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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21 views

Laplace to Z-transform [closed]

How is the Impulse Invariance transformation of $$\frac{(s+a)}{(s+a+jb)(s+a-jb)}$$ is $$\frac{(1-e^{-aT} cos(bT) z^{-1})}{(1-e^{(-a-jb)T} cos(bT) z^{-1})(1-e^{(-a+jb)T} cos(bT) z^{-1})}$$ ?
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36 views

Help me with z transform

So the question is basically z transform the given system. $(y[n+2] + 3y[n+1] - 4y[n])=(x[n+2] - 5x[n+1])$ I've to find h[z] first then it's really easy to solve it. So that's what I got so far; $z^2y(...
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18 views

z-transformation of difference equation

Let $$x_{n+1}-x_{n}=3x_{n}+2$$ I want to find the z-transformation, in the solution, it is stated that $y(z)=\frac{2}{\frac{1}{z}-3}$ I get a completely different result after multiplying everything ...
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9 views

Cross-Correlation property in Laplace transform?

Like convolution is a multiplication in Laplace transform, what is correlation in Laplace transform? Is there a property for it in Z transform?
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Find the mistake of my answer, finding the inverse of $Z$-transformation.

Find inverse of $Z$-transform of $X(z)=\dfrac{z}{(z-2)(z^2+6z+9)}$. I have tried as follows. Let \begin{align*} X(z)&=\dfrac{z}{(z-2)(z^2+6z+9)} =\dfrac{z}{(z-2)(z+3)^2}\\ &=\dfrac{A}{z-...
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24 views

Cannot find the inverse of Z-transform.

Find the invers of $Z$-transform of $X(z)=\dfrac{z}{(z-2)(z^2+6z+9)}$. I try as below. Let \begin{align*} X(z)&=\dfrac{z}{(z-2)(z^2+6z+9)} =\dfrac{z}{(z-2)(z+3)^2}\\ &=\dfrac{A}{z-2}+\...
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73 views

Inverse $z$-transform of $\frac{z_{1}^{-1}}{1 - az_{1}^{-1}z_{2}^{-2}}$

I'd like to know how to calculate the inverse $z$-transform of $$X(z_{1},z_{2}) = \frac{z_{1}^{-1}}{1 - az_{1}^{-1}z_{2}^{-2}},\quad |z_{1}|\cdot |z_{2}|^{2} > |a|.$$ I`ve tried to make the change $...
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18 views

Z transform or Laplace transform

Is there any app which can perform $z$-transform or can we perform this operation using a scientific calculator?
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16 views

Going in Reverse From Z-Domain to a Difference Equation

I am attempting the task of beginning with a $z$-domain equation, and then reverse engineering a difference equations from it. The eventual goal is to solve the difference equation, and then go back ...
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Generating Functions VS Z-Transforms as Solutions to Recurrence Relations

In a Discrete Mathematics video, recurrence relations are solved by applying generating functions to each term, doing algebra, and extracting coefficients of the result. There is no mention of Z-...
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How to calculate Z-transform from a series

If we have the sequence $\{1,1,\dots\}$, how would I go about calculating the Z-transform? Such that we find $Z\{1,1 \dots\}(z)$.
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Z-transform and region of convergence of $x(n)=\left(\frac{1}{3}\right)^{|n|}-\left[\left(\frac{1}{2}\right)^{n} u(n)\right]$

How can we find the Z-transform of $$ x(n)=\left(\frac{1}{3}\right)^{|n|}-\left[\left(\frac{1}{2}\right)^{n} u(n)\right] $$ and it's region of convergence (ROC) ?? where the Z-transform of x(n) is $X(...
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For $G(x)=\sum_{n=1}^{\infty} \exp \left(-\pi x n^{2}\right)$ proove that $\frac{1+2 G(x)}{1+2 G\left(\frac{1}{x}\right)}=\frac{1}{\sqrt{x}}$

How can we prove rigorously that for $G(x)=\sum_{n=1}^{\infty} \exp \left(-\pi x n^{2}\right)$ , x>0 we have the equality $$ \frac{1+2 G(x)}{1+2 G\left(\frac{1}{x}\right)}=\frac{1}{\sqrt{x}}?? $$ ...
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59 views

Inverse $\mathcal{Z}$ transform of $\frac{1}{z^2(z+1)}$ without delta function

While finding inverse $\mathcal{Z}$ transform of $\frac{1}{z^2(z+1)}$ i split it into three parts using partial fractions: $$\frac{a}{z} + \frac{b}{z^2}+\frac{c}{z+1}$$ and so answer comes as $$a \...
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(SOLVED) Inverse Z transform of a complicated fractional expression

Problem Consider the following recurrence equation: $$y[n+2]+ \frac{1}{4}y[n] = \cos(n \frac{\pi}{3})$$ $$\text{Where }y[0] = y[1] = 0$$ $$\text{What is }y[n]?$$ My attempt Using $Z$ transform for ...
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Create Transfer Function with Known Input Delay and Specified Attributes

