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Simulation Results

Figure 10.24: Real Gaussian-windowed chirp (time domain).

Figure 10.24 shows the waveform of 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);
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.

Figure 10.25: Real Gaussian-windowed chirp (spectrogram).

Figure 10.26: Gaussian-windowed chirp (frequency domain).

Figure 10.27: Downgoing chirp.

Figure 10.28: Short chirp--time waveform.
Figure 10.29: Short chirp--spectrum.

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Tightening the IFFTs
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Chirplet Frequency-Rate Estimation