Kadomtsev–Petviashvili equation

The star transform, or starred transform is a discrete-time variation of the Laplace transform that represents an ideal sampler with period of time T. The star transform is similar to the Z transform with a simple change of variables, but the star transform explicitly identifies each sample in terms of the sampling period (T), while the Z transform only refers to each sample by integer index value.

The star transform is so named because it is frequently represented by an asterisk or "star" in the notation.

The inverse star transform represents a signal that has been sampled at interval T. The inverse star transform is not the original signal, x(t), but is instead a sampled version of the original signal. The following shows the relationship between the various representations:

$x(t)\to X^{*}(s)\to x^{*}(t)$

Definition

The star transform can be formally defined as such:

$X^{*}(s)=\sum _{k=0}^{\mathrm{\infty }}x(kT)e^{-kTs}$

Relation to Laplace transform

The star transform can be related to the Laplace transform, by taking the residues of the Laplace transform of a function, as such:

$X^{*}(s)=\sum [\text{residues of }X(\lambda )\frac{1}{1-e^{-T(s-\lambda )}}{]}_{\text{at poles of }X(\lambda )},$

or,

$X^{*}(s)=\frac{1}{T}\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}X(s+jm{\omega }_s)+\frac{x(0)}{2}$

Where ${\omega }_s$ is the radian sampling frequency such that ${\omega }_s=\frac{2\pi }{T}$

Relation to Z transform

The star transform can be related to the Z transform, by making the following change of variables:

$z=e^{Ts}$

Note that in the Z-transform domain, the information in T is lost.

Properties of the star transform

Property 1. $X^{*}(s)$ is periodic in $s$ with period $j{\omega }_s$.

$X^{*}(s+jm{\omega }_s)=X^{*}(s)$

Property 2. If $X(s)$ has a pole at $s=s_1$, then $X^{*}(s)$ must have poles at $s=s_1+jm{\omega }_s$ where $m=0,\pm 1,\pm 2,...$

References

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