In my application a filtering in the frequency space is desired. I designed two high-pass filters: $h_1$ Butterworth IIR filter, and $h_2$ FIR. I designed them using Python's SciPy functions. Using signal.freqz I obtained the frequency response $H_1(k)$ and $H_2(k)$.

I now want to filter signal in a real-time, buffer-wise. I read buffers of certain size, say $N=2048$, and do a DFT and obtain $F(k)$. I then want to multiply the Fourier coefficient with the two filters frequency response: $G_i(k) = F(K) H_i(k)$ for $i=1,2$. Then I do inverse DFT to get the filtered signal. (Remark: in practice I use overlap/add in the buffering, and so recovering back the filtered signal in time domain requires windowing and is more complicated.)

Now for the questions:

Multiplying $F(k)$ by $|H_1(k)|$ seems to work, even in real-time. But multiplying by the absolute value is like performing zero-phase filtering, which by theory is non-causal. However, since I apply it buffer-wise, this seems to be the loophole. Am I correct?

I also filtered using $F(k) H_2(K)$ and in Python it works just fine. However, giving the list of coefficient $\{ H_2(k) \in \mathbb{C} \mid k=0,...,N/2 \}$ for a colleague, for her to filter in another application/software, the filtering didn't seem to work. The filtered signal was a total mess (the unfiltered signal is a short speech by a woman). I verified with her that she applied the filter with the "mirror conjugate" completion so $\tilde{H}_2 = (H_2(k), \overline{H_2(N/2+1-k)}) \in \mathbb{C}^N$. Is there a theoretical reason for it not to work? (so we can narrow the reason to a bug in her code)

  • $\begingroup$ Are you sure that it is zero phase and not linear phase due to the delay of the buffer? $\endgroup$ – fibonatic Sep 2 at 16:13
  • $\begingroup$ Good question. I am not sure of the answer, but think that since we multiply in a positive real frequency response, the filtering introduces 0 phase. $\endgroup$ – Triceratops Sep 3 at 12:05

It's not a loophole. A causal linear phase filter is identical to a non-causal zero phase filter in series with a delay of half the filter length. If you check the results of your filter with an impulse input, you will probably see such a delay.

So it's not really real-time if you count the delay, but is often close enough to seem real-time if the filter's impulse response and/or block size is short relative to your real-time requirements.

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  • $\begingroup$ But why the zero-phase filtering $F(k)|H_1(k)|$ of the IIR works? $\endgroup$ – Triceratops Sep 7 at 14:50

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