Questions tagged [z-transform]

The Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.

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44 views

Z-Transform vs. Fourier Transform convergence

Are there signals for which the Fourier transform is known to exist (perhaps including singularities) and for which the z-transform does not converge?
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Find the $\mathcal Z$-transform of this function?

I need to find the $\mathcal Z$-transform of $x(n)=a^{-n} u(n)$. Assume, $a$ is a positive constant , but the power of $a$ is negative. Looking at the transform table, I found that $\mathcal Z$-...
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Impulse response of an LTI system given the input and output signals

I have been given the input and output signals of an LTI system as: $x[n] = (\frac{1}{2})^nu[n] + 2^nu[-n-1]$ $y[n] = 6(\frac{1}{2})^nu[n] - 6(\frac{3}{4})^nu[n]$ I have found the system function $...
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Do all systems in z domain are filters?

Normally a system transfer function is represented by H(z) I want to know wether all those system transfer functions are only representing a digital filter? Or their any other thing/entity in z ...
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Using ROC to find stability of system in specific example

I've started learning about finding the ROC from the transfer function, but I'm confused about an example. $$H(Z) = \frac{2Z + 1}{Z^2 + Z - \frac{5}{16}}$$ I understand the poles lie at $z = \frac{-...
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Discrete Filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$

Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
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Transfer Function Block Diagram Confirmation

Can someone confirm if this is the correct block diagram for the following transfer function? The original equation provided was: y[n+1] = y[n] + 0.01x[n] Which I rearranged into H(z) = Y(z)/X(z) = ...
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In z-transform, if z means “delay”, why do we talk about the value of z?

I've been watching MIT's signals course and trying to understand $z$-transform. The course introduces $z^{-1}$ as an operator that delays the signal by $1$ time unit (which works very well with the ...
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1answer
428 views

When to use Fourier, Laplace and Z transforms?

If we have an LTI system, with an input signal $x(t)$, impulse response $h(t)$ and output $y(t)$, I was under the assumption that if the input and impulse response were continuous in time, then you ...
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1answer
66 views

Factored form vs partial fraction form?

I have already understood partial fraction and here is link for my relevant DSP SE question Finding inverse z transform for two sided ROC? But now i want to know, is there any difference between ...
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Is $\cos(\pi \sqrt n)u[n]$ stable?

I took a z transform and got a double pole at $z=1$, but I don't know if that's correct. I'm lost because I don't know if $\cos(\theta)$ converges or diverges or what that means for $h[n]$ being ...
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Necessary Conditions for stability in z domain?

What are the necessary(must) conditions for stability in z domain? I am sure about one(ROC must include unit circle) Is there any other such condition which states that there shouldn't be any poles in ...
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Inverse z-transform. Where is mistake?

I've already wrote about that trouble (link here), but I don't understand where I've made a mistake. Full description of the task is as follows: Z-transform of sequence {x(k)} describe by the ...
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Finding inverse z transform for two sided ROC?

I have a z transform $Y(z)=(z^2-z) /(z^2+1.3z+0.3)$ In this case i have two poles ,one at -0.3 and one at -1.0 I want to find inverse z transform of Y(z) Following is my matlab code: ...
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Potential issues arising from too stable discretization

When numerically simulating a system, usually some kind of discretization is necessary, obtained by some kind of z-transform, such as, for instance, the bilinear transform $s\mapsto \frac{2}{\triangle ...
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1answer
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Z transform getting different answer for transform of rational function

Why am i getting different answers and which one is correct?? In this also
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1answer
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Determing inverse Z-transform using impulse response?

