The log magnitude Fourier transform is given by
and the phase is
Note that both log-magnitude and (unwrapped) phase are parabolas in .
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'').
Chirplet Frequency-Rate Estimation
The chirp rate may be estimated from the relation as follows:
denote the measured (or known) curvature at the
midpoint of the analysis window
averages of the measured curvatures
along the log-magnitude and phase of a spectral peak,
- Then the chirp-rate
estimate may be estimated from the
spectral peak by
fs = 8000; x = chirp([0:1/fs:0.1],1000,1,2000); M = length(x); n=(-(M-1)/2:(M-1)/2)'; w = exp(-n.*n./(2*sigma.*sigma)); xw = w(:) .* x(:);
Figure 10.25 shows the same chirplet in a time-frequency plot. Figure 10.26 shows the spectrum of the example chirplet. Note the parabolic fits to dB magnitude and unwrapped phase. We see that phase modeling is most accurate where magnitude is substantial. If the signal were not truncated in the time domain, the parabolic fits would be perfect. Figure 10.27 shows the spectrum of a Gaussian-windowed chirp in which frequency decreases from 1 kHz to 500 Hz. Note how the curvature of the phase at the peak has changed sign.
Audio Filter Banks
Modulated Gaussian-Windowed Chirp