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.

Thu Jan 8 19:50:27 EST 1998