I have a transfer function $$G(s) = e^{-5.8s}\cdot \frac{5}{s+5}$$ How to get the $\frac{dx}{dt} = Ax+Bu$ form out of it , The example is taken from Mathwork site And when I Try to find the Z ...
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Solving a recurrence equation using $\mathcal{Z}$-transform

The question is: solve $x_{n+1}-2x_{n} = 2n$ where $x(0) = 1$. I'm trying to solve this equation, here is where I got stuck: $zX(z)-x(0).z-2X(z)=\frac{2z}{(z-1)^2}$ $ X(z)[z-2]=\frac{2z}{(z-1)^2}+z$ $...
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Inverse z-transform by convolution method

I've been asked to find the poles, impulse response of a system (in digital signal processing). The transfer function was, $$H(z) = \frac{z^2+z}{z^2-z+0.5}$$ I solved it by the partial fractions ...
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19 views

Property of Z transform

Is there any direct relationship exist between p times differentiation of F(z) and f(n) similar to Laplace transform where n times differentiation directly related to its time domain counterpart ? ...
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$Z$-transform of floor($\frac{n}{5}$) [closed]

I found problem about floor array that I can't solve. Find Z transform of f(n) = floor(n/5). I tried writing this array and for n from 1 to infinity, I got n/5 = 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 6/5... ...
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Solve the following Z-Transform Difference Equation

We have this given equation $$y_{n+1}- 2.5y_{n} = -1.1$$ With the initial condition of $$y_{0} = \frac{1}{200}$$ All examples of this found on the internet have $y_{0}$ as an integer. How can this be ...
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How to solve $x[n]\ast x[n]=u[n]$?

I came across the following question: The causal discrete-time signal $x[n]$ satisfies: $$x[n]\ast x[n]=u[n]$$ Where $\ast$ is the convolution operator and $u[n]$ is the discrete unit step function. ...
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1answer
33 views

Relation between $r^n x[n]$ and $X(z)$

We know that if $x[n]$ has Z-transform, and an RoC of $\alpha < |z|<\beta$ then the z transform of $p^n x[n]$ is $X(\frac{z}{p})$ with an RoC of $ p\alpha < |z| < p\beta$. But I've seen ...
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If $a_{n+1}=2a_n −n^2+n$ Define a sequence $a_n$ that satisfy the recurrence relation as described above, with $a_1 = 3$

If $$a_{n+1}=2a_n −n^2+n$$ Define a sequence $a_n$ that satisfy the recurrence relation as described above, with $a_1 = 3$ Find the value of $$\dfrac{ |a_{20} - a_{15} | }{18133} $$ Attempt First ...
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20 views

Find the exact decay from a discrete sum of powers

In a discrete function of the form $h_n=\sum_k{A_k\,p_k^n}$, with $h_n$ being an unknown vector of real values, $A_k$, $p_k$ vectors of complex numbers, and $n$ integer, is it possible to find out the ...
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13 views

Signal where the bilateral z-transform exists, but the unilteral z-transform doesn't for some value?

Like the title says, I'm looking for a signal such that the bilateral Z-transform exists, but the unilateral transform does not exist for some arbitrary value or set of values for z. So, in more ...
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38 views

Why does the Z transform represent a delay?

I'm studying the Z-transform. I recently did by hand the Z transform of an discrete impulse delayed $\mathcal{z}\{\delta[n-k]\} = z^{-k}$ I get that this means that any signal can be represented as ...
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28 views

Finding inverse Z Transform

Find the inverse Z transform: I have done the solution but my answer does not match with the one given in the textbook. What I may have done wrong?
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Determine the Z-Transform for the following sequence: $ |n|(\frac{1}{2})^{|n|} $

Determine the Z-Transform for the following sequence: $$ |n|(\frac{1}{2})^{|n|} $$ I have tried to solve the above problem. However, the answer that I got is the negative of what is given in the ...
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constraint on $n_0$ for $ x[n] = (-1)^nu[n] + a^nu[-n-n_0] $ for a given ROC.

Let $$ x[n] = (-1)^nu[n] + a^nu[-n-n_0] $$ Determine the constraints on the complex number $a$ and the integer $n_0$, given that the ROC of X(z) is: $$ 1<|z|<2 $$ My question is that what ...
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Determine the constraint on $ r = |z| $ for the sum to converge: $ \sum_{n=0}^{n= \infty} [\frac{1+(-1)^n}{2} ] z^{-n} $

Determine the constraint on $ r = |z| $ for the following sum to converge: $$ \sum_{n=0}^{n= \infty} [\frac{1+(-1)^n}{2} ] z^{-n} $$ Solution: $$ \frac{1}{2} \sum_{n=0}^{n= \infty} z^{-n} + \frac{...
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Discrete LTI systems with complex inputs?

