Advanced z-transform

In mathematics and signal processing, the advanced z-transform is an extension of the z-transform, to incorporate ideal delays that are not multiples of the sampling time. It takes the form

F ( z , m ) = ∑ k = 0 ∞ f ( k T + m ) z − k {\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}}

where

  • T is the sampling period
  • m (the "delay parameter") is a fraction of the sampling period [ 0 , T ] . {\displaystyle [0,T].}

It is also known as the modified z-transform.

The advanced z-transform is widely applied, for example to accurately model processing delays in digital control.

Properties

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

Linearity

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).}

Time shift

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).}

Damping

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).}

Time multiplication

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).}

Final value theorem

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).}

Example

Consider the following example where f ( t ) = cos ⁡ ( ω t ) {\displaystyle f(t)=\cos(\omega t)} :

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}}}

If m = 0 {\displaystyle m=0} then F ( z , m ) {\displaystyle F(z,m)} reduces to the transform

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}},}

which is clearly just the z-transform of f ( t ) {\displaystyle f(t)} .

References

  • Jury, Eliahu Ibraham (1973). Theory and Application of the z-Transform Method. Krieger. ISBN 0-88275-122-0. OCLC 836240.
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