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## Time-Frequency Analysis of Modulated Signals

This example shows how to use the continuous wavelet transform (CWT) to analyze signals jointly in time and frequency.

Load a quadratic chirp signal and plot its spectrogram. The signal's frequency begins at approximately 500 Hz at t = 0, decreases to 100 Hz at t=2, and increases back to 500 Hz at t=4. The sampling frequency is 1 kHz.

fs = 1000;

Obtain a time-frequency plot of this signal using the CWT with a bump wavelet. The bump wavelet is a good choice for the CWT when your signals are oscillatory and you are more interested in time-frequency analysis than localization of transients.

The CWT clearly shows the time evolution of the quadratic chirp's frequency. The quadratic chirp is a frequency-modulated signal. While that signal is synthetic, frequency and amplitude modulation occur frequently in natural signals as well. Use the CWT to obtain a time-frequency analysis of an echolocation pulse emitted by a big brown bat (Eptesicus Fuscus). The sampling interval is 7 microseconds. Use the bump wavelet with 32 voices per octave. Thanks to Curtis Condon, Ken White, and Al Feng of the Beckman Center at the University of Illinois for the bat data and permission to use it in this example.

t = 0:DT:(numel(batsignal)*DT)-DT;
[cfs,f] = cwt(batsignal,'bump',1/DT,'VoicesPerOctave',32);
helperCWTTimeFreqPlot(cfs,t.*1e6,f./1e3,'surf','Bat Echolocation (CWT)',...
'Microseconds','kHz')

Obtain and plot the STFT of the bat data.

[S,F,T] = spectrogram(batsignal,50,48,128,1/DT);
helperCWTTimeFreqPlot(S,T.*1e6,F./1e3,'surf','Bat Echolocation (STFT)',...
'Microseconds','kHz')

For both the simulated and natural modulated signals, the CWT provides results similar to the STFT.

For the final example, obtain a time-frequency analysis of some seismograph data recorded during the 1995 Kobe earthquake. The data are seismograph (vertical acceleration, nm/sq.sec) measurements recorded at Tasmania University, Hobart, Australia on 16 January 1995 beginning at 20:56:51 (GMT) and continuing for 51 minutes at 1 second intervals. Use a bump wavelet.