I'm reading and pondereing about the convolution sumation, properties and how this is related to discrete LTI systems. I'm using the book Signals and Systems by Alan V. Oppenheim, and on the chapter ...
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one-to-one mapping of z-transform data

I am trying to make a simple example of a data set of (say) 10 numbers and then its one-to-one corresponding z-transformed data set of 10 numbers. No idea how can I do it? Can I take any 10 random ...
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Z transform (residue method)

I have a big problem that gives me headaches. I have to find out inverse Z transform (by residue method) of $$\frac{(z^2+1)}{(z^2-z+1)}.$$ Please, help me! :( *the response is in the picture that I ...
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How do I generalize a certain Markov model?

This question is a further attempt to generalize a certain Markov model of limit & market orders arriving in a financial exchange as first proposed in [1]. See Solving another non-trivial ...
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Another bizarre convolutional identity.

Let $\lambda^M \ge0$ and $\Lambda\ge 0$ and $q \in (0,1)$. Now define another three numbers $(a,b,c)$ by solving the following set of non-linear equations below: \begin{eqnarray} c \cdot (c-2) &=&...
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How do I find the magnitude response of a filter given only the ROC as a diagram?

The figures below show the pole-zero plots of two filters, H1(z ) and H2(z ). The poles and the zeros lie on circles of radius a and 1/a, with a < 1. How can I sketch a correct magnitude response? ...
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Computing the Transfer Function of open-loop-gain equation (Z-Transform)

I am designing a closed-loop control system. I need to figure out how I should design the compensator so that the system is stable for a wide range of inputs. The problem is, the open-loop-gain ...
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1answer
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Inverting a digital filter

I want to understand a technique for calculating the inverse of a digital filter. Let A be a function from $R$ to $R$ such that $$ A(0) = 1;\quad A(1) = a_1;\quad A(2) = a_2; \quad\text{, otherwise } ...
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How can we add Dirac function to the equation to solve the problem.

I am currently trying to understand this problem, I have come across with this while studying Z-Transform. The problem with this \begin{equation} x[n] = u[n]+\bigg(-\frac{3}{4}u[-n]\bigg) \\ x[n] = ...
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Suppose the $Z$-transform of $x(n)$ is $X(z)$. What is the $Z$-transform of $x(2n)$?

Suppose the $Z$-transform of $x(n)$ is $X(z)$. What is the $Z$-transform of $x(2n)$? Here is my thought process: $X_2(z) =\displaystyle\sum_{- \infty}^{\infty}x_2(n)z^{-n}=\sum_{- \infty}^{\infty}x(...
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74 views

Solving another non-trivial recurrence relation

This is a generalization of question Yet another non-trivial recurrence relation to solve. . Let $E \in {\mathbb R}$ and $\lambda^{C} \ge 0$, $\lambda^{M} \ge 0$ and $\Lambda \ge 0$.In addition to ...
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70 views

Yet another non-trivial recurrence relation to solve.

This question is related to a certain Markov chain with variable transition probabilities described in 1. Let $E \in {\mathbb R}$ and $\lambda^{C} \ge 0$, $\lambda^{M} \ge 0$ and $\Lambda \ge 0$ . ...
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1answer
82 views

Why is a Toeplitz matrix a representation of Laurent series?

I am reading the 2006 book Spectra and Pseudospectra by Trefethen. On pages 50-51, it is stated that for the following Laurent series $$f(z)=\sum_k a_k z^k=2z^{-3}-z^{-2}+2i z^{-1}-4z^2-2iz^3$$ and ...
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1answer
192 views

Calculate sum of sum using Z transform

$$S = \sum_{n=0}^{\infty} \frac{1}{3^n} \sum_{k=0}^{n} \frac{k}{2^k}$$ We know that if one signal y can be written as $$y(n) = \sum_{k=0}^{n} x(k)$$ then it's Z transform is $$ \frac{z}{z-1} X(z) $$ ...
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103 views

Z-Transform on Dirac

I have a question to do with doing a Z-Transform on a Dirac function for signal processing. So if we have $x(n)=\delta (n)+\delta (n-1)$ the transform should look like $X(z)=\sum (\delta (n)+\delta (...
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1answer
19 views

How to convert a rational transfer function from the Z-domain to the time domain and back?

I'm in the process of studying z-transform for a project involving audio processing. I already asked a related of question on dsp.stackexchange.com, but I'm having a somewhat hard time understanding ...
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1answer
18 views

Unable to get the right answer through convolution in z transform

Through the method of convolution, I found that the inverse z transform of $$\frac{z^2}{(z-2)^2}$$ as $2^{n} \, (n+1)$. But, when I try to use the same formula of convolution to find the inverse of $...
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1answer
22 views

Unable to find the z transform of this function

My question is to find the z transform of the function I am unable to find a series which matches what we get when we expand the summation.
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1answer
31 views

Multiplication Property of Z-Transform

I have searched a lot online and have yet to find anything that proves the multiplication property of the z-transform ie $$ x_1[n]x_2[n] \iff \frac{1}{2 \pi j} \oint X_1(u)X_2(\frac{z}{u})u^{-1}du $$...

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