If you have a two-pole Butterworth bandpass filter described by a particular Q value (Q>0.5) and a resonant center frequency "f", is there any equation that will tell you the phase/time delay induced by this filter at "f"?
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Electrical – Calculating Phase/Time Delay induced by a 2-pole Bandpass filter? ie. Group Delay
band passdelaydistortionfilterphase
Best Answer
The formulas for phase (\$pd(\omega)\$) and group (\$gd(\omega)\$) delays of a H(s) transfer function are related to the phase:
$$\Phi(\omega)=\arctan{\frac{\Im\left(H(s)\right)}{\Re\left(H(s)\right)}}$$ $$pd(\omega)=-\frac{\Phi(\omega)}{\omega}$$ $$gd(\omega)=-\frac{\text{d}\Phi(\omega)}{\text{d}\omega}$$
However, you'll find that these don't really hold up to the reality, when the most certain way to determine them is to simply measure the relevant delay of input vs the output. Also, the direct measurement will most likely tell you the phase delay, while the group delay will need special considerations, in that you'll have to amplitude modulate your input signal with an envelope of the desired frequency, then measure the delay of the whole group compared to the input. Quite often, reality will show inconsitencies with the calculations.