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If a digital LTI filter has phase response on the form $arg[H(e^{jw})] = -\alpha\omega$, what could one say about it's causality (and consequently, the causality of it's impulse response)? Also is Linear Phase a characteristic of only Causal FIR Symmetric Systems?

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For a generic value of the group delay $\alpha$, the filter has a 2-sided infinite impulse, and is therefore not causal.

Causal linear phase systems are (almost) always FIR systems with any of the 4 types of symmetry.

In these cases, the group delay is multiple of 1/2 (0, 1/2, 1, 3/2, etc.).

You can have causal linear phase IIR filters, but these are not of finite-order (cannot be implemented with a difference equation), so they are of little use in practice.

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  • $\begingroup$ I'm not sure what you mean by causal linear phase IIR filters. If you have an infinitely long impulse response with the constraint $h(n)=0$ for $n<0$ (causality), and a symmetry constraint due to the linear phase requirement, then I'd say you can't have a causal IIR filter with exactly linear phase. However, you might mean something else. Could you elaborate? $\endgroup$ – Matt L. Jun 2 '14 at 7:15
  • $\begingroup$ If you have symmetric denominator you will have poles on or outside the unit circle. So while they are causal and linear phase, they will not be stable. What Juancho refers to might be some non-trivial structures where you run the data in blocks forward and backward and therefore ends up with linear phase (no good explanation, but it is quite non-trivial...). $\endgroup$ – Oscar Jun 2 '14 at 8:44
  • $\begingroup$ @Oscar: OK, I was actually implying stability because otherwise the systems are not useful. And the forward/backward trick is in fact non-causal. So I'm still not sure what Juancho meant. $\endgroup$ – Matt L. Jun 2 '14 at 14:34
  • $\begingroup$ @Matt L.: Sorry I missed the "'t" in "can't"... Hard to know the level of the questions asked sometimes... $\endgroup$ – Oscar Jun 2 '14 at 15:31
  • $\begingroup$ I would like to add that, while the classification into four categories is technically correct, for practical purpose there are filters that have an arbitrary group delay. A fractional sample shift of any of the mentioned types gives any desired group delay but at the same time makes the impulse response infinite. Machine precision truncation does still give a finite response and is practically indistinguishable from a real linear phase filter. $\endgroup$ – Jazzmaniac Jun 2 '14 at 21:23
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You cannot really tell whether the filter is causal.

However, you can tell whether the filter is definitely not causal (at least for FIR filters). We have

  • $\alpha < 0$: not causal (whether it is acausal or anti-causal is not known).
  • $\alpha \neq 0,1/2,1,3/2,2,\ldots$: acausal (but not anti-causal)
  • $\alpha = 0,1/2,1,3/2,2,\ldots$: unknown (can be causal or acausal)

The reason is (as Juancho) already pointed out, the symmetry in the impulse response for linear phase filters (which makes the group delay $\alpha$ equal to half of the length of the impulse response minus one ($(N-1)/2$).

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  • $\begingroup$ Just clarifying: (length-1)/2, i.e. order/2. (Rather than length/2 - 1, which the sentence also could be interpreted as.) $\endgroup$ – Oscar Jun 2 '14 at 8:48
  • $\begingroup$ @Oscar, thanks. I meant that, but I agree it is ambiguous. $\endgroup$ – Andreas H. Jun 3 '14 at 3:17

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