University of Illinois in Urbana-Champaign.
Download our Matlab scripts for computing reassigned spectra and spectrograms of sampled signals (last updated 23 April 2007).
Read more about the method of time-frequency reassignment.
An excellent and thorough review and analysis of the method of time-frequency reassignment by Stephen Hainsworth and Malcolm Macleod is available as Technical Report CUED/FINFENG /TR.459 from the Cambridge University Engineering Department, and may be downloaded from Citeseer.
François Auger, Patrick Flandrin, Olivier Lemoine, and Paulo Gonçalvès have developed the Time-Frequency Toolbox, an extensive collection of Matlab M-files for analyzing non-stationary signals using a variety of time-frequency representations. It includes an extensive tutorial document and user's guide, along with richly-documented M-files for signal generation and analysis and visualization. Visit the Time-Frequency Toolbox website to read more about this valuable resource.