# FIR estimator for IIR system

Suppose that we have a dynamical system of which the impulse responses are infinite (IIR). Now I found methods on papers (http://dx.doi.org/10.1109/9.839942) estimating states or outputs of such a system with a FIR estimator. So this FIR estimator gives me approximately the same (state) output as my original system.

I find it hard to conceive how an estimator with only poles in $z_i=0$ is able to duplicate a system with poles $|z_i|<1$ (but not necessarily $z_i=0$). I think it has something to do with interpolation conditions, but I would really appreciate it if someone could make this clear to me. Thanks :)

I understand that for a stable dynamical system IIR will decay to zero and thus can be truncated to FIR. But my question remains how a FIR could 'mimic' the poles of an IIR?

• Could you give some references to the papers you mentioned? Also note that the impulse response of any stable IIR filter must be decaying sufficiently fast, so it's clear that it can be approximated arbitrarily closely by an FIR system with a sufficiently long impulse response. – Matt L. Jan 30 '16 at 16:37
• what-is-the-best-first-order-iir-approximation-to-a-moving-average-filter is the other way around, but has plaots and tables comparing the two. – denis Nov 20 '16 at 14:49

Simply truncating the impulse response of an IIR filter will give the optimal $\ell_2$ FIR approximation to the IIR system.

Whether this form of approximation is good enough to "mimic" the IIR system depends on what you mean by that.

Note that approximation of an impulse response by a FIR usually requires a lot more zeros and poles (at z = 0) than the nearly equivalent IIR. Enough extra degrees of freedom in the amount and placement of zeros might allow shaping a FIR response arbitrarily close to a stable IIR response (with more degrees of freedom in the placement of poles), at the cost of a lot more (often magnitudes more) zeros. So this "duplication" doesn't come for free.

This isn't a direct answer to your paraphrased question, "How can an FIR system mimic the poles of an IIR system", but this should give you an idea how well it can be done.

In short, download both the IIR and FIR filter programs from this site and design the same Inverse Chebyshev filter in both programs. Use the Edit | Copy and Edit | Paste commands to copy the response from the IIR into the FIR program and see how many FIR taps are required to "mimic" an IIR response. I think you will understand the point Hotpaw2 is making rather quickly.

If you don't want to download these programs, at least take a look at this web page to see how the tap count affects the duplication of an IIR filter response.

In short, the answer to your question is; if you don't have a limit on FIR tap count, you can emulate any IIR response with an FIR system. Or said a bit differently, if you have an infinite number of taps, you can easily emulate an infinitely long IIR response.