A pole-zero plot shows the location in the complex plane of the poles and zeros of the transfer function of a dynamic system, such as a controller, compensator, sensor, equalizer, filter, or communications channel. By convention, the poles of the system are indicated in the plot by an X while the zeros are indicated by a circle or O.
A pole-zero plot is plotted in the plane of a complex frequency domain, which can represent either a continuous-time or a discrete-time system:
Continuous-time systems use the Laplace transform and are plotted in the s-plane:
Real frequency components are along its vertical axis (the imaginary line where )
Discrete-time systems use the Z-transform and are plotted in the z-plane:
Real frequency components are along its unit circle
The region of convergence (ROC) for a given continuous-time transfer function is a half-plane or vertical strip, either of which contains no poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is causal or anti-causal.
This system has no (finite) zeros and two poles:
and
The pole-zero plot would be:
Notice that these two poles are complex conjugates, which is the necessary and sufficient condition to have real-valued coefficients in the differential equation representing the system.
The region of convergence (ROC) for a given discrete-time transfer function is a diskorannulus which contains no uncancelled poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is causal or anti-causal.
If the ROC extends outward from the pole with the largest (but not infinite) magnitude, then the system has a right-sided impulse response. If the ROC extends outward from the pole with the largest magnitude and there is no pole at infinity, then the system is causal.
If the ROC extends inward from the pole with the smallest (nonzero) magnitude, then the system is anti-causal.
The ROC is usually chosen to include the unit circle since it is important for most practical systems to have BIBO stability.