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

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

where $\delta _{T}(t)$ 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 $x^{{*}}(t)$, 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 $X^{{*}}(s)={\mathcal {L}}[x^{{*}}(t)]$, where:

${\begin{aligned}x^{*}(t)\ {\stackrel {{\mathrm {def}}}{=}}\ x(t)\cdot \delta _{T}(t)&=x(t)\cdot \sum _{{n=0}}^{\infty }\delta (t-nT).\end{aligned}}$

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

$X^{{*}}(s)={\frac {1}{2\pi j}}\int _{{c-j\infty }}^{{c+j\infty }}{X(p)\cdot {\frac {1}{1-e^{{-T(s-p)}}}}\cdot dp}.$

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:

$X^{{*}}(s)=\sum _{{\lambda ={\text{poles of }}X(s)}}\operatorname {Res}\limits _{{p=\lambda }}{\bigg [}X(p){\frac {1}{1-e^{{-T(s-p)}}}}{\bigg ]}.$

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 ${\frac {1}{1-e^{{-T(s-p)}}}}$ in the right half-plane of p. The result of such an integration would be:

$X^{{*}}(s)={\frac {1}{T}}\sum _{{k=-\infty }}^{\infty }X\left(s-j{\tfrac {2\pi }{T}}k\right)+{\frac {x(0)}{2}}.$

Relation to Z transform

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

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

This substitution restores the dependence on T.

Properties of the starred transform

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

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

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

Citations

1. 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.
2. 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