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(z) = \frac{P(z)}{Q(z)}

where

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

Contents

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(z) = \frac{z%2B2}{z^2%2B\frac{1}{4}}

The only zero is located at: -2, and the two poles are located at: \pm\frac{i}{2}.

The pole–zero plot would be:

Interpretation

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

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.

See also

Bibliography