# Filtering and Differentiating phase-modulated signals

For a project work, I need to demodulate data from a Laser Doppler Vibrometer. The distance information is phase-modulated, therefore the velocity information is frequency-modulated: $$x \propto \varphi \; \Leftrightarrow \; v = \dot{x} \propto \dot{\varphi} = \omega$$

After demodulation the (velocity) signal still is quite noisy, therefore low pass filtering is useful. I've tried filtering both in the distance domain (before differentiating) and in the velocity domain (after differentiating the distance signal). The results seem to be quite similar. What's the difference, exactly?

Specifically, I'd like to know:

1. What's the unit of the velocity/distance spectrum, exactly?
2. What is the relation between the (maximum) velocity and the filter's cutoff frequency? Could I specify a "cutoff velocity" instead?
3. What method is to prefer? Filtering in distance or velocity domain?
• Once you've demodulated to a distance measurement, anything you do will be linear and filterable --- in essence, it makes little difference whether you filter the distance signal or the velocity signal. You may want to consider using a(n Extended) Kalman filter to do joint demodulation and smoothing. A partial implementation of this idea is written up here. If I have time later today I'll try to write a proper answer. – Peter K. Jun 17 '16 at 14:41