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 [math]\displaystyle{ x(t) }[/math], which is transformed to a function [math]\displaystyle{ X^{*}(s) }[/math] in the following manner:

[math]\displaystyle{ \begin{align} X^{*}(s)=\mathcal{L}[x(t)\cdot \delta_T(t)]=\mathcal{L}[x^{*}(t)], \end{align} }[/math]

where [math]\displaystyle{ \delta_T(t) }[/math] 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 [math]\displaystyle{ x^{*}(t) }[/math], which is the output of an ideal sampler, whose input is a continuous function, [math]\displaystyle{ x(t) }[/math].

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

[math]\displaystyle{ \begin{align} x^*(t)\ \stackrel{\mathrm{def}}{=}\ x(t)\cdot \delta_T(t) &= x(t)\cdot \sum_{n=0}^\infty \delta(t-nT). \end{align} }[/math]

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

[math]\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}. }[/math]

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:

[math]\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]. }[/math]

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

[math]\displaystyle{ X^{*}(s) = \frac{1}{T}\sum_{k=-\infty}^\infty X\left(s-j\tfrac{2\pi}{T}k\right)+\frac{x(0)}{2}. }[/math]

Relation to Z transform

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

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

This substitution restores the dependence on T.

It's interchangeable,

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

[math]\displaystyle{ \bigg. X(z) = X^*(s)\bigg|_{\displaystyle s = \frac{\ln(z)}{T}} }[/math]  

Properties of the starred transform

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

[math]\displaystyle{ X^*(s+j\tfrac{2\pi}{T}k) = X^*(s) }[/math]

Property 2:  If [math]\displaystyle{ X(s) }[/math] has a pole at [math]\displaystyle{ s=s_1 }[/math], then [math]\displaystyle{ X^{*}(s) }[/math] must have poles at [math]\displaystyle{ s=s_1 + j\tfrac{2\pi}{T}k }[/math], where [math]\displaystyle{ \scriptstyle k=0,\pm 1,\pm 2,\ldots }[/math]

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