SherbyNet - /page/wikibook/entry.mu

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <a href="Signal_processing" title="Signal processing">signal processing</a> and <a href="Control_theory" title="Control theory">control theory</a>, a <b>pole–zero plot</b> is a graphical representation of a <a href="Rational_function" title="Rational function">rational</a> <a href="Transfer_function" title="Transfer function">transfer function</a> in the <a href="Complex_plane" title="Complex plane">complex plane</a> which helps to convey certain properties of the system such as:

</p>

<ul><li><a href="BIBO_stability" title="BIBO stability">Stability</a></li>

<li><a href="Causal_system" title="Causal system">Causal system</a> / <a href="Anticausal_system" title="Anticausal system">anticausal system</a></li>

<li><a href="Radius_of_convergence" title="Radius of convergence">Region of convergence (ROC)</a></li>

<li><a href="Minimum_phase" title="Minimum phase">Minimum phase</a> / non minimum phase</li></ul>

<p>A pole-zero plot shows the location in the complex plane of the <a href="Zeros_and_poles" title="Zeros and poles">poles and zeros</a> of the <a href="Transfer_function" title="Transfer function">transfer function</a> of a <a href="Dynamic_system" class="mw-redirect" title="Dynamic system">dynamic system</a>, such as a controller, compensator, sensor, equalizer, <a href="Filter_(signal_processing)" title="Filter (signal processing)">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.

</p><p>A pole-zero plot is plotted in the plane of a complex frequency <a href="Domain_of_a_function" title="Domain of a function">domain</a>, which can represent either a continuous-time or a discrete-time system:

</p>

<ul><li>Continuous-time systems use the <a href="Laplace_transform" title="Laplace transform">Laplace transform</a> and are plotted in the <i>s-plane</i>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\sigma +j\omega }">

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<annotation encoding="application/x-tex">{\displaystyle s=\sigma +j\omega }</annotation>

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</math></span><img src="./52b9a99cfe16a2003294604462f41acc7863fe66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.763ex; height:2.509ex;" alt="{\displaystyle s=\sigma +j\omega }" loading="lazy"></span>

<ul><li>Real frequency components are along its <a href="Vertical_axis" class="mw-redirect" title="Vertical axis">vertical axis</a> (the imaginary line <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s{=}j\omega }">

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<annotation encoding="application/x-tex">{\displaystyle s{=}j\omega }</annotation>

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</math></span><img src="./31891e5759c023f2f7e028cbe081112f637d052d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle s{=}j\omega }" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma {=}0}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma {=}0}</annotation>

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</math></span><img src="./c0a87d283c892d25f5fe934f0aa5f8f7c834cd66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.3ex; height:2.176ex;" alt="{\displaystyle \sigma {=}0}" loading="lazy"></span>)</li></ul></li>

<li>Discrete-time systems use the <a href="Z-transform" title="Z-transform">Z-transform</a> and are plotted in the <i>z-plane</i>: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=Ae^{j\phi }}">

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<annotation encoding="application/x-tex">{\displaystyle z=Ae^{j\phi }}</annotation>

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</math></span><img src="./7ad17dbb9d8d095c0e953aa12f70409d8d0191d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.903ex; height:2.676ex;" alt="{\displaystyle z=Ae^{j\phi }}" loading="lazy"></span>

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

<div class="mw-heading mw-heading2"><h2 id="Continuous-time_systems">Continuous-time systems</h2></div>

<p>In general, a <a href="Rational_function" title="Rational function">rational</a> transfer function for a continuous-time <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system</a> has the form:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}}">

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<annotation encoding="application/x-tex">{\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}}}}</annotation>

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</p><p>where

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> are polynomials in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math></span><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"></span>,</li>

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> is the order of the numerator polynomial,</li>

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<annotation encoding="application/x-tex">{\displaystyle b_{m}}</annotation>

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</math></span><img src="./330e36691c8285b9c890d82a7c11855f3259dd58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.673ex; height:2.509ex;" alt="{\displaystyle b_{m}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span><sup>th</sup> coefficient of the numerator polynomial,</li>

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is the order of the denominator polynomial, and</li>

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<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>

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</math></span><img src="./790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span><sup>th</sup> coefficient of the denominator polynomial.</li></ul>

<p>Either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> or both may be zero, but in real systems, it should be the case that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\leq N}">

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<annotation encoding="application/x-tex">{\displaystyle M\leq N}</annotation>

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</math></span><img src="./e49d1bd89c5a14156a19cc822cd6837e74e06573.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.604ex; height:2.343ex;" alt="{\displaystyle M\leq N}" loading="lazy"></span>; otherwise the gain would be unbounded at high frequencies.

