Harman Patil (Editor)

Starred transform

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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 X ∗ ( s ) in the following manner:

Contents

X ∗ ( s ) = L [ x ( t ) ⋅ δ T ( t ) ] = L [ x ∗ ( t ) ] ,

where δ T ( t ) 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 x ∗ ( t ) , 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 X ∗ ( s ) = L [ x ∗ ( t ) ] , where:

x ∗ ( t )   = d e f   x ( t ) ⋅ δ T ( t ) = x ( t ) ⋅ ∑ n = 0 ∞ δ ( t − n T ) .

Then per the convolution theorem, the starred transform is equivalent to the complex convolution of L [ x ( t ) ] = X ( s ) and L [ δ T ( t ) ] = 1 1 − e − T s , hence:

X ∗ ( s ) = 1 2 π j ∫ c − j ∞ c + j ∞ X ( p ) ⋅ 1 1 − e − T ( s − p ) ⋅ d p .

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 ) = ∑ λ = poles of  X ( s ) Res p = λ ⁡ [ X ( p ) 1 1 − e − T ( s − p ) ] .

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 1 1 − e − T ( s − p ) in the right half-plane of p. The result of such an integration would be:

X ∗ ( s ) = 1 T ∑ k = − ∞ ∞ X ( s − j 2 π T k ) + x ( 0 ) 2 .

Relation to Z transform

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

X ∗ ( s ) = X ( z ) | z = e s T  

This substitution restores the dependence on T.

Properties of the starred transform

Property 1:   X ∗ ( s ) is periodic in s with period j 2 π T .

X ∗ ( s + j 2 π T k ) = X ∗ ( s )

Property 2:  If X ( s ) has a pole at s = s 1 , then X ∗ ( s ) must have poles at s = s 1 + j 2 π T k , where k = 0 , ± 1 , ± 2 , …

References

Starred transform Wikipedia
(Text) CC BY-SA