The STFT consists of a correlation of the signal with
constant-size portions of a wave, while the wavelet transform
consists of correlations with a
constant-Q family of functions. The two transforms,
however, are in some ways similar. Although the former is generally
thought of as a time-frequency method,
and the latter, a time-scale method, both attempt to localize the signal
in the time-frequency plane. In a rather loose sense, both the modulated
window of the STFT, and the
wavelet
of the wavelet transform, may be regarded as ``portions of waves''.
Chirplets, in a similar manner, may be
regarded as ``portions of chirps''.
We generally use complex-valued chirplets
to avoid the mirroring in the f=0 axis that results from using
only real-valued chirplets.
Figure 2 provides a comparison in terms of real and imaginary components as well as time-frequency distributions, between a wave, wavelet, chirp, and chirplet.
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY
In Fig. 3, we provide the same comparison with a 3-D particle-rendering, where the three coordinate axes are the function's real value, imaginary value, and time. Discrete samplings of four chirplets are shown: the top two have chirprate set to zero, and the leftmost two have an arbitrarily large window.
Figure: FIGURE GOES SOMEWHERE IN THIS GENERAL VICINITY