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:

• Stability
• Causal system / anticausal system
• Region of convergence (ROC)
• Minimum phase / non minimum phase

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: ${\displaystyle s=\sigma +j\omega }$
  • Real frequency components are along its vertical axis (the imaginary line ${\displaystyle s{=}j\omega }$ where ${\displaystyle \sigma {=}0}$)
• Discrete-time systems use the Z-transform and are plotted in the z-plane: ${\displaystyle z=Ae^{j\phi }}$
  • Real frequency components are along its unit circle