# 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 , which is transformed to a function in the following manner:[1]

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, .

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:

Then per the convolution theorem, the starred transform is equivalent to the complex convolution of and , hence:[1]

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:

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:

## Relation to Z transform

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

This substitution restores the dependence on *T*.

It's interchangeable,

## Properties of the starred transform

**Property 1:** is periodic in with period

**Property 2:** If has a pole at , then must have poles at , where

## Citations

- 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. - Bech, p 9

## 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