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