The problem:

Suppose you had a 6 second sample that consisted of a continuous 10 Hz and 13 Hz signal in addition to an 80 Hz signal that turned off and on briefly at one second and then off entirely at 3 seconds.

A Fourier transform of this signal would show three peaks at 10 Hz, 13 Hz and 80 Hz but give no indication of the duration of each part of the signal:

Potential improvement? 

• Collect a series of Fourier Transforms for short overlapping time intervals (a windowed Fourier transform).

• Plot the intensity of the FFT for each frequency and each time as a density plot (called a spectrogram).

But!

• If the window is too small you cannot resolve low frequencies well:
  

• If the window is too big you do not get good time resolution:

Solution:

• Use a continuously varying window size. Here a spectrogram of the same data made using the Morlet wavelet window:
  

Examples of the CWT:

• The author singing the notes "do", "re", "me":

• Screen shot of Chris Lang's C++ program of whistle spectrogram:
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• Brief sketch of how a CWT works (in pdf format)
  
• C++ code for a CWT
• Compiled CWT program
  
• sample data file for a test run
• C++ code that generated the sample data