But I would like to find such a FIR filter which delays each frequency by the time of one actual period -- i.e. so that 1000hz is delayed by 1 ms, 100Hz is delayed by 10 ms and so on.

• All sine waves regardless of amplitude, frequency and phase remain unchanged (altering phase by 360° doesn't do anything)
• The kind of filter we're talking about is a linear mapping of signals.
• Any finite-length signal can be represented as a linear combination of different sine waves.

Don't you realize that this filter actually does nothing at all?

Heh. Would delaying by a quarter-period be ok for your application?

It does nothing to the signal as long as the input is static. But pops and clicks can't come out unchanged because each frequency band is excited at a different time.

Quote from: Axon on 2008-07-12 19:07:25

Heh. Would delaying by a quarter-period be ok for your application?

Well, it's one of more possibilities. If you can suggest how I could do that, it will certainly help me understand the matter a lot better.

This is where you're wrong. It contradicts point #2 and #3 of the proof.

Look up Hilbert transforms.

In that case I probably wasn't exact enough in describing what filter I'm looking for. The effect I want to get is that if the input signal contains a short pure sine "click" at 500Hz starting at the position of 2 seconds, the output should contain the same click starting at 2.002 seconds (i.e. delayed by 1/500 of a second). And this amount of delay would be different depending on what frequencies would occur in the input.

Quote from: Axon on 2008-07-12 21:01:24

Look up Hilbert transforms.

Hilbert transforms shift the phase by 90 degrees but the delay is the same for the entire frequency range -- i.e. if the impulse itself were 2 seconds long, then everything would be delayed by 1 second. What I'm trying to find is a filter which makes different frequencies delayed by different amounts of time.

• creating many zero-delay band pass filters (try ten 1/4-octave-bands covering a spectrum from 750 to 24000 Hz for example)
• shifting their impulse responses according to the desired frequency-dependant delays (ie based on the band's center frequency)
• summing the results and checking the resulting impulse response

There's an intrinsic tradeoff between frequency resolution and time resolution here. If you evaluated this over wider bands, say 1/4 octave, your idea would totally make sense, and would be relatively easy to implement - just put together a filter bank and delay each band by a different amount on reconstruction. But of course you lose a certain degree of smoothness in the delays that way.

Petr, I got you the first time. I'm simply trying to explain to you that this specific frequency-dependant delay thingy you're talking about and the identity mapping are the same.. If you're not convinced try approximating your filter by

• creating many zero-delay band pass filters (try ten 1/4-octave-bands covering a spectrum from 750 to 24000 Hz for example)
• shifting their impulse responses according to the desired frequency-dependant delays (ie based on the band's center frequency)
• summing the results and checking the resulting impulse response

Compare the impulse response to the zero-delay band pass filter for 750-24000 Hz. If I'm right they should be similar.

edit: I just realized that this is close to what Axon wrote

Cheers,
SG

I think you've lost me.
If I understand your words well, that means that this method may make audible artefacts if I stretch the filter to, for example, 10 time the original size (i.e. delaying 20Hz by 500 ms).

If I understand you well, do you want to tell me it's actually impossible to make such a filter with the FIR method?

If I understand you well, do you want to tell me it's actually impossible to make such a filter with the FIR method?

It's perfectly possible and pretty easy to do as well because the filter you try to implement doesn't do anything.

Obviously, the filter WOULD shift the (non-periodic) signal along the time axis

function [y,x] = fddelay(radius)
   x = -floor(radius):radius;
   f2 = 1;
   y = x.*0;
   while (f2>0.01) %% cutoff frequency
      f1 = f2 * 0.95; %% narrow bands
      fcenter = (f1+f2)/2;
      delayinsamples = 2/fcenter;
      y = y + bandpass(x-delayinsamples,f1,f2);
      f2 = f1;
   end;
return;

function y = bandpass(x,f1,f2)
   y = lowpass(x,f2)-lowpass(x,f1);
return;

function y = lowpass(x,freq)
   y = sinc(x.*freq).*freq;
return

[y,x] = fddelay(999);
plot(x,y);

Quote from: Axon on 2008-07-12 21:01:24

Look up Hilbert transforms.

Hilbert transforms shift the phase by 90 degrees but the delay is the same for the entire frequency range -- i.e. if the impulse itself were 2 seconds long, then everything would be delayed by 1 second. What I'm trying to find is a filter which makes different frequencies delayed by different amounts of time.

What I'd like to get is something very similar only with the exception that the phases should stay preserved.

This seems to be exactly what you want - a filter with a frequency dependent group delay which delays different frequencies by different amounts of time. There are several methods for designing such allpass filters, many of them fairly conceptually simple.

No, a hilbert transformer shifts every frequency by 90 degrees. So it would turn y=sin(x)+sin(4*x) into y=cos(x)+cos(4*x). It's not just a simple quarter period delay.

Yeah, he wants varying group delay. He also wants H(z)=1 for |z|=1. Since the group delay is the derivative of the phase response "Theta" and "Theta" is constant zero for H(z)=1 these two requirements contradict each other.

Hilbert transforms shift the phase by 90 degrees but the delay is the same for the entire frequency range [...]

No, a hilbert transformer shifts every frequency by 90 degrees. So it would turn y=sin(x)+sin(4*x) into y=cos(x)+cos(4*x). It's not just a simple quarter period delay. [...]

I love this counter intuitive example.

One thing I don't understand: your Octave/MATLAB script uses log spacing.

If you switch to linear spacing, then you get a clean 4 sample delay. It doesn't matter how fine the spacing, that's what you get. What's happening?

   while (f2>0.01) %% cutoff frequency
      f1 = f2 - 0.001;