Chirp Sinusoid

Now, let's make the phase quadratic, and thus non-linear with respect to time.

$$\theta(t) = 2\pi\mu t^2 + 2\pi f_0 t + \phi.$$

The instantaneous frequency (the derivative of the phase $\theta$), becomes

$$\omega_i(t) = \frac{d}{dt}\theta(t) = 4\pi\mu t+ 2\pi f_0,$$

which in Hz becomes

$$f_i(t) = 2\mu t + f_0.$$

Notice the frequency is no longer constant but changing linearly in time.

To create a sinusoid with frequency sweeping linearly from $f_1$ to $f_2$, consider the equation for a line $y = mx + b$ to obtain

$$f(t) = \frac{f_2-f_1}{2T}t + f_1,$$

where $T$ is the duration of the sweep.