# Can I determine a system's $z$-domain transfer function from its pole-zero plot?

Is it possible to generate the z-domain transfer function from a pole-zero plot diagram?

• Yes it is possible. – user2718 Apr 3 '13 at 17:53
• Is there a methodology that we have to follow? – 20317 Apr 3 '13 at 18:03

Without knowing specifics (ignoring a proportionality constant we'll call K as inidicated in the more compete answer above). Note that K does not affect the dynamic behavior of the system, so it may not be of particular interest.

it goes like this...

H(z) is a rational function of the form $$H(z) = \frac{(z-Z1)(z-Z2)...(z-Zm)}{(z-P1)(z-P2)...(z-Pn)}$$

where Z1,Z2...Zm are your zero values and P1,P2...Pn are your pole values.

I assume you know the exact values of you're poles and zeros. If you don't, you'll have to determine approximations from the PZP.

No, it is not possible to generate the $z$-domain transfer function uniquely and solely from the pole-zero plot. The reason is because you can only generate something like:

$$H(z) = K \frac{(z-Z_1)(z-Z_2)...(z-Z_m)}{(z-P_1)(z-P_2)...(z-P_n)}$$

from the pole-zero diagram, and there is nothing in that diagram to tell you what the gain term, $K$, is.