# Questions tagged [z-transform]

The $z$-transform is a discrete analogue to the Laplace transform, in that it maps a time domain signal into a representation in complex frequency plane.

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### How do summations/integrals like Fourier, Laplace, z-transforms preserve all the information about the original signal?

In normal summations, like 2+3=5, the information about the original numbers is lost. But in infinite summations like integral transforms, no information is lost and the function can still be ...
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### DFT LT and ZT of an array of data

Let us say I have an array of $10$ elements $= \{1, 5, 10, 15, 20, 25, 30, 35, 40, 45\}$. How can I get its Fourier transformed array, Laplace transformed array and $Z$-transformed array? Do I need ...
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### Why do we take a x(k+1)-x(k) for proving the final value theorem of z transform?

I am studying z transform. Now, I am in final value theorem proof. enter link description here The given link says that we take the function x(k+1)-x(k) for no apparent reason. Can we prove it by ...
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### Discrete State Space Representation

I'm looking at a continuous state space system and I want to discretize it. I've seen what others have done that works. However, I saw this method but the justification was not given. Please can ...
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### Find $\mathscr{Z}$ transform of $\begin{cases}(\frac{1}{2})^{-n}&\text{if$n$is a multiple of 3},\\1&\text{otherwise}\end{cases}$

Find $\mathscr{Z}$ transform of the following discrete signal: $$x[n]=\begin{cases}\left(\displaystyle\frac{1}{2}\right)^{-n}&\text{if n is a multiple of 3},\\1&\text{otherwise}.\end{cases}$$...
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### How do you solve this matrix simultaneous equations?

This is from discrete systems analysis, you are given 2 matrix equations that you apply z transform to and then you need to solve for matrix Y. I have forgotten how to do these equations. Can someone ...
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### Relationship Between Laplace and $z$-Transforms.

I've recently come across the relation $s= \frac{2(z-1)}{T(z+1)}$ between the Laplace and $z$-Transforms with inverse $z= \frac{2+sT}{2-sT}$ in some lecture slides, however there was no elaboration ...
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### polynomial long division quick question about the algorithm

I was making use of polynomial long division in inverse Z transform and I got stuck in a brainfart in one stage of the polynomial long division. I posted the original question into digital signal ...
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### Does any discrete signal has z-transform (even noise)?

I faced to the following discrete $$y[n+1]=Ay[n]+Bx[n]+\eta[n]$$ where $y[n] \in \mathbb{R}^n$, $A$ and $B$ are matrices with appropriate dimension, and $\eta[n]$ is noise. I have no problem with ...
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### BIBO Stability in Z-domain

I'd really appreciate it if someone could please explain to me the condition for a LTI system to be BIBO stable, in z-domain. I have a background in control, and in linear control for example, if we ...
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### How to convolve $u(n)$ and $u(-n-5)$ without using $z$ transform

As the title says, I wanted to test the property of the $z$ transform where: $$z[x(n) \cdot x(h)] = z[x(n)] z [h(n)].$$ I have already solved the right hand part. All that remains to show is the ...
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### Z-transform of $nu_{n}$ and $u_{n}/n$

I am studying z transform and I couldn't get how to derive these two formulas $$Z[ nu_{n}] =-z \frac{d}{dz} Z(u_{n})$$ $$Z\left[ \frac{1}{n} u_{n}\right] =- \int_0^z z^{-1}Z(u_{n})$$ These ...
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### A case of discrete filter factorization.

Based on this question, where in the language of signal processing the difference of a self convolution and a lazy filter becomes the convolution of two other filters. One slightly longer and one ...
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### open loop transfer function

a block diagram system is given as per the attached image. what is the open loop transfer function? i say it is found by opening the loop and multiplying all of the terms. So that: U = KFYX. Then ...