# Where does this star came from? [EDIT - Detailed Question]

First of all, it's a rare topic in google, so I can't find intuitive explanation about step-invariant, so I don't understand actually what it is, except to solve zero order hold transfer function.

In this pdf, page 15 to 18 I've been learning about zero order hold (ZOH). http://www.engr.usask.ca/classes/EE/480/notes/myee480-p4-sampleHold.pdf

As you can see above, suddenly the laplace transform has become starred, why does it happened?

EDIT

First thought, I think it was like distributive starred property. But why suddenly it has a star while you still have starred curly bracket? Is it a typo?

And also, from the wikipedia
https://en.wikipedia.org/wiki/Starred_transform
It's written that $X^*(s) = L[x(t).dirac_T(t)] = L[x^*(t)]$
The star is only on the big X and small x, and it's not on the L one, so why the creator made it star on the L one?

Furthermore, if you also take a look at page 17 (top right). First, (1-e^-Ts) are converted into laplace transformation form, and then it gets starred, and then it is separated, and then it return to the first original (1-e^-Ts) form. What's the use? It's seems illogical.

If you don't mine, please also explain intuitively what is step-invariant z-transform and why the creator use step-invariant z-transform method to find the transfer function and not the other method.

• These are three separate questions, and shouldn't be asked in the same question. Please focus on one question, and save the other two for a later point in time. Commented Feb 1, 2018 at 12:36
• @MarcusMüller alright done, do you know the answer sir? Commented Feb 1, 2018 at 12:55

## 1 Answer

This seems as a starred transformation. It is a discrete version of the Laplace transformation. Laplace transformation is continuous by nature. Starred transformation is discrete, and the star denotes difference between continuous and analog domain.

This is commonly used when teaching about the Z transform. It tends to be confusing, so the teachers typically just skim over it.

Here are some explanation about it: https://en.wikipedia.org/wiki/Starred_transform

www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=934

• Alright, First thought, I think it was like distributive starred property. But why suddenly it has a star while you still have starred curly bracket? Commented Feb 1, 2018 at 23:05
• Do you think it's a typo? Commented Feb 1, 2018 at 23:10
• Furthermore, if you also take a look at page 17 (top right). First, (1-e^-Ts) are converted into laplace transformation form, and then it gets starred, and then it is separated, and then it return to the first original (1-e^-Ts) form. What's the use? It's seems illogical Commented Feb 1, 2018 at 23:10
• And also, from the wiki it said that X*(s) = L[x(t).dirac(t)] = L[x*(t)], the star is on big X and small x, not on the L one, so why the creator made it star on the L one? Commented Feb 1, 2018 at 23:27
• don't know, since I have not gone through the math. Sorry not much time at my hand at the moment Commented Feb 2, 2018 at 16:43