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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 timelags we analyze selfcorrelation with timeshift, frequencyshift, and chirprate. We call this generalization of autocorrelation the `autochirplet ambiguity function'. If, for example, the signal contains timeshifted versions of itself, modulated versions of itself, dilated versions of itself, timedependent frequencyshifted versions of itself, or frequencydependent timeshifted 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 twodimensional (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 coordinatetransformed (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 threeparameter ``timeshiftfrequencyshiftscaleshift'' subspace of (15) and the timefrequencyscale subspace of the chirplet transform. Extending this relation to the entire fiveparameter CCT would give us the autochirplet transform. This extension is one of our current research areas in the continued development of the chirplet theory.