Starred transform

In applied mathematics, the starred transform, or star transform, converts a sequence of samples, x(nT), into a Laplace transform, X^*(s), whose inverse, x^*(t), is the product of a continuous function, x(t), and a Dirac comb function. Such a product is a convenient mathematical abstraction that represents the function of an ideal sampler. Thus, the starred transform can also be explained as the Laplace transform of the output of an ideal sampler.

The starred transform is also a de-normalized version of the one-sided Z-transform, because it restores the dependence on sampling parameter T. The starred transform is so-named because of the asterisk or "star" in the customary notation.

Definition

Given a continuous time function x(t) and its one sided Laplace transform:

$X(s)={\mathcal {L}}\{x(t)\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{0}^{\infty }{x(t)\cdot e^{-st}dt},$

a periodic summation of X(s) can be constructed from just a discrete set of samples of x(t), taken at multiples of a sampling interval, T. This function is known as the starred transform of the discrete sequence:

$X^{*}(s)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{n=0}^{\infty }x(nT)\cdot \underbrace {e^{-snT}} _{{\mathcal {L}}\{\delta (t-nT)\}}={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X\left(s-j{\tfrac {2\pi }{T}}k\right)+{\frac {x(0)}{2}}.$  

The inverse Laplace transform is:

${\begin{aligned}x^{*}(t)\ {\stackrel {\mathrm {def} }{=}}\ {\mathcal {L}}^{-1}\{X^{*}(s)\}&=\sum _{n=0}^{\infty }x(nT)\cdot \delta (t-nT)\\&=\sum _{n=0}^{\infty }x(t)\cdot \delta (t-nT),\\&=x(t)\cdot \sum _{n=0}^{\infty }\delta (t-nT).\end{aligned}}$

This mathematical abstraction is called an impulse sampled function.

Relation to Z transform

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

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

This substitution restores the dependence on T.

Properties of the star transform

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

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

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

References

• 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