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I measured that a 2-pole band pass filter, given imput audio of different frequencies, outputs an audio signal with different time delay.

At 440Hz, the time delay between of the input audio and the output audio is something in the order of (1/440)*200000 = 454 samples and at 1Khz it's about (1/1000)*200000 = 200 samples.

What is the theoretical correlation between filter peak frequency and the delay time of the output signal?

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  • $\begingroup$ It depends on the exact filter you're using. The quantity that is important here is the group delay, which measures the amount of delay that the filter has as a function of frequency. $\endgroup$ – Jason R May 4 '15 at 15:09
  • $\begingroup$ Thankyou. Is there a way to make an estimate? is the group delay a complex non linear function, is it an integer? it's a BP2 filter in the reaktor program, it's quite a confusing code. i should ask on the reaktor forum perhaps. $\endgroup$ – com.prehensible May 4 '15 at 15:22
  • $\begingroup$ You'll need to know the transfer function of the filter in order to estimate it. If you have the filter coefficients that are used, you can calculate it directly. $\endgroup$ – Jason R May 4 '15 at 15:55
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    $\begingroup$ @JasonR: Note that since the OP talks about sinusoidal input signals (at 440 Hz, and at 1kHz) it is not the group delay that is important but the phase delay. A sinusoid of (angular) frequency $\omega$ experiences a delay of $-\phi(\omega)/\omega$, where $\phi(\omega)$ is the phase of the system's complex frequency response. $\endgroup$ – Matt L. May 4 '15 at 17:26
  • $\begingroup$ A good rule of thumb is the delay is inversely proportional to the bandwidth. See a description of the rule of thumb here $\endgroup$ – MrMas Sep 4 '15 at 19:06
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The lower the frequency, the longer the period of oscillation. In any filter using memory (IIR or causal FIR), the filter will have to evaluate a signal for a longer period of time in order to gather information about the same number of periods of a lower frequency signal component than for a higher frequency signal component. And a filter can't filter a particular spectral component unless it has gathered enough information about that spectral component to "know" whether that particular component is in the stop-band or pass-band, which takes longer (other factors being roughly equivalent) for a lower frequency signal component.

Or you could design a linear phase FIR with the same delay for a lower frequency bandpass, but since it has gathered less periods of information about that lower frequency, it's quality of filtering (width of the transition band, etc.) will be less.

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