In Matlab there is a command iztrans for finding inverse Z-transform. But how can we find inverse Z-transform using impulse response? The Matlab command ...
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What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
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1answer
42 views

Z-transform of not quite an upsampler

I know the z-transform of an upsampler is: $$ y[n] = \begin{cases} x(n/L) &n=0,\pm L, \pm 2L, ...\\ 0&otherwise \end{cases} \longrightarrow Y(z)= X(z^{L}) $$ if $x[n]_L$ is defined to zero ...
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1answer
88 views

Z Transform problem

I have a class exercise of an inverse Z transform and I have some trouble. I will render an example to make my point. Let's asume the Z transform pairs: $$a^n \cdot u[n] \Leftrightarrow \frac{1}{1-az^{...
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factoring poles / zeros: off by constant gain compared with textbook

(From Schaum's DSP outline, 2nd edition, problem 5.32) Book says factor it and extract H(z) from the factored product: $$ H(z)H(z^{-1})= \frac{ \frac{5}{4} - \frac{1}{2}z - \frac{1}{2}z^{-1} }{ \...
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What is the $\mathcal{Z}$-transform of a constant?

The Fourier transform of a constant exists. Can anyone please tell me what the $\mathcal{Z}$-transform of a constant is? Thanks in advance.
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Book request for Z-Transform

I am at the moment self studying Z-Transform. I need some references on this subject with some applications ( electrical circuits or else). What are good references on this subject. Thanks in advance.
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First order low pass filter

I am trying to better understand the first-order low pass filter: Summary: Per wikipedia, a first order low pass filter yields the following in discrete time: $$ \frac{Y(s)}{U(s)}= \frac{\omega_{c}}...
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ROC of the function in the problem 9.14 of Oppenheim's Signals and Systems textbook

I have solved the problem 9.14 in Oppenheim's Signals and Systems textbook, but my solution and the one in Slader is different. Problem is given above. And Slader solution is here. I have also ...
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1answer
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One more inverse z transform … a bit more complicated [closed]

Thank you for all the help teaching me. I am looking at one more inverse z transform and not understanding what to do with it. $F(t) = m \cdot a(t)$ (i.e. Force = mass $\times$ acceleration) $F(s) = ...
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Causality of z-transform $a^nu[n+1]$

To preface, this is not a homework related question but purely for self-study purposes. I'm try to do the analyse of z-transform of $a^nu[n+1]$. It is clearly a non-causal signal, I try to explain it ...
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1answer
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Derive the Forward Euler substitution for transfer function

In the book "The control handbook. Volume 1 " by Levine, the author shows that the transfer function: can be aproximated and discretized in the transfer function: using the forwar euler integral ...
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1answer
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Proof of Forward Euler for discretizing a transfer function

In Levine book "The control handbook" it is shown that, for discretizing a transfer function $\frac{1}{s}$ using Forward Euler i simply have to replace s with $\frac{z-1}{T}$. How can extend the ...
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Forward Euler Discretization

I don't understand why the substitution $s=\frac{z-1}{T}$ allows us to discretize a transfer function from laplace to z-transform through Forward Euler Discretization. Can you explain to me ?
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Non-polynomial Z-transform

My prof said that when a transfer function described by a z-transform is not polynomial, then i can't perform the anti-transformation. But, what does it means to be not polynomial ? Can you explain to ...
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1answer
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Impulse response of a system in z domain

The question 3-23 in the "Discrete-Time Signal Processing - Second Edition" is: and the solution is: I cannot understand the solution. In the second row of the answer when I multiply (-4) with ...
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1answer
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why z-transform differentiation is needed?

Find the z-transform and sketch pole-zero plot and ROC for $$x(n)=|n|\left(\frac{1}{2}\right)^{|n|}$$ Right now, i can get the following, $$x(n)=\begin{cases} n(\frac{1}{2})^n, & n \geq 0 \\ -n(...
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1answer
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Inverse Z transform of a left sided signal [closed]

Please help me to find the impulse response $h(n)$ of a Transfer function $H(z) = \dfrac{1+\frac{1}{6}z^{-1}}{1-\frac{1}{4}z^{-1}}$ given $h(n)$ is left sided.
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transfer function and 'causal' signal - evaluate transfer function or use z-transform of input?

