How to use a FIR on a signal with a drifting central frequency?
Without understanding more detailed results on what i(t) is, one solution is to estimate either the phase or frequency first, and then pass that result through a filter. 

An approach to do the frequency estimation digitally directly on the I (real) and Q (imaginary) samples of x(t) is the cross product frequency discriminator. After hard limiting x(t) to remove the AM components (which would result in samples on the unit circle in the complex plane), the frequency can be estimated using:

$$F \propto  I_1Q_2-I_2Q_1$$    

Where $I_1$,$Q_1$ and $I_2$,$Q_2$ represent successive samples. This is because the cross product is proportional to the phase between two vectors, so in this case the phase is the phase versus time, and therefore frequency. This is shown in the math and diagram below.

Frequency is the change in phase versus time ($d\phi/dt$). To determine the phase between two hard limited successive samples in time $V_1$ and $V_2$ given by:

$V_1 = e^{j\phi_1} = I_1+jQ_1$

$V_2 = e^{j\phi_2} = I_2+jQ_2$

For small angles, the phase $\phi_2-\phi_1$ can be reasonably approximated by the imaginary component of the conjugate product $(V^*_1) (V_2)$:

$V^*_1V_2 = e^{-j\phi_1}e^{j\phi_2} =e^{j(\phi_2-\phi_1)}$

For small angles $cos(\phi) \approx \phi$,

So therefore for small angles

$\phi_2-\phi_1 \approx Im(e^{-j\phi_1}e^{j\phi_2})$

$= Im((I_1-jQ_1)(I_2+jQ_2))$

$=I_1Q_2-I_2Q_1$





The above is extremely simple with low processing requirements useful for hardware implementation but depends on the small angle criteria for accurate angle estimation. This will work well as long as the sampling rate is high enough that the phase between samples doesn't deviate significantly. Alternatively if further processing is feasible, an ATAN2 operation can be instead be done on the two successive I,Q samples for full 4 quadrant accurate phase estimation (and therefore frequency given the phase difference between two successive samples over a much wider frequency range).

A simple delay and multiply frequency discriminator could also be used, followed by a  filtering of the resulting frequency versus time waveform. I have further detailed an example of using a simple delay and multiply frequency discriminator in this recent answer: Demodulation of FSK signal

Both approaches require hard limiting your waveform first to remove any amplitude variation of your signal. A differentiator as done through the differencing in these approaches is a high pass filter, the hard limiter removes all AM noise, and the post detection filtering will remove the noise induced "jumps" that you see in the result.