Consider again the Fourier transform of a complex Gaussian in (10.27):

(11.33) |

Setting gives

(11.34) |

The

(11.35) |

and the phase is

(11.36) |

Note that

In practice, it is simple to estimate the *curvature* at a
spectral peak using *parabolic interpolation*:

We can write

Note that the window ``amplitude-rate''
is always positive.
The ``chirp rate''
may be positive (increasing frequency) or
negative (downgoing chirps). For purposes of chirp-rate estimation,
there is no need to find the true spectral peak because the curvature
is the *same* for all
. However, curvature estimates are
generally more reliable near spectral peaks, where the signal-to-noise
ratio is typically maximum.
In practice, we can form an estimate of
from the known FFT
analysis window (typically ``close to Gaussian'').

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University