</p>

<div class="mw-heading mw-heading3"><h3 id="Poles_and_zeros">Poles and zeros</h3></div>

<ul><li>the zeros of the system are roots of the numerator polynomial: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\{\beta _{m}\mid m\in 1,\ldots M\}}">

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<li>the poles of the system are roots of the denominator polynomial: <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\{\alpha _{n}\mid n\in 1,\ldots N\}}">

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<annotation encoding="application/x-tex">{\displaystyle s=\{\alpha _{n}\mid n\in 1,\ldots N\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle A(s)|_{s=\alpha _{n}}=0.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Region_of_convergence">Region of convergence</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Laplace_transform#Region_of_convergence" title="Laplace transform">Laplace transform § Region of convergence</a></div>

<p>The <a href="Region_of_convergence" class="mw-redirect" title="Region of convergence">region of convergence</a> (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 <a href="Causal_system" title="Causal system">causal</a> or anti-causal.

</p>

<ul><li>If the ROC includes the <a href="Imaginary_axis" class="mw-redirect" title="Imaginary axis">imaginary axis</a>, then the system is <a href="BIBO_stability" title="BIBO stability">bounded-input, bounded-output (BIBO) stable</a>.</li>

<li>If the ROC extends rightward from the pole with the largest <a href="Complex_number#Definition" title="Complex number">real-part</a> (but not at infinity), then the system is causal.</li>

<li>If the ROC extends leftward from the pole with the smallest real-part (but not at negative infinity), then the system is anti-causal.</li></ul>

<p>The ROC is usually chosen to include the imaginary axis since it is important for most practical systems to have <a href="BIBO_stability" title="BIBO stability">BIBO stability</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {25}{s^{2}+6s+25}}}</annotation>

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</p><p>This system has no (finite) zeros and two poles:

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</p><p>The pole-zero plot would be:

</p><p><span class="mw-default-size" typeof="mw:File"></span>

</p><p>Notice that these two poles are <a href="Complex_conjugate" title="Complex conjugate">complex conjugates</a>, which is the necessary and sufficient condition to have real-valued coefficients in the differential equation representing the system.

</p>

<div class="mw-heading mw-heading2"><h2 id="Discrete-time_systems">Discrete-time systems</h2></div>

<p>In general, a rational transfer function for a discrete-time <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">LTI system</a> has the form:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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}}}}">

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<annotation encoding="application/x-tex">{\displaystyle 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}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle b_{m}}</annotation>

</semantics>

</math></span><img src="./330e36691c8285b9c890d82a7c11855f3259dd58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.673ex; height:2.509ex;" alt="{\displaystyle b_{m}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span><sup>th</sup> coefficient of the numerator polynomial,</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> is the order of the denominator polynomial, and</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation>

</semantics>

</math></span><img src="./790f9209748c2dca7ed7b81932c37c02af1dbc31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" loading="lazy"></span> is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span><sup>th</sup> coefficient of the denominator polynomial.</li></ul>

<p>Either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> or both may be zero.

</p>

<div class="mw-heading mw-heading3"><h3 id="Poles_and_zeros_2">Poles and zeros</h3></div>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=\beta _{m}}</annotation>

</semantics>

</math></span><img src="./b799d42c6d573ccdc7a9869975edbbe9343afcb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.177ex; height:2.509ex;" alt="{\displaystyle z=\beta _{m}}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(z)|_{z=\beta _{m}}=0}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(z)|_{z=\beta _{m}}=0}</annotation>

</semantics>

</math></span><img src="./b165c81e940363d0f241d103d343f631afcc8254.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.097ex; height:3.343ex;" alt="{\displaystyle P(z)|_{z=\beta _{m}}=0}" loading="lazy"></span> are the <a href="Zero_(complex_analysis)" class="mw-redirect" title="Zero (complex analysis)">zeros</a> of the system</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=\alpha _{n}}</annotation>

