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# Pole–zero plot

In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as:

In general, a rational transfer function for a discrete LTI system has the form:

$X\left(z\right) = \left\{\left\{P\left(z\right)\right\}over\left\{Q\left(z\right)\right\}\right\}$

where

• $z_i$ such that $P\left(z_i\right) = 0$ are the zeros of the system
• $z_j$ such that $Q\left(z_j\right) = 0$ are the poles of the system

In the plot, the poles of the system are indicated by an x while the zeroes are indicated by an o.

## Example

If P(z) and Q(z) are completely factored, their solution can be easily plotted in the Z-Plane. For example, given the following transfer function:

$X\left(z\right) = \left\{\left\{\left(z+2\right)\right\}over\left\{\left(z^2+1/4\right)\right\}\right\}$

The only zero is located at: $-2$ The two poles are located at: $-\left\{\left\{i\right\}over\left\{2\right\}\right\}$, $+\left\{\left\{i\right\}over\left\{2\right\}\right\}$

The pole-zero plot would be:

## Interpretation

The region of convergence for a given transfer function is a disk or annulus which contains no poles.

• If the disc includes the unit circle, then the system is BIBO stable.
• If the region of convergence extends outward from the largest pole (not at infinity), then the system is right-sided.
• If the region of convergence extends inward from the smallest nonzero pole, then the system is left-sided.

It should be noted that the choice of ROC is not unique, however the ROC is usually chosen to include the unit circle since it is important for most practical systems to have Bounded Input, Bounded Output (BIBO) stability.

## Bibliography

• Haag, Michael. Understanding Pole/Zero Plots on the Z-Plane. Connexions. 22 June 2005
• Eric W. Weisstein. "Z-Transform." From MathWorld--A Wolfram Web Resource.
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