# Abs(poles) < 1 by what margin for a stable filter?

I checked the literature for recent algorithms used to design a digital filter that is a minimax approximation of a desired frequency response. All the articles I found work out examples where all the poles have magnitude less than 0.92, or less that 0.89, etc. I havn't seen a published example with a pole having magnitude 0.95 and certainly not 0.9995. If a filter is implemented with double precision machine arithmetic, how close to the unit circle can a pole get to get a useful filter? Would it be a bad idea to have a pole with magnitude 0.95?

It would most likely depend upon the order of the filter, but having a pole at position $z_p$ where $|z_p| = 0.95$ shouldn't pose a stability problem if you're using double-precision floating-point arithmetic with a filter of reasonable length.

Since I don't know anything about your specific application, one other effect of the pole magnitude that might be relevant is the resulting filter's impulse response. Recall that poles in a system's transfer function correspond to decaying exponential terms in the system's impulse response. As Wikipedia notes:

$$a^nu[n] \Leftrightarrow \frac{1}{1-az^{-1}}$$

where $u[n]$ is the discrete unit step function, used here to express that the impulse response is causal ($0\ \forall\ n < 0$). Therefore, if your filter has a pole at $z = a$, then there will be a corresponding $a^n$ term in the filter's impulse response.

• For small $|a|$, this term will decay out in a relatively small number of samples. As $|a| \rightarrow 1$, the amount of time (measured in samples) required for the exponential function to decay increases.

• When you reach the "critically stable" point of $|a| = 1$, the exponential never decays and the system impulse response does not decay to zero.

• If $|a| > 1$ (i.e. the pole lies outside the unit circle), then the exponential function diverges and the impulse response blows up to infinity; this is why a discrete-time system is not BIBO stable if it contains poles outside the unit circle.

Given the above, the other concern that you might have is the overall effective time duration of the filter's impulse response. Although, as its name would suggest, an IIR filter theoretically has an impulse response of infinite length, in practice the response will usually decay to a negligible level after a finite time period. If your application is sensitive to this characteristic, then it makes sense to choose pole locations that are further from the unit circle. There will be corresponding tradeoffs in frequency-domain characteristics, as placing poles near the unit circle can help to make transition regions sharper and more narrow, as is often desirable.