# How to realize Poles and zeros at infinity??especially through transfer function?

I have a question regarding the poles and zeros at infinity I often read here in DSP SE and also in some textbooks about poles and zeros at infinity

This question also answers somehow (but not in much easy/simple way) Pole/Zero existence at infinity

I am confused about how to realize the poles and zeros at infinity especially from observing/looking at system transfer function??

It's actually quite straightforward: positive powers of $$s$$ (or, in discrete-time, $$z$$), correspond to poles at infinity. Negative powers give you zeros at infinity.

Let's look at some examples. In continuous time, an ideal differentiator has the transfer function

$$H(s)=s\tag{1}$$

Clearly, $$\lim_{s\to\infty}H(s)=\infty$$, hence you have a pole at infinity (and a zero at $$s=0$$).

The transfer function of an ideal integrator is

$$H(s)=\frac{1}{s}\tag{2}$$

Here we have $$\lim_{s\to\infty}H(s)=0$$, i.e., a zero at infinity (and a pole at $$s=0$$).

In discrete-time, a delay of one sample corresponds to

$$H(z)=z^{-1}\tag{3}$$

and you get $$\lim_{z\to\infty}H(z)=0$$.

A (non-causal) advance by one sample is represented by the transfer function

$$H(z)=z\tag{4}$$

In this case you have a pole at infinity: $$\lim_{z\to\infty}H(z)=\infty$$

Note that in continuous time as well as in discrete time, systems with poles at infinity cannot be causal and stable.

• what do you mean by last line note??system with poles at infinity cannot be stable??can't be even marginally stable??as in the case of differentiator?? – abtj Sep 29 '19 at 11:12
• @abtj: Exactly that, they can't be causal and stable. An ideal differentiator is unstable. – Matt L. Sep 29 '19 at 11:24
• @MattL. can the systems with poles at infinity be marginally stable?? – engr Sep 13 at 17:35
• @engr: No, marginal stability means that there are single poles on the stability border, which in continuous time is the imaginary axis, while in discrete time it is the unit circle. – Matt L. Sep 13 at 18:55