# Range of difference equation coefficients in practical FIR/IIR filters

In the design of practical FIR and IIR filters using difference equations, what is the range of coefficients that is employed for practical filters. How many orders might be used? What might the range of coefficients be?

IIR filters are typically implemented as cascaded second order sections. For the filter to be stable, the poles must be inside the unit circle, that means the denominator coefficient for each section are $$a_0 = 1$$, $$|a_1| < 2$$, and $$|a_2| < 1$$.

If the IIR filter is minimum phase, the same holds for the numerator coefficients as well, except for an overall gain, which can be all over the place.

IIR filter order varies but it's generally low. Orders higher than 20 are pretty rare.

For FIR filters I don't think there is anything "typical". A simple differentiator has an order of 2, room impulse response can have an order of 100s of thousands.

You can always scale any filter to any order of magnitude you want.

• Very kind of you to answer. The reason I asked is for the purpose of testing a system identification toolkit. What range of coefficients would provide a suitable set of inputs to test the toolkit against realistic filters? Jan 29, 2022 at 19:04

I can provide additional insights specific to FIR filters to add to Hilmar's good response for IIR filters (pun intended!):

For FIR Filter Designs below are some common estimators used as guidance for the number of taps needed that would provide the insight you seek. The main takeaway is that the tighter the transition band is, the longer number of coefficients are needed for a given rejection in the stopband (and ripple in the passband).

Note as demonstrated in this other post, the actual number of multiplications required can be significantly reduced through further operations (linear phase filters reduce the number by 2, half-band filters can reduce by another factor of 2 by putting zeros in every other coefficient) and polyphase filters can efficiently reduce operations when the passband is significantly smaller than the sampling rate.

• Thank you. See my comment on Hilmar's reply for the reason for my asking. Jan 29, 2022 at 19:05
• What is "reasonable" varies widely for any given application and often part of a higher tradespace with frequency planning of sampling rates with cost and resources. I don't think we could say that either 100 or 1000 is more reasonable than the other. What I suggest is testing with a wide range and indicating your assumptions together with your results. Jan 29, 2022 at 19:07