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<title>Advanced z-transform</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Advanced z-transform</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Signal_processing" title="Signal processing">signal processing</a>, the <b>advanced z-transform</b> is an extension of the <a href="Z-transform" title="Z-transform">z-transform</a>, to incorporate ideal delays that are not multiples of the <a href="Sampling_rate" class="mw-redirect" title="Sampling rate">sampling time</a>. The advanced z-transform is widely applied, for example, to accurately model processing delays in <a href="Digital_control" title="Digital control">digital control</a>. It is also known as the <b>modified z-transform</b>.

</p><p>It takes the form

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<annotation encoding="application/x-tex">{\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}</annotation>

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</math></span><img src="./913589e3306b1580d16c4f2092eb498a494e0c54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.827ex; height:7.009ex;" alt="{\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}" loading="lazy"></span></dd></dl>

<p>where

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<ul><li><i>T</i> is the sampling period</li>

<li><i>m</i> (the "delay parameter") is a fraction of the sampling period <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 [0,T].}">

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<annotation encoding="application/x-tex">{\displaystyle [0,T].}</annotation>

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</math></span><img src="./d49a2b0474d5ee6d0e1967879a5489d3978f828c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle [0,T].}" loading="lazy"></span></li></ul>

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

<p>If the delay parameter, <i>m</i>, is considered fixed then all the properties of the z-transform hold for the advanced z-transform.

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

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F_{k}(z,m).}</annotation>

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</math></span><img src="./5c99602c0d5cd8f51d64851cc21fe54c43677cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.769ex; height:7.509ex;" alt="{\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F_{k}(z,m).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Time_shift">Time shift</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}</annotation>

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</math></span><img src="./ea69426441f5b29711d43ffd14c9df8e6c4d5d9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.189ex; height:3.009ex;" alt="{\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).}" loading="lazy"></span></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}</annotation>

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</math></span><img src="./bed78fc31b0f407559130b121b5a6c4826d0c8d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.733ex; height:3.343ex;" alt="{\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Time_multiplication">Time multiplication</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}</annotation>

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</math></span><img src="./d749b1bc7701279ad0cab37ecc90f91ad615ba5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.793ex; height:6.176ex;" alt="{\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Final_value_theorem">Final value theorem</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).}</annotation>

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</math></span><img src="./978d36f2cee234074a7c4ccba8c8c1e782fe7135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.274ex; height:4.343ex;" alt="{\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).}" loading="lazy"></span></dd></dl>

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

<p>Consider the following example 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 f(t)=\cos(\omega t)}">

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<annotation encoding="application/x-tex">{\displaystyle f(t)=\cos(\omega t)}</annotation>

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</math></span><img src="./94a59e2b7c92e8e86434586de811fa7c990bd486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.232ex; height:2.843ex;" alt="{\displaystyle f(t)=\cos(\omega t)}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(z,m)&={\mathcal {Z}}\left\{\cos \left(\omega \left(kT+m\right)\right)\right\}\\&={\mathcal {Z}}\left\{\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right\}\\&=\cos(\omega m){\mathcal {Z}}\left\{\cos(\omega kT)\right\}-\sin(\omega m){\mathcal {Z}}\left\{\sin(\omega kT)\right\}\\&=\cos(\omega m){\frac {z\left(z-\cos(\omega T)\right)}{z^{2}-2z\cos(\omega T)+1}}-\sin(\omega m){\frac {z\sin(\omega T)}{z^{2}-2z\cos(\omega T)+1}}\\&={\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}.\end{aligned}}}</annotation>

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</math></span><img src="./7b8f06d99bcc9e89defa9a1e9891edeb18548a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.499ex; margin-bottom: -0.172ex; width:71.889ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}F(z,m)&={\mathcal {Z}}\left\{\cos \left(\omega \left(kT+m\right)\right)\right\}\\&={\mathcal {Z}}\left\{\cos(\omega kT)\cos(\omega m)-\sin(\omega kT)\sin(\omega m)\right\}\\&=\cos(\omega m){\mathcal {Z}}\left\{\cos(\omega kT)\right\}-\sin(\omega m){\mathcal {Z}}\left\{\sin(\omega kT)\right\}\\&=\cos(\omega m){\frac {z\left(z-\cos(\omega T)\right)}{z^{2}-2z\cos(\omega T)+1}}-\sin(\omega m){\frac {z\sin(\omega T)}{z^{2}-2z\cos(\omega T)+1}}\\&={\frac {z^{2}\cos(\omega m)-z\cos(\omega (T-m))}{z^{2}-2z\cos(\omega T)+1}}.\end{aligned}}}" loading="lazy"></span></dd></dl>

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

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</math></span><img src="./e57f21007575fd03e3be0da20af34d25829cc9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}" loading="lazy"></span> then <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 F(z,m)}">

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

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</math></span><img src="./2bf655b8a9ac8320cd64e49648543ec9d7bafef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.713ex; height:2.843ex;" alt="{\displaystyle F(z,m)}" loading="lazy"></span> reduces to the transform

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<annotation encoding="application/x-tex">{\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}},}</annotation>

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</math></span><img src="./26ad638663b78516d9e3b02a83bdd36fcc11bb04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.044ex; height:6.676ex;" alt="{\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}},}" loading="lazy"></span></dd></dl>

<p>which is clearly just the <i>z</i>-transform of <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 f(t)}">

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

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</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span>.

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

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</style><cite id="CITEREFJury1973" class="citation book cs1"><a href="Eliahu_Ibraham_Jury" class="mw-redirect" title="Eliahu Ibraham Jury">Jury, Eliahu Ibraham</a> (1973). <i>Theory and Application of the z-Transform Method</i>. Krieger. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-88275-122-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/836240">836240</a>.</cite></li></ul>

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</style><div id="Digital_signal_processing96" style="font-size:114%;margin:0 4em"><a href="Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Detection_theory" title="Detection theory">Detection theory</a></li>

<li><a href="Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete signal</a></li>

<li><a href="Estimation_theory" title="Estimation theory">Estimation theory</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sub-fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li>

<li><a href="Digital_image_processing" title="Digital image processing">Digital image processing</a></li>

<li><a href="Speech_processing" title="Speech processing">Speech processing</a></li>

<li><a href="Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">Statistical signal processing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<ul>

<li><a href="Matched_Z-transform_method" title="Matched Z-transform method">Matched Z-transform method</a></li></ul></li>

<li><a href="Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li>

<li><a href="Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li>

<li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> (DCT)</li>

<li><a href="Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li>

<li><a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li>

<li><a href="Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

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

<li><a href="Post's_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li>

<li><a href="Starred_transform" title="Starred transform">Starred transform</a></li>

<li><a href="Zak_transform" title="Zak transform">Zak transform</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aliasing" title="Aliasing">Aliasing</a></li>

<li><a href="Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a></li>

<li><a href="Downsampling_(signal_processing)" title="Downsampling (signal processing)">Downsampling</a></li>

<li><a href="Nyquist_rate" title="Nyquist rate">Nyquist rate</a> / <a href="Nyquist_frequency" title="Nyquist frequency">frequency</a></li>

<li><a href="Oversampling" title="Oversampling">Oversampling</a></li>

<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li>

<li><a href="Sampling_rate" class="mw-redirect" title="Sampling rate">Sampling rate</a></li>

<li><a href="Undersampling" title="Undersampling">Undersampling</a></li>

<li><a href="Upsampling" title="Upsampling">Upsampling</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-09-01" href="https://en.wikipedia.org/wiki/?title=Advanced_z-transform&oldid=1243380059">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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