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:
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 zeroes are indicated by a circle or O.
A pole-zero plot can represent either a continuous-time (CT) or a discrete-time (DT) system. For a CT system, the plane in which the poles and zeros appear is the s plane of the Laplace transform. In this context, the parameter s represents the complex angular frequency, which is the domain of the CT transfer function. For a DT system, the plane is the z plane, where z represents the domain of the Z-transform.
In general, a rational transfer function for a continuous-time LTI system has the form:
where
Either M or N or both may be zero, but in real systems, it should be the case that ; otherwise the gain would be unbounded at high frequencies.
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