Simulation Results

Figure 9.20 shows the time waveform for a Gaussian-windowed chirp (``chirplet'') generated by the following matlab code:

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 9.21 shows the same chirplet in a time-frequency plot. Figure 9.22 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 9.23 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.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.