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Please tell me. Also I dont know why phase is linear with FIR filters. I would like quantitative analysis. And why linear phase is not achieved by IIR filters ?

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Causal IIR filters cannot have linear phase, non-causal IIR filters can. As in $H(e^{j\omega})=e^{-j\omega\alpha}$ – Parsa Oct 29 '14 at 2:46

2 Answers 2

The impulse response of a linear phase filter must be symmetric. If the impulse response is infinitely long, then the center of the impulse is an infinite distance away from the beginning, giving the symmetric IIR filter infinite delay.

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"The impulse response of a linear phase filter must be symmetric" ... though not necessarily symmetric about zero? – endolith Apr 22 at 22:53

For digital filters, linear phase places the following requirement on the transfer function:

$$ H(z) = H(z^{-1}).$$

That restriction implies a linear phase IIR filter would need to have poles both inside and outside the unit circle, making it unstable. Similar arguments apply for analog filters.

That being said, there are any number of approximations which may be "close enough" to linear phase, depending on the application--especially if the causality of filter is sacrificed. For a review of techniques, see the introduction to this paper:

S.R. Powell, P.M.Chau, A Technique for Realizing Linear-Phase IIR Filters, IEEE Trans. Signal Processing, Vol 39, No 11, Nov 1991, pp 2425-2435.

The algorithm in that paper achieves linear phase with acausal block processing, rather than the usual offline "Forward-Backward" zero-phase approach.

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Shouldn't your first sentence say "zero phase", not "linear phase"? when "the causality of filter is sacrificed", you are converting zero-phase to delayed linear phase, no? – endolith Apr 22 at 22:51

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