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Next: CHIRPLET TRANSFORM SUBSPACES Up: THE CHIRPLET Previous: The pyramidal (multi-resolution) CCT

Autochirplet and Cross Chirplet Transforms

If, in (11) we choose the mother chirplet to be the signal itself:

S_t_c,f_c,(_t),c,d = C_t_c,f_c,(_t),c,d s(t) | s(t)  

then we have a generalization of the autocorrelation function, where, instead of only analyzing time-lags we analyze self-correlation with time-shift, frequency-shift, and chirprate. We call this generalization of autocorrelation the `autochirplet ambiguity function'. If, for example, the signal contains time-shifted versions of itself, modulated versions of itself, dilated versions of itself, time-dependent frequency-shifted versions of itself, or frequency-dependent time-shifted versions of itself, then this structure will become evident when examining the `autochirplet ambiguity function'. The `autochirplet ambiguity function' is not new, but, rather, was proposed by Berthonberthon as a generalization of the radar ambiguity function. Note that the radar ambiguity functionwoodward1,skolnik is a special case of (15).

It is well known that the power spectrum is the Fourier transform of the autocorrelation function, and that the Wigner distribution is the two-dimensional (rotated) Fourier transform of the radar ambiguity function. Recent work has also shown that there is a connection between the wideband ambiguity function and an appropriately coordinate-transformed (to a logarithmic frequency axis) version of the Wigner distributionbertrand, where the connection is based on the Mellin transform. This connection gives us a link between the three-parameter ``time-shift--frequency-shift--scale-shift'' subspace of (15) and the time-frequency-scale subspace of the chirplet transform. Extending this relation to the entire five-parameter CCT would give us the autochirplet transform. This extension is one of our current research areas in the continued development of the chirplet theory.

Steve Mann
Thu Jan 8 19:50:27 EST 1998