A chirp signal is one that sweeps in frequency from ω1 = 2πf1 to ω2 = 2πf2 as time goes fr...

A chirp signal is one that sweeps in frequency from ω1 = 2πf1 to ω2 = 2πf2 as time goes from t = 0 to t = T2. The general formula for a chirp is

where

The derivative of ψ(t) is the instantaneous frequency (in rad/s), which is also the frequency heard if the frequencies are in the audible range.

(a) For the chirp in (3.46), determine formulas for the beginning frequency (ω1) and the ending frequency (ω2) in terms α, β, and T2.

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(b) For the chirp signal

 

derive a formula for the instantaneous frequency versus time.

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(c) Make a plot of the instantaneous frequency (in Hz) versus time over the range 0 ≤ t ≤ 1 s.