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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35 views

Unit impulse response of a discrete-time LTI system

The problem: Consider a discrete-time LTI system. If the output signal is: $$ y[n]=5 \left( \frac{1}{5} \right) ^n u[n] -2^{-n} u[n] $$ , then the input signal will be: $$ x[n]=\left( \frac{...
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Characteristic Polynomial Equation

after using the z transformation, I get the characteristic polynomial equation following $(z-1)^2+(\alpha(z-1)+\beta)=0$, where z is the complex number, and $\alpha$ and $\beta$ are parameters. I ...
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41 views

Quadratic First Order Difference Equation

I'm studying a variant of the discrete Nerlove Arrow model \begin{equation} x_{t} = \lambda x_{t-1} + a_t \end{equation} which can be easily shown to be decomposed as \begin{equation} x_{t} = \sum_{k=...
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Ways of calculating Z Transform of a geometric serie

If you have a function $x(n) = 2^nu(n+1)$ And $u(n) = 1 \quad if \quad n > 0$ And you need the Z-Transform you'll have to go through a sum, knowing: $$\sum_{n=0}^\infty 2^nz^{-n} = \frac{1}{1-{...
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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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Z transform of the below equation

Below question from the filter design derivation problem on digital signal processing using Billinear Transformation method. Equation is as below: $$y(nT)- y(nT-T)+\frac{aT}{2}y(nT)+\frac{aT}{2}y(nT-...
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How to make use of the modified z-transform

The modified/advanced z-transform was introduced to analyze the behavior of sampled data systems between the samples. I understand how to derive the z-transform of a given continous transfer function....
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Help understanding the change of variables in a derivation

I am reading the textbook queueing systems 1 by Kleinrock. In appendix 1, while finding the z-transform of the unit function, the author proceeds from the step $z^{-n}\sum_{m} u_{n+m}z^{n+m} \...
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57 views

Why define the z-transform differently from the Laplace transform?

The mapping between the $z$-plane and the $s$-plane is defined by $$z=e^{sT}$$ where $T$ is the sampling period. A result of this mapping is that the shapes of the planes are quite different, e.g., ...
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What does z transform gives?

I'm pretty familiar with solving z transform, Region of convergence, all I've read throughout my semester while working with digital signal. But intuitively i somehow lack what does z-transform gives ...
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Inverse $Z$-transform $X (z) = \log \left( \frac{z}{z-a} \right)$

I need help to find the inverse $Z$-transform of the following function $$X (z) = \log \left( \frac{z}{z-a} \right)$$ I get to the point $$x[n] = \frac{\delta[n]}{n}+\frac{\epsilon[n]*a^n}{n}$$ ...
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How to find Inverse Z transform for a Z transform having region of convergence as ring?

I have done some examples of finding inverse Z transform using long divison when the given Z transform is right sided.Also there where some examples where left sided region of convergence was given. I ...
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Stuck on finding the Inverted z transformation

I have the following equation : $X(z) = \frac{1-z^{-1}}{1-0.25z^{-1}}$ The question I must answer is pretty simple : I need to find the inverted z transform and then create a graph for that. The ...
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z transform of $n^2 (1/3)^n$

How can i find the z transform of $n^2(1/3)^n$ using the properties? I refer to the two sided z transfrom. There is a property where $nx[n]$ becomes $-zdX(z)/z$ after z-transform is applied.But how ...
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z transform adding terms linearity

i am very sorry as this is an easy question but my confidence is very low. in a text book the author has the following signal and z transform. the signal is, $$f(n) = 2u(n) + 3u(n)$$ the author then ...
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MATLAB c2d giving incorrect z transform?

I have a signal $f(t) = e^{-t} $. I know that the Laplace of this should be $\dfrac{1}{1+s}$ and the z transform should be $\dfrac{z}{z+e^{-T}}$. I am a little confused, first off, on what the z ...
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Necessary and sufficient conditions for z-transform convergence

In some handwritten notes I've received, I have the following theorem: Let $f[k]$ be a signal and its z-transform be given by $$F[z] = \sum_{k=0}^\infty f[k]z^{-k}$$ Then, if there exist $c > 0$ ...
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Expand $X(z) = \frac{1}{1+az}$ into a causal sequence

For a HW problem, I'm told to expand $\frac{1}{1+az}$ into a causal and noncausal sequence. I found the noncausal sequence by long division (the result is $1-az+(az)^2-\dots$) and found the region of ...
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How to deduce Fourier / Z transform from Laplace transform through a CAS-Calculator.

I have a calculator with CAS ( https://en.wikipedia.org/wiki/Computer_algebra_system ) and in particular I have a Casio Algebra FX 2.0 Plus ( https://en.wikipedia.org/wiki/...
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307 views

DTFT (Discrete Time Fourier Transform) duality property applied to cos / sin

Normal DTFT table contains: $$ \cos(\omega_0 n) \xrightarrow{DTFT\ 2\pi} \pi \delta[\omega - \omega_0] + \pi \delta[\omega + \omega_0] $$ $$ \sin(\omega_0 n) \xrightarrow{DTFT\ 2\pi} i\ \pi\ \...
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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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using z transform pairs to solve the question.

