Pole–zero plot

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