# could anybody explains me why IIR filters do not have linear phase?

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 ?

• 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
• Not all FIR filters have linear phase. Most of the FIR filters you tend to design have linear phase by design. One example of a filter that is not necessarily linear phase is from Adaptive Filtering/Equalisation - there usually isn't a constraint to impose phase linearity. – David Mar 31 '20 at 1:56

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.

• 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 '15 at 22:51
• @endolith Isn’t a flat line still a line? – user28715 Nov 10 '17 at 23:06
• @StanleyPawlukiewicz The requirement specified is zero phase. Filters can be linear phase and not meet this requirement. – endolith Nov 11 '17 at 20:43

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.

• "The impulse response of a linear phase filter must be symmetric" ... though not necessarily symmetric about zero? – endolith Apr 22 '15 at 22:53
• Your question is more about if the signal needs to be even, since that is the sharpest form of symmetry. Regarding hotpaw2's comment I wonder by what logic it needs be symmetric? – Starhowl Apr 29 '17 at 7:43

Clements and Pease have shown that causal infinite-duration impulse responses can also have Fourier transforms with generalized linear phase. The corresponding system functions, however, are not rational, and thus, the systems cannot be implemented with difference equations.

So basically the IIR filter transfer function is $$H(z)=\pm H(1/z)$$. so if the poles are inside the unit circle,reciprocally poles are outside the unit circle so the system is unstable and non causal that's why linear phase is not displayd by IIR filter