Starred transform

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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{align}
X^{*}(s)=\mathcal{L}[x(t)\cdot \delta_T(t)]=\mathcal{L}[x^{*}(t)],
\end{align}

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[edit]

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


\begin{align}
x^*(t)\ \stackrel{\mathrm{def}}{=}\ x(t)\cdot \delta_T(t) &= x(t)\cdot \sum_{n=0}^\infty \delta(t-nT).
\end{align}

Then per the convolution theorem, the starred transform is equivalent 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[edit]

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[edit]

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[edit]

  1. ^ a b 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[edit]

  • 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