Star transform
From Wikipedia, the free encyclopedia
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, it should be noted, 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:
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[edit] Definition
The star transform can be formally defined as such:
[edit] 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 \bigg[residues\ of\ X(\lambda)\frac{1}{1-e^{-T(s-\lambda)}}\bigg]_{at\ poles\ of\ E(\lambda)},](../../../math/2/0/6/20604a88de5e62ed218eca85953c7a42.png)
or,
Where 
is the radian sampling frequency such that 
[edit] Relation to Z transform
The star transform can be related to the Z transform, by making the following change of variables:
Note that in the Z-transform domain, the information in T is lost.
[edit] Properties of the star transform
Property 1. 
is periodic in 
with period 
.
Property 2. If 
has a pole at 
, then must have poles at 
where 
[edit] References
- Phillips and Nagle, "Digital Control System Analysis and Design", 4rth Edition, Prentice Hall, 1995. ISBN 0-13-309832-X
