Pole–zero plot

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Signal_processing/Npaja6zb">signal processing</a> and <a href="/facts/Control_theory/aIuERpn6">control theory</a>, a pole–zero plot is a graphical representation of a <a href="/facts/Rational_function/2nJGu0Ko">rational</a> <a href="/facts/Transfer_function/slDsh1lH">transfer function</a> in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> which helps to convey certain properties of the system such as:

<ul><li><a href="/facts/BIBO_stability/RxD8h2f5">Stability</a></li>
<li><a href="/facts/Causal_system/68Dch3YW">Causal system</a> / <a href="/facts/Anticausal_system/23uHUoS4">anticausal system</a></li>
<li><a href="/facts/Radius_of_convergence/4Un0ANfP">Region of convergence (ROC)</a></li>
<li><a href="/facts/Minimum_phase/a9WcqXXc">Minimum phase</a> / non minimum phase</li></ul>
A pole-zero plot shows the location in the complex plane of the <a href="/facts/Zeros_and_poles/gnA3U8VP">poles and zeros</a> of the <a href="/facts/Transfer_function/slDsh1lH">transfer function</a> of a <a href="/facts/Dynamic_system/782gREl5">dynamic system</a>, such as a controller, compensator, sensor, equalizer, <a href="/facts/Filter_(signal_processing)/vydJCNle">filter</a>, 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 <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>, which can represent either a continuous-time or a discrete-time system:

<ul><li>Continuous-time systems use the <a href="/facts/Laplace_transform/0Ge4MIc9">Laplace transform</a> and are plotted in the s-plane: 
 
 
 
 s
 =
 σ
 +
 j
 ω
 
 
 {\displaystyle s=\sigma +j\omega }

<ul><li>Real frequency components are along its <a href="/facts/Vertical_axis/lkTqvMmB">vertical axis</a> (the imaginary line 
 
 
 
 s
 
 =
 
 j
 ω
 
 
 {\displaystyle s{=}j\omega }
 
 where 
 
 
 
 σ
 
 =
 
 0
 
 
 {\displaystyle \sigma {=}0}
 
)</li></ul></li>
<li>Discrete-time systems use the <a href="/facts/Z-transform/ERNvDJqt">Z-transform</a> and are plotted in the z-plane: 
 
 
 
 z
 =
 A
 
 e
 
 j
 ϕ
 
 
 
 
 {\displaystyle z=Ae^{j\phi }}

<ul><li>Real frequency components are along its <a href="/facts/Unit_circle/VsTffjPK">unit circle</a></li></ul></li></ul>

Pole–zero plot open-in-new

Pole–zero plot