To see why the derivative of the angle function would be the instantaneous frequency, repe...

To see why the derivative of the angle function would be the instantaneous frequency, repeat the experiment of Section 3-7.2.

(a) Use the following parameters to define a chirp signal:

 

Determine α and β in (3.46) to define x(t) so that it sweeps the specified frequency range.

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(b) The rest of this problem is devoted to a MATLAB experiment that demonstrates why the derivative of the angle function is the “correct” definition of instantaneous frequency. First, make a plot of the instantaneous frequency fi(t) (in Hz) versus time.

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(c) Make a plot of the signal synthesized in (a). Pick a time-sampling interval that is small enough so that the plot is very smooth. Put this plot in the middle panel of a 3 × 1 by using subplot, subplot (3, 1, 2).

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(d) Generate a 4 Hz sinusoid, and plot it in the upper panel of a 3 × 1 by using subplot, subplot (3, 1, 1).

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(e) Generate an 8 Hz sinusoid, and then plot it in the lower panel of a 3 × 1 by using subplot, subplot (3, 1, 3).

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(f) Compare the three signals and comment on the apparent frequency of the chirp in the time range 1.6 ≤ t ≤ 2 s. Which sinusoid matches the chirp in this time region? Compare the expected fi(t) in this region with 4 Hz and 8 Hz.