Starred transform

{{For|the topological transformation in circuit networks|star-triangle 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 {{nowrap| $X^{*}(s)$ }} in the following manner: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.

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

where {{nowrap| $\delta_T(t)$ }} 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 {{nowrap| $x^{*}(t)$ }}, 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 {{nowrap| $X^{*}(s)=\mathcal{L}[x^{*}(t)]$ }}, 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 to the complex convolution of {{nowrap| $\mathcal{L}[x(t)]=X(s)$ }} and {{nowrap| $\mathcal{L}[\delta_T(t)]=\frac{1}{1-e^{-Ts}}$ }}, hence:

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

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 {{nowrap| $\frac{1}{1-e^{-T(s-p)}}$ }} 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:

: $\bigg. X^*(s) = X(z)\bigg|_{\displaystyle z = e^{sT}}$ Bech, p 9

This substitution restores the dependence on T.

It's interchangeable,{{Citation needed|reason=Need a proof|date=February 2018}}

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

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

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 {{nowrap| $X(s)$ }} has a pole at $s=s_1$ , then {{nowrap| $X^{*}(s)$ }} must have poles at $s=s_1 + j\tfrac{2\pi}{T}k$ , where $\scriptstyle k=0,\pm 1,\pm 2,\ldots$

Citations

References

{{cite web|last=Bech|first=Michael M.|title=Digital Control Theory|url=http://homes.et.aau.dk/mmb/DigitalControlTheory/WS3/WP3_2.pdf|publisher=AALBORG University|accessdate=5 February 2014}}
{{cite book|last=Gopal|first=M.|title=Digital Control Engineering|date=March 1989|publisher=John Wiley & Sons|isbn=0852263082}}
Phillips and Nagle, "Digital Control System Analysis and Design", 3rd Edition, Prentice Hall, 1995. {{ISBN|0-13-309832-X}}

Category:Transforms