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: 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 == Continuous-time systems == In general, a rational transfer function for a continuous-time LTI system has the form: H ( s ) = B ( s ) A ( s ) = ∑ m = 0 M b m s m s N + ∑ n = 0 N − 1 a n s n = b 0 + b 1 s + b 2 s 2 + ⋯ + b M s M a 0 + a 1 s + a 2 s 2 + ⋯ + a ( N − 1 ) s ( N − 1 ) + s N {\displaystyle H(s)={\frac {B(s)}{A(s)}}={\displaystyle \sum _{m=0}^{M}{b_{m}s^{m}} \over s^{N}+\displaystyle \sum _{n=0}^{N-1}{a_{n}s^{n}}}={\frac {b_{0}+b_{1}s+b_{2}s^{2}+\cdots +b_{M}s^{M}}{a_{0}+a_{1}s+a_{2}s^{2}+\cdots +a_{(N-1)}s^{(N-1)}+s^{N}}}} where B {\displaystyle B} and A {\displaystyle A} are polynomials in s {\displaystyle s} , M {\displaystyle M} is the order of the numerator polynomial, b m {\displaystyle b_{m}} is the m {\displaystyle m} th coefficient of the numerator polynomial, N {\displaystyle N} is the order of the denominator polynomial, and a n {\displaystyle a_{n}} is the n {\displaystyle n} th coefficient of the denominator polynomial.