Star 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 in the following manner:Jury, Eliahu I. ''Analysis and Synthesis of Sampled-Data Control Systems''., Transactions of the American Institute of Electrical Engineers- Part I: Communication and Electronics, 73.4, 1954, p. 332-346.
:
\begin
X^(s)=\mathcal (t)\cdot \delta_T(t)\mathcal ^(t)
\end

where 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 , 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 , where:
:$\begin x^*(t)\ \stackrel\ x(t)\cdot \delta_T(t) &= x(t)\cdot \sum_^\infty \delta(t-nT). \end$

Then per the convolution theorem, the starred transform is equivalent to the complex convolution of and , hence:

:$X^(s) = \frac \int_^.$

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_\operatorname\limits_\bigg (p)\frac\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 in the right half-plane of ''p''. The result of such an integration would be:
:$X^(s) = \frac\sum_^\infty X\left(s-j\tfrack\right)+\frac.$

Relation to Z transform

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

:$\bigg. X^*(s) = X(z)\bigg, _$  Bech, p 9

This substitution restores the dependence on ''T''.

It's interchangeable,
:$\bigg. X(z) = X^*(s)\bigg, _$  
:$\bigg. X(z) = X^*(s)\bigg, _$  

Properties of the starred transform

Property 1:  $X^*(s)$ is periodic in $s$ with period $j\tfrac.$

:$X^*(s+j\tfrack) = X^*(s)$

Property 2:  If has a pole at $s=s_1$, then must have poles at $s=s_1 + j\tfrack$, where $\scriptstyle k=0,\pm 1,\pm 2,\ldots$

Citations

References

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*Phillips and Nagle, "Digital Control System Analysis and Design", 3rd Edition, Prentice Hall, 1995. {{ISBN, 0-13-309832-X

Transforms