From my studying difference equations and transfer functions, I understand that when a complex exponential input $x[n]=z_1^n$ is applied to an LTI system with transfer function $H(z)$, determining the ...
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2answers
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How do I convert this simple Laplace equation to Z-domain?

A basic model of coupled strings (eg. piano) is provided here as: The principle is that it has two identical string simulations each formed by a delay line and LPF. The outputs of these are summed at ...
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1answer
239 views

Impulse response of a second order LTI

I have a set of measurement which I want to model with 2nd-order difference equations (first order eqs don't model well enough). The equation is $$y[n] = \alpha_1 y[n-1] + \alpha_2 y[n-2] + \beta_0 ...
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LCCDE in simple words?

What is LCCDE?I only know its abbreviation/full form :linear constant-coefficient difference equation I know that in s domain we have differential equations and in z domain we have difference ...
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1answer
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Effect of sign change on ROC & z transform?

When we take ztransform of unit step it is z/[z-1] And our ROC is |z|>1 But if some how the minus sign between z and 1 changes to +, will our ROC still be same as old ( |z| >1 ) or will it be ...
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1answer
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How we can find ROC(Region of convergence) given a signal in z domain?

I read that ROC is a region,which is a set of values where z transform is defined ,that is it converges Lets say i have a discrete time time signal $x[n]=n^2 u(n)$ and i want to find its ROC(Region ...
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Are discrete time functions expected to be this inaccurate?

I am attempting to test some discrete time functions against their continuous counterparts and I am surprised by how inaccurate the discrete versions are coming out. I am modeling the displacement, ...
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461 views

Design discrete controller for zero steady state error

I have the following system where $$G(s)=\frac{0.5}{s+1}+\frac{5}{s+10}$$ How can I design the C(z) controller so that the steady state error for a step input r(t)=1(t) is zero? I know that this ...
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1answer
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How to do this inverse Z-transform? Is this correct?

It was explained to me here that I need to do an inverse z-transform on a z-based equation in order to get something I can use. That example was quite simple. But I'm not sure how to do the same ...
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What does z^(-1) represent here?

I understand that when you have a Laplace function, you can do a bilinear or forward/backward Euler substitution for $s$ to phrase it in terms of $z^{-1}$. In a typical filter, $z^{-1}$ represents the ...
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What type of z transform is this?

I am trying to understand this equation: It comes from: $Force = mass * acceleration$ $F(t) = m * a(t)$ $F(s) = m * (s^2 * y(s) - s*y0 - y0)$, where $y0=0$ $F(s) = m * s^2 * y(s)$ $y(s) = F(s)/(...
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1answer
186 views

Why do we care about the multiplicity of poles and zeros in rational Z-transforms?

In the DSP class I'm taking, a lot of the questions ask me to list the multiplicity of a pole or zero in a rational Z-transform. For example, the multiplicity of the zero at $z=1$ in :$\frac{(z-1)^2}{(...
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1answer
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How can one know the polynome equation based on the output of a system?

I came across this situation in my textbook: However I have no clue about how you can (starting from the stepresponse on the left), get the polynome equation on the right. Could someone please ...
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ROC of Z Transform of $x(n) = 2(3)^nu(-n)$

Using definition, I got its Z transform as $X(z) = \dfrac{2}{1-\dfrac{z}{3}}$ and the summation converges only when $|z|<\frac{1}{3}$. So its ROC is $|z|<\frac{1}{3}$. But my question is: for ...
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What are the missing steps in the derivation of this equation?

I am trying to understand the model of a piano hammer described here. The relevant excerpt is: As I understand it, they are saying in (12a) that if the hammer spring's uncompressed end's y position ...
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s-Domin or z-Domain - What to Use for Mixed systems

I recently had to deal with a power electronic system where I had to implement a dynamic model of a power converter in order to design a suitable controller for that converter. The control will be ...

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