</semantics>

</math></span><img src="./9612b31bb59bd04db1e1c1977e057f6763f62f85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.893ex; height:2.009ex;" alt="{\displaystyle z=\alpha _{n}}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(z)|_{z=\alpha _{n}}=0}">

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(z)|_{z=\alpha _{n}}=0}</annotation>

</semantics>

</math></span><img src="./34555a9ea5327ef2ccc9bbe538b9b804f47c0c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.941ex; height:3.176ex;" alt="{\displaystyle Q(z)|_{z=\alpha _{n}}=0}" loading="lazy"></span> are the <a href="Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of the system.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Region_of_convergence_2">Region of convergence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Z-transform#Region_of_convergence" title="Z-transform">Z-transform § Region of convergence</a></div>

<p>The <a href="Region_of_convergence" class="mw-redirect" title="Region of convergence">region of convergence</a> (ROC) for a given discrete-time transfer function is a <a href="Disk_(mathematics)" title="Disk (mathematics)">disk</a> or <a href="Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a> 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 <a href="Causal_system" title="Causal system">causal</a> or anti-causal.

</p>

<ul><li>If the ROC includes the <a href="Unit_circle" title="Unit circle">unit circle</a>, then the system is <a href="BIBO_stability" title="BIBO stability">bounded-input, bounded-output (BIBO) stable</a>.</li>

<li>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.</li>

<li>If the ROC extends inward from the pole with the smallest (nonzero) magnitude, then the system is anti-causal.</li></ul>

<p>The ROC is usually chosen to include the unit circle since it is important for most practical systems to have <a href="BIBO_stability" title="BIBO stability">BIBO stability</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(z)}</annotation>

</semantics>

</math></span><img src="./2ace59286128b75d9170a45c011806a670f36469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.643ex; height:2.843ex;" alt="{\displaystyle P(z)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(z)}</annotation>

</semantics>

</math></span><img src="./60a5534f44e8f3f5ad54f80348d19a6252b36bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.736ex; height:2.843ex;" alt="{\displaystyle Q(z)}" loading="lazy"></span> are completely factored, their solution can be easily plotted in the <a href="Z-transform" title="Z-transform">z-plane</a>. For example, given the following transfer function:

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {z+2}{z^{2}+{\frac {1}{4}}}}}</annotation>

</semantics>

</math></span></span>

</p><p>The only (finite) zero is located at: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=-2}</annotation>

</semantics>

</math></span><img src="./4b453160b71ef2af97c4907904f3199c920678e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="{\displaystyle z=-2}" loading="lazy"></span>, and the two poles are located at: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\pm {\frac {j}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=\pm {\frac {j}{2}}}</annotation>

</semantics>

</math></span><img src="./8272fd6dbce2b3379e657530c6d1b2ac94d1a0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.993ex; height:5.343ex;" alt="{\displaystyle z=\pm {\frac {j}{2}}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

</p><p>The pole–zero plot would be:

</p><p><span class="mw-default-size" typeof="mw:File"></span>

</p>

<ul><li><a href="Root_locus" class="mw-redirect" title="Root locus">Root locus</a></li>

<li><a href="Laplace_transform" title="Laplace transform">Laplace transform</a></li>

<li><a href="Z-transform" title="Z-transform">Z-transform</a></li>

<li><a href="Rational_function" title="Rational function">Rational function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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</style><cite id="CITEREFHaag,_Michael2005" class="citation web cs1">Haag, Michael (June 22, 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180612141209/https://cnx.org/contents/3ktflSan@13/Understanding-PoleZero-Plots-o">"Understanding Pole/Zero Plots on the Z-Plane"</a>. <i><a href="OpenStax_CNX" title="OpenStax CNX">OpenStax CNX</a></i>. Archived from <a rel="nofollow" class="external text" href="https://cnx.org/contents/3ktflSan@13/Understanding-PoleZero-Plots-o">the original</a> on June 12, 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">June 9,</span> 2018</span>.</cite></li>

<li><cite id="CITEREFEric_W._Weisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Z-Transform.html">"Z-Transform"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">January 24,</span> 2010</span>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-08" href="https://en.wikipedia.org/wiki/?title=Pole%E2%80%93zero_plot&oldid=1289353342">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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