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Reference.org
Advanced z-transform
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<h2 id="properties">Properties</h2> <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. </p> <h3>Linearity</h3> Z { ∑ k = 1 n c k f k ( t ) } = ∑ k = 1 n c k F k ( z , m ) . {\displaystyle {\mathcal {Z}}\left\{\sum _{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum _{k=1}^{n}c_{k}F_{k}(z,m).} <h3>Time shift</h3> Z { u ( t − n T ) f ( t − n T ) } = z − n F ( z , m ) . {\displaystyle {\mathcal {Z}}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).} <h3>Damping</h3> Z { f ( t ) e − a t } = e − a m F ( e a T z , m ) . {\displaystyle {\mathcal {Z}}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).} <h3>Time multiplication</h3> Z { t y f ( t ) } = ( − T z d d z + m ) y F ( z , m ) . {\displaystyle {\mathcal {Z}}\left\{t^{y}f(t)\right\}=\left(-Tz{\frac {d}{dz}}+m\right)^{y}F(z,m).} <h3>Final value theorem</h3> lim k → ∞ f ( k T + m ) = lim z → 1 ( 1 − z − 1 ) F ( z , m ) . {\displaystyle \lim _{k\to \infty }f(kT+m)=\lim _{z\to 1}(1-z^{-1})F(z,m).} <h2 id="example">Example</h2> <p>Consider the following example where f ( t ) = cos ( ω t ) {\displaystyle f(t)=\cos(\omega t)} : </p> F ( z , m ) = Z { cos ( ω ( k T + m ) ) } = Z { cos ( ω k T ) cos ( ω m ) − sin ( ω k T ) sin ( ω m ) } = cos ( ω m ) Z { cos ( ω k T ) } − sin ( ω m ) Z { sin ( ω k T ) } = cos ( ω m ) z ( z − cos ( ω T ) ) z 2 − 2 z cos ( ω T ) + 1 − sin ( ω m ) z sin ( ω T ) z 2 − 2 z cos ( ω T ) + 1 = z 2 cos ( ω m ) − z cos ( ω ( T − m ) ) z 2 − 2 z cos ( ω T ) + 1 . {\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}}} <p>If m = 0 {\displaystyle m=0} then F ( z , m ) {\displaystyle F(z,m)} reduces to the transform </p> F ( z , 0 ) = z 2 − z cos ( ω T ) z 2 − 2 z cos ( ω T ) + 1 , {\displaystyle F(z,0)={\frac {z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}},} <p>which is clearly just the <i>z</i>-transform of f ( t ) {\displaystyle f(t)} . </p> <ul><li><a href="/facts/Eliahu_Ibraham_Jury/swB1xcjw">Jury, Eliahu Ibraham</a> (1973). <i>Theory and Application of the z-Transform Method</i>. Krieger. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-88275-122-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/836240">836240</a>.</li></ul>