Phase of the chirp signal

Question

I'm having a tough time understanding the phase increments which are quadratic in nature in the chirp signal. When I see consecutive samples, what will I see in terms of phase and if I have to measure the phase difference between the two samples, what will I see ?

Answer 1

Take the chirp signal

$$ x(t) = e^{j\pi\alpha t^2} = e^{j\phi(t)} $$

Where $\alpha$ is the chirp rate of the signal. You can see that the function describing the phase, $\phi(t)$, is in the form of a quadratic equation. So you can plot

$$ \phi(t) = \pi\alpha t^2 $$

and see the quadratic nature of the phase at any point in time. The actual phase-change between each sample depends on what your sample rate is.

Answer 2

It may help to know that frequency is the derivative of phase. A change in phase versus a change in time is frequency by definition. A phase that keeps growing in the positive direction linearly represents a positive frequency. If you looked at this on the complex plane you would see a phasor rotating counter clockwise: constant magnitude and linearly increasing phase. If the phase rotates $2\pi$ it will have completed 1 cycle. Thus we get to cycles/sec and Hz. To complete this visual picture, a similar phasor that rotated clockwise would be a negative frequency: the phase is increasing in the negative direction with time. Add the two of them together and you get a result that stays on the real axis (so actually has a phase that is only 0 and $\pi$) which is a real cosine given by Euler's Identity:

$$cos(\theta) = \frac{e^{j\theta} + e^{-j\theta}}{2}$$

Note that $e^{j\theta}$ is the same thing as $1\angle \theta$, so when you have the expression $e^{j\omega t}$ you have the spinning phasor I described with a magnitude one and a phase that is growing linearly with time at rate $\omega$.

With that in mind hopefully it is much clearer to you now how the phase of a chirp signal relates to its frequency: Phase versus time is the integral of frequency versus time. If the frequency was constant (positive) with time, the phase would be increasing linearly with time. Similarly if you have a frequency ramp (frequency increasing linearly with time which is a chirp), the phase would be quadratically increasing with time.