Starred transform

In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace 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 $x(t)$, which is transformed to a function $X^{*}(s)$ in the following manner:

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

where ${\delta }_T(t)$ 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 $x^{*}(t)$, which is the output of an ideal sampler, whose input is a continuous function, $x(t)$.

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

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

$\begin{array}{rl}{x}^{*}(t)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}x(t)\cdot {\delta }_T(t)& =x(t)\cdot \sum _{n=0}^{\mathrm{\infty }}\delta (t-nT).\end{array}$

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

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

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:

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

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 $\frac{1}{1-e^{-T(s-p)}}$ in the right half-plane of p. The result of such an integration would be:

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

Relation to Z transform

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

$X^{*}(s)=X(z){|}_{z=e^{sT}}$  

This substitution restores the dependence on T.

It's interchangeable,

$X(z)=X^{*}(s){|}_{e^{sT}=z}$  

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

Properties of the starred transform

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

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

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

References

• Bech, Michael M.. "Digital Control Theory". AALBORG University. http://homes.et.aau.dk/mmb/DigitalControlTheory/WS3/WP3_2.pdf. 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