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Starred transform

${\displaystyle {\begin{aligned}X^{*}($s)={\mathcal {L}}[x(t)\cdot \delta _{T}(t)]={\mathcal {L}}[x^{*}(t)],\end{aligned}}}

where ${\displaystyle \delta _{T}($t)} is a Dirac comb function, with period of time T.

The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an impulse sampled function ${\displaystyle x^{*}($t)} , which is the output of an ideal sampler, whose input is a continuous function, ${\displaystyle x($t)} .

The starred transform is similar to the Z transform, with a simple change of variables, where the starred transform is explicitly declared in terms of the sampling period (T), while the Z transform is performed on a discrete signal and is independent of the sampling period. This makes the starred transform a de-normalized version of the one-sided Z-transform, as it restores the dependence on sampling parameter T.

Relation to Laplace transform[edit]

Since ${\displaystyle X^{*}($s)={\mathcal {L}}[x^{*}(t)]} , where:

${\displaystyle {\begin{aligned}x^{*}($t)\ {\stackrel {\mathrm {def} }{=}}\ x(t)\cdot \delta _{T}(t)&=x(t)\cdot \sum _{n=0}^{\infty }\delta (t-nT).\end{aligned}}}

Then per the convolution theorem, the starred transform is equivalent to the complex convolution of ${\displaystyle {\mathcal {L}}[x($t)]=X(s)} and ${\displaystyle {\mathcal {L}}[\delta _{T}($t)]={\frac {1}{1-e^{-Ts}}}} , hence:^[1]

${\displaystyle X^{*}($s)={\frac {1}{2\pi j}}\int _{c-j\infty }^{c+j\infty }{X(p)\cdot {\frac {1}{1-e^{-T(s-p)}}}\cdot dp}.}

This line integration is equivalent to integration in the positive sense along a closed contour formed by such a line and an infinite semicircle that encloses the poles of X(s) in the left half-plane of p. The result of such an integration (per the residue theorem) would be:

${\displaystyle X^{*}($s)=\sum _{\lambda ={\text{poles of }}X(s)}\operatorname {Res} \limits _{p=\lambda }{\bigg [}X(p){\frac {1}{1-e^{-T(s-p)}}}{\bigg ]}.}

Alternatively, the aforementioned line integration is equivalent to integration in the negative sense along a closed contour formed by such a line and an infinite semicircle that encloses the infinite poles of ${\displaystyle {\frac {1}{1-e^{-T(s-p)}}}}$ in the right half-plane of p. The result of such an integration would be:

${\displaystyle X^{*}($s)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X\left(s-j{\tfrac {2\pi }{T}}k\right)+{\frac {x(0)}{2}}.}

Relation to Z transform[edit]

Given a Z-transform, X(z), the corresponding starred transform is a simple substitution:

${\displaystyle {\bigg .}X^{*}($s)=X(z){\bigg |}_{\displaystyle z=e^{sT}}}  ^[2]

This substitution restores the dependence on T.

It's interchangeable,^{[citation needed]}

${\displaystyle {\bigg .}X($z)=X^{*}(s){\bigg |}_{\displaystyle e^{sT}=z}}  

${\displaystyle {\bigg .}X($z)=X^{*}(s){\bigg |}_{\displaystyle s={\frac {\ln(z)}{T}}}}  

Properties of the starred transform[edit]

Property 1:  ${\displaystyle X^{*}($s)} is periodic in ${\displaystyle s}$ with period ${\displaystyle j{\tfrac {2\pi }{T}}.}$

${\displaystyle X^{*}(s+j{\tfrac {2\pi }{T}}k)=X^{*}($s)}

Property 2:  If${\displaystyle X($s)} has a pole at ${\displaystyle s=s_{1}}$, then ${\displaystyle X^{*}($s)} must have poles at ${\displaystyle s=s_{1}+j{\tfrac {2\pi }{T}}k}$, where ${\displaystyle \scriptstyle k=0,\pm 1,\pm 2,\ldots }$

Citations[edit]

References[edit]

• Bech, Michael M. "Digital Control Theory" (PDF). AALBORG University. Retrieved 5 February 2014.
• Gopal, M. (March 1989). Digital Control Engineering. John Wiley & Sons. ISBN 0852263082.
• Phillips and Nagle, "Digital Control System Analysis and Design", 3rd Edition, Prentice Hall, 1995. ISBN 0-13-309832-X