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 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 and are polynomials in , is the order of the numerator polynomial, is the m-th coefficient of the numerator polynomial, is the order of the denominator polynomial, and is the n-th coefficient of the denominator polynomial. 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. the zeros of the system are roots of the numerator polynomial: such that the poles of the system are roots of the denominator polynomial: such that The region of convergence (ROC) for a given CT 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. If the ROC includes the imaginary axis, then the system is bounded-input, bounded-output (BIBO) stable.