View source for Pole–zero plot

{{Short description|Diagram showing the singularities of a given control system's transfer function}}
In [[mathematics]], [[signal processing]] and [[control theory]], a '''pole–zero plot''' is a graphical representation of a [[rational function|rational]] [[transfer function]] in the [[complex plane]] which helps to convey certain properties of the system such as:

* [[BIBO stability|Stability]]
* [[Causal system]] / [[anticausal system]]
* [[Radius of convergence|Region of convergence (ROC)]]
* [[Minimum phase]] / non minimum phase

A pole-zero plot shows the location in the complex plane of the [[zeros and poles|poles and zeros]] of the [[transfer function]] of a [[dynamic system]], such as a controller, compensator, sensor, equalizer, [[Filter (signal processing)|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 of a function|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'': <math>s = \sigma + j \omega</math>
** Real frequency components are along its [[vertical axis]] (the imaginary line <math>s {=} j \omega</math> where <math>\sigma{=}0</math>)
* Discrete-time systems use the [[Z-transform]] and are plotted in the ''z-plane'': <math>z = A e^{j\phi}</math>
** Real frequency components are along its [[unit circle]]

== Continuous-time systems ==
In general, a [[rational function|rational]] transfer function for a continuous-time [[LTI system theory|LTI system]] has the form:

<math display="block">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}</math>

where
* <math>B</math> and <math>A</math> are polynomials in <math>s</math>,
* <math>M</math> is the order of the numerator polynomial,
* <math>b_m</math> is the <math>m</math>{{Sup|th}} coefficient of the numerator polynomial,
* <math>N</math> is the order of the denominator polynomial, and
* <math>a_n</math> is the <math>n</math>{{Sup|th}} coefficient of the denominator polynomial.

Either <math>M</math> or <math>N</math> or both may be zero, but in real systems, it should be the case that <math>M \le N</math>; otherwise the gain would be unbounded at high frequencies.

===Poles and zeros===

* the zeros of the system are roots of the numerator polynomial: <math display="block">s = \{ \beta_m \mid m \in 1, \ldots M \}</math> such that <math display="block">B(s)|_{s = \beta_m} = 0 </math>
* the poles of the system are roots of the denominator polynomial: <math display="block">s = \{ \alpha_n \mid n \in 1, \ldots N \}</math> such that <math display="block">A(s)|_{s = \alpha_n} = 0.</math>

===Region of convergence===
{{See also|Laplace transform#Region of convergence}}
The [[region of convergence]] (ROC) for a given continuous-time 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 system|causal]] or anti-causal.

* If the ROC includes the [[imaginary axis]], then the system is [[BIBO stability|bounded-input, bounded-output (BIBO) stable]].
* If the ROC extends rightward from the pole with the largest [[complex number#Definition|real-part]] (but not at infinity), then the system is causal.
* If the ROC extends leftward from the pole with the smallest real-part (but not at negative infinity), then the system is anti-causal.

The ROC is usually chosen to include the imaginary axis since it is important for most practical systems to have [[BIBO stability]].

===Example===

<math display="block">H(s) = \frac{ 25 }{ s^2 + 6s + 25 } </math>

This system has no (finite) zeros and two poles:
<math display="block"> s = \alpha_1 = -3 + 4j </math>
and
<math display="block"> s = \alpha_2 = -3 - 4j </math>

The pole-zero plot would be:

[[File:Pole-zero plot.png|P-z plot of previously created example]]

Notice that these two poles are [[complex conjugate]]s, which is the necessary and sufficient condition to have real-valued coefficients in the differential equation representing the system.

== Discrete-time systems ==
In general, a rational transfer function for a discrete-time [[LTI system theory|LTI system]] has the form:

<math display="block">
H(z) = \frac{P(z)}{Q(z)} = \frac
{    \displaystyle\sum_{m=0}^M {b_m z^{-m}}}
{1 + \displaystyle\sum_{n=1}^N {a_n z^{-n}}}
= \frac
{b_0 + b_1 z^{-1} + b_2 z^{-2} \cdots + b_M z^{-M}}
{1   + a_1 z^{-1} + a_2 z^{-2} \cdots + a_N z^{-N}}
</math>

where
* <math>M</math> is the order of the numerator polynomial,
* <math>b_m</math> is the <math>m</math>{{Sup|th}} coefficient of the numerator polynomial,
* <math>N</math> is the order of the denominator polynomial, and
* <math>a_n</math> is the <math>n</math>{{Sup|th}} coefficient of the denominator polynomial.

Either <math>M</math> or <math>N</math> or both may be zero.

===Poles and zeros===
* <math>z = \beta_m</math> such that <math>P(z)|_{z = \beta_m} = 0 </math> are the [[zero (complex analysis)|zero]]s of the system
* <math>z = \alpha_n</math> such that <math>Q(z)|_{z = \alpha_n} = 0</math> are the [[pole (complex analysis)|pole]]s of the system.

===Region of convergence===
{{See also|Z-transform#Region of convergence}}
The [[region of convergence]] (ROC) for a given discrete-time transfer function is a [[Disk (mathematics)|disk]] or [[Annulus (mathematics)|annulus]] which contains no uncancelled poles. In general, the ROC is not unique, and the particular ROC in any given case depends on whether the system is [[causal system|causal]] or anti-causal.

* If the ROC includes the [[unit circle]], then the system is [[BIBO stability|bounded-input, bounded-output (BIBO) stable]].
* If the ROC extends outward from the pole with the largest (but not infinite) magnitude, then the system has a right-sided impulse response. If the ROC extends outward from the pole with the largest magnitude and there is no pole at infinity, then the system is causal.
* If the ROC extends inward from the pole with the smallest (nonzero) magnitude, then the system is anti-causal.

The ROC is usually chosen to include the unit circle since it is important for most practical systems to have [[BIBO stability]].

===Example===
If <math>P(z)</math> and <math>Q(z)</math> are completely factored, their solution can be easily plotted in the [[Z-transform|z-plane]]. For example, given the following transfer function:
<math display="block">H(z) = \frac{z+2}{z^2+\frac{1}{4}}</math>

The only (finite) zero is located at: <math>z = -2</math>, and the two poles are located at: <math>z = \pm\frac{j}{2}</math>, where <math>j</math> is the [[imaginary unit]].

The pole–zero plot would be:

[[Image:PoleZeroPlot.png]]

==See also==
*[[Root locus]]
*[[Laplace transform]]
* [[Z-transform]]
* [[Rational function]]

==Bibliography==
* {{cite web |url=https://cnx.org/contents/3ktflSan@13/Understanding-PoleZero-Plots-o |title=Understanding Pole/Zero Plots on the Z-Plane |author=Haag, Michael |date=June 22, 2005 |work=[[OpenStax CNX]] |accessdate=June 9, 2018}}{{Dead link|date=January 2023}}
* {{cite web |url=http://mathworld.wolfram.com/Z-Transform.html |title=Z-Transform |author=Eric W. Weisstein |author-link=Eric W. Weisstein |work=[[MathWorld]] |accessdate=January 24, 2010}}

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[[Category:Signal processing]]