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 x ( t ) {\displaystyle x(t)} , which is transformed to a function X ∗ ( s ) {\displaystyle X^{*}(s)} in the following manner:^[1]

X ∗ ( s ) = L [ x ( t ) ⋅ δ T ( t ) ] = L [ x ∗ ( t ) ] , {\displaystyle {\begin{aligned}X^{*}(s)={\mathcal {L}}[x(t)\cdot \delta _{T}(t)]={\mathcal {L}}[x^{*}(t)],\end{aligned}}}

where δ T ( t ) {\displaystyle \delta _{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 ) {\displaystyle x^{*}(t)} , which is the output of an ideal sampler, whose input is a continuous function, x ( t ) {\displaystyle 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.

Since X ∗ ( s ) = L [ x ∗ ( t ) ] {\displaystyle X^{*}(s)={\mathcal {L}}[x^{*}(t)]} , where:

x ∗ ( t )   = d e f   x ( t ) ⋅ δ T ( t ) = x ( t ) ⋅ ∑ n = 0 ∞ δ ( t − n T ) . {\displaystyle {\begin{aligned}x^{*}(t)\ {\stackrel {\mathrm {def} }{=}}\ x(t)\cdot \delta _{T}(t)&=x(t)\cdot \sum _{n=0}^{\infty }\delta (t-nT).\end{aligned}}}

Then per the convolution theorem, the starred transform is equivalent to the complex convolution of L [ x ( t ) ] = X ( s ) {\displaystyle {\mathcal {L}}[x(t)]=X(s)} and L [ δ T ( t ) ] = 1 1 − e − T s {\displaystyle {\mathcal {L}}[\delta _{T}(t)]={\frac {1}{1-e^{-Ts}}}} , hence:^[1]

X ∗ ( s ) = 1 2 π j ∫ c − j ∞ c + j ∞ X ( p ) ⋅ 1 1 − e − T ( s − p ) ⋅ d p . {\displaystyle X^{*}(s)={\frac {1}{2\pi j}}\int _{c-j\infty }^{c+j\infty }{X(p)\cdot {\frac {1}{1-e^{-T(s-p)}}}\cdot dp}.}

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 ) ] . {\displaystyle X^{*}(s)=\sum _{\lambda ={\text{poles of }}X(s)}\operatorname {Res} \limits _{p=\lambda }{\bigg [}X(p){\frac {1}{1-e^{-T(s-p)}}}{\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 1 1 − e − T ( s − p ) {\displaystyle {\frac {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 . {\displaystyle X^{*}(s)={\frac {1}{T}}\sum _{k=-\infty }^{\infty }X\left(s-j{\tfrac {2\pi }{T}}k\right)+{\frac {x(0)}{2}}.}

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

X ∗ ( s ) = X ( z ) | z = e s T {\displaystyle {\bigg .}X^{*}(s)=X(z){\bigg |}_{\displaystyle z=e^{sT}}}  ^[2]

This substitution restores the dependence on T.

It's interchangeable,^{[citation needed]}

X ( z ) = X ∗ ( s ) | e s T = z {\displaystyle {\bigg .}X(z)=X^{*}(s){\bigg |}_{\displaystyle e^{sT}=z}}  

X ( z ) = X ∗ ( s ) | s = ln ⁡ ( z ) T {\displaystyle {\bigg .}X(z)=X^{*}(s){\bigg |}_{\displaystyle s={\frac {\ln(z)}{T}}}}  

Property 1:   X ∗ ( s ) {\displaystyle X^{*}(s)} is periodic in s {\displaystyle s} with period j 2 π T . {\displaystyle j{\tfrac {2\pi }{T}}.}

X ∗ ( s + j 2 π T k ) = X ∗ ( s ) {\displaystyle X^{*}(s+j{\tfrac {2\pi }{T}}k)=X^{*}(s)}

Property 2:  If X ( s ) {\displaystyle X(s)} has a pole at s = s 1 {\displaystyle s=s_{1}} , then X ∗ ( s ) {\displaystyle X^{*}(s)} must have poles at s = s 1 + j 2 π T k {\displaystyle s=s_{1}+j{\tfrac {2\pi }{T}}k} , where k = 0 , ± 1 , ± 2 , … {\displaystyle \scriptstyle k=0,\pm 1,\pm 2,\ldots }

• 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