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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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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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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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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62 views

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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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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32 views

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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329 views

Justification of bilinear transform

I would like to understand the "justification" for the bilinear transform. The basic idea as I understand it is that by integration rule of Laplace transform we have for continuous $y(t)$: $$\mathcal{...
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363 views

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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Frequency warping when integrators are replaced with backward-euler and forward-euler integration

Resonant controllers are used in the power industry. The transfer function is $G_{res}(s) = \frac{K_i s}{s^2 +2\omega_c s+ \omega_o^2}$ The "textbook" discrete implementation is depicted in ...
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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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Help with my first (simple) Z-transform

I need to transform this Laplace function to the z-domain: From the answer I received: $s=(1-z^{−1})/T$ Then substitution into my Laplace function would give: $t(z) = 2R/(m*(1-z^{−1})/T + 2R)$ Is ...
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I often see here formula expressed in term of $z$. But what is $z$?

While searching resources for generating pink noise (and with your help in the comments and answers of other questions), I came to such kind of formula: $$ H(z) = { .041 - .096z^{-1} + .051z^{-2} - ...
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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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The z factor in polyphase decomposition

I'm a beginner in DSP and I'm going through the textbook of Oppenheim's Discrete Time Signal Processing. There are two figures in the text, one which I can visualize, and the other I can't. The first ...
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Is it impossible to determine the inverse Z-transform without any other information?

Suppose I give you this as my transfer function $H(z)$: $$ H(z) = \frac{1} { 1 - az^{-1}}$$ With no other information given, is it even possible to determine the inverse Z-transform? The reason I'...
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How can I obtain the response signal for this question?

In particular I am having trouble with 6b). From what I understand, we can split a difference LTI equation into two sums, the sum of the previous responses, and the sum of the previous inputs. ...
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ROC and impulse response

For the LTI system given below, there are three regions of convergence. $$H(z)=\frac{5-3z^{-1}}{1-\frac53z^{-1}-\frac23z^{-2}}$$ a) Find all possible regions of convergence for this filter. b) For ...
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Find the length of the impulse response of a Linear Phase Type 4 FIR filter

The length should be found such that the group delay is minimum It is given that the impulse response is real. One zero of the Transfer function is at $0.6e^{j\frac{\pi}{4}}$, and another one is at -...
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transient and steady-state response of first order system

Considering this general 1st order transfer function $$ H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}} $$ How to find (analytically) the transient and steady-state responses? With steady-state response I ...
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Does separation of variables of a transfer function followed by a z-domain transform work?

I tried taking a low pass filter transfer function in the form $A/(s^2+Bs+C)$ and separating it into multiple fractions over each root $(D/(s+r1))+(E/(s+r2))$. Then I substituted $s =((2/T)*((z-1)/(z+...
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Understanding convolution of Chirp z algorithm

I dont´t understand how works the convolution part of the Chirp z. I understand how the DFT is transformed \begin{align*} x(k) = \sum_{n=0}^{N-1} x(n) W_N^{kn} \end{align*} to this expresion: \...
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Understanding a lowpass - comb filter implementation

I try to understand the implementation of the low-pass comb filter of the Freeverb reverberation algorithm: https://ccrma.stanford.edu/~jos/pasp/Lowpass_Feedback_Comb_Filter.html The original ...
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293 views

Two different answers while doing Inverse Z-transform

Given, $$X(z) = \frac{z}{3z^2 - 4z + 1}$$ Question 1 I need to calculate inverse z-transform for ROC $|z|>1$ When I try to calculate inverse $z$-transform using partial fraction of $X(z)$ and ...
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non-uniform antenna array design

There are tons of papers discussed this topics and most of them are related to fancy optimization techniques (convex/non-convex). I am wondering if we can have a simpler way to find the antenna ...
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Why there is Difference between shapes of ROC of z domain and s domain?

ROC(region of convergence) of Z domain is shown by a circular region while ROC in S domain is shown by a rectangular(approximately looking like rectangle) region What is the reason of this difference ...
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Powers of $z$ in a discrete transfer function

I have been working with discrete-time systems and I am not yet able to understand the reason behind using powers of $z^{-1}$ in transfer functions. I get that $z^{-1}$ means a delay, but when we ...
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What is prewhitening filter mode?

In this paper, the following prewhitening filter is described: $$ C(z) = \sum_{k=0}^n c_{k}z^{-k} $$ where $n$ and $c_k$ are known. The paper also describes the values $C(\lambda_{k})$, with $\...
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Why not use a complex number in the exponent in the z-transform?

I am trying to comprehend how the z-transform has come to be similar but different from its continuous counterpart, namely the Laplace transform. It seems to me that the most parallel and intuitive ...
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60 views

Missing delay in heavyside step function

I found the following task that was inspired by an example in the book A. V. Oppenheim and R. W. Schafer, "Discrete-Time Signal Processing", 3rd Edition, 2014. Task: Consider the 2nd-order IIR ...
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How to detect algebraic loop in a system

I came across a system (screen below) that looks like it should result in a algebraic loop, but when writing the equations using the Z-transform, it obviously does not (calculation below too). But I ...
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Where to start in the design of my filter to remove 50 Hz

pick any 3 random files from this database, : https://physionet.org/pn3/ecgiddb/ They are subject to 50 Hz powerline interference. We wish to convert from time domain to frequency domain and remove ...
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130 views

Usefulness of Matched $z$ transform Method

I'm aware that the matched $z$ transform method maps between the continuous $s$ plane and the discrete/digital $z$ plane but my question is - when would this be necessary? Why would we need to convert ...
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Z transform - Inverse System function - Why number of poles and zeros myst be equal?

I know that if a system is causal then the system function H(z) must have : a) a ROC that spans from the exterior of the most distant