# Transfer function of a PLL Loop Filter that can support a linearly increasing (chirping) frequency

What is a transfer function for a the loop filter of a PLL that can track a sinusoid of the form:

$$x(t)=\cos\left(2\pi\left(\frac{1}{2}c_0t^2 + f_0t + \phi_0\right)\right)$$ where $c_0$ is the chirp (units of Hz/s), $f_0$ is the center frequency (units of Hz) and $\phi_0$ (units of cycles) is the initial phase?

I am currently using a loop filter followed by a direct digital synthesize (DDS). The loop filter I am currently using is a proportional plus integrator loop filter with transfer function $$F_{\text{loop}}(z) = k_1 + \frac{k_2}{1-z^{-1}}.$$ The DDS has a transfer function of $$F_{\text{DDS}}(z) = k_0\frac{z^{-1}}{1-z^{-1}}.$$

I would like to replace the loop filter with one that, combined with the DDS, can track the aforementioned sinusoid. The feedback loop is configured in the usual manner, namely, conjugate-multiplying the output of the DDS with the input sinusoid and then the output of that is fed into the loop filter, and, finally, the output of the loop filter is fed into the DDS. (Sorry I don't have a block diagram.)

This PLL works well with no noise present and $f_0 < 10$ and so, is clearly not very useful. I need something that can handle a phase offset, a phase ramp (nonzero frequency), and a frequency ramp (nonzero chirp) simultaneously, in addition to some noise.

NOTE: I have simplified this down to tracking a sinusoid. I'm actually tracking the phase of a modulated signal and so, rather than conjugate multiplying, I feed the match filter and the DDS outputs into a phase error detection function. However, for my purposes here, this simplified approach should suffice.

(I thought about posting this to the Electrical Engineering stack exchange, but since I'm working with a completely digital implementation, my question felt more natural here.)