Inapplied mathematics, thestarred transform, orstar transform, is a discrete-time variation of theLaplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. The transform is an operator of a continuous-time function${\displaystyle x(t)}$, which is transformed to a function${\displaystyle X^{\ast }(s)}$ in the following manner:^[1]

${\displaystyle \begin{array}{r}{X}^{\ast }(s)=\mathcal{L}[x(t)\cdot {\delta }_T(t)]=\mathcal{L}[x^{\ast }(t)],\end{array}}$

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

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

The starred transform is similar to theZ 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 ade-normalized version of the one-sidedZ-transform, as it restores the dependence on sampling parameter T.

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

Then per theconvolution 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]

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

Thisline 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 ofp. The result of such an integration (per theresidue theorem) would be:

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

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 ofp. The result of such an integration would be:

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

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

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

This substitution restores the dependence onT.

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

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

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

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

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

Property 2:  If${\displaystyle X(s)}$ has a pole at${\displaystyle s=s_1}$, then${\displaystyle X^{\ast }(s)}$ must have poles at${\displaystyle s=s_1+j\frac{2\pi }{T}k}$, where${\displaystyle k=0,\pm 1,\pm 2,\dots }$

• Bech, Michael M."Digital Control Theory"(PDF). AALBORG University. Retrieved5 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

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