Following is a question of z transform: $$ (-\frac{1}{3})^n u[-n-2] $$ Now I know that a transform pair similar to this is: $$ -\alpha^n u[-n-1]=\frac{1}{1-\alpha z^{-1}} ...
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inverse z-transform of $\frac{1}{(1-z^{-1})^{2}}$

how to find inverse-z-transform of: $$ x[n] = Z-Transform\left\{ \frac{1}{(1-z^{-1})^{2}} \right\} $$ I could convert it to convolution... but then i'm stuck with solving convolution...which i'm not ...
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Approximating Differential Equations Using Difference Equations/The $z$-Transform.

I'm aware that difference equations and the $z$-Transform method can be used to approximate differential equations but I'm wondering how exactly it does this as I don't have a clue? All I've come ...
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Weird thing about z-transform and difference

here is my doubt: we were told that the ROC of the Z-transform of the sum of two sequences is the intersection of the respective ROCs as the two of them are limited only if both of them are. Now I had ...
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Why does $x[k] = k^k$ not have a Z-transform?

The signal $x[k] = k^k, k=1,2,3...$ does not have a Z-transform. Why? The definition of the Z-transform is: $$X[z] = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$
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difference equation, LTI system, Z-transform, impulse response

exercises 2 and 3 instructions are in picture Here is a z-table that I can use to make inverse z-transforms I did questions 2a and 2b, but I don't know much about how to do the final 2c (I got stuck ...
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Solving a difference equation via z-transforms

I'm trying to figure out if there is a "nice" solution to this difference equation $$\alpha p_t = p_{t-1} + \beta x_t(x_t - x_{t-1}).$$ Using a $z$ transform, I get $$P_z(\alpha - z^{-1}) = \beta ...
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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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86 views

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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1answer
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Z transform and non-$\ell^{1}$ causal sequences

Consider a complex sequence $h[n]$, such that: if $n < 0 \Rightarrow h[n] = 0 $ $\sum_{n=0}^{+\infty} \left|h[n]\right|$ diverges $H(z) = \sum_{n=0}^{+\infty} h[n]z^{-n}$ converges uniformly $\...
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Maximum value or bounds of a sequence based on its Z tranform

Consider a sequence $\{a_i\}_{i=0}^{\infty}$ of real numbers and its Z tranform $$A(z)=\sum_{i=0}^{\infty}a_iz^{-i}$$ We assume that all poles of $A(z)$ lie in the interior of the unit circle. Thus, ...
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1answer
92 views

Z-transform of $\sin{(n\frac{\pi}{2}(-1)^{n})}$

I was trying to do this z transform: $\sin{(n\frac{\pi}{2}(-1)^{n})}$ but I don't have the solution. I have divided in 4 cases: for n=0 0 for n=1 -1 for n=2 0 for n=3 1 so 4k, 4k+1, 4k+2 ...
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486 views

Partial Fraction Expansion Methods for inverse z transform

I am following PFE for control system from a book. The author says that first it is best to divide by z for both sides, then multiply back through later. the thing is, that i get a different answer ...
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1answer
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Need help with z-transform question.

The question is to find the z-transform of $$ x(n) = 3n\cdot u(n-2)$$ So far I have seen a question asking to find z-transformation of $x(n)=3^n u(n-2)$ and I know the solution. Howver for the ...
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148 views

Z transform of real-time input to discrete transfer function

I am trying to implement a first order discrete transfer function to model a small centrifugal pump to control its output flow rate. I started with a continuous first order transfer function which ...
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131 views

Derivation of Fibonacci sequence by difference equation/Z transform

I'm trying to derive the Fibonacci sequence. I have the following problem: $$N(t) = N(t-1)+ N(t-2) \quad \quad \quad \quad (I)$$ With initial conditions $N(1) = 2$ and $N(2) = 3$. Using: $$N(t+2) =...
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Is there a way to deduce Z-Transform initial and final value theorems from Laplace Transform?

Given that Laplace Transform and Z-Transform are closely related, I wonder if there's a way to deduce Z-Transform final value theorem: $$\lim_{n\rightarrow \infty} f[n] = \lim_{z \rightarrow 1}(1-z^{-...
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223 views

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 ...
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93 views

Simple Inverse Z-Transform

I'm kind of confused on how to get the inverse Z-transform of basic things. We were told to basically break down the function into smaller parts that we know the z-transform of but I can't find a list ...
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1answer
34 views

Z Transform of n-varying function

I've been doing some reading on z-transforms and I'm still fairly new to the topic. I understand finding the transforms very basic signals. But the approach to finding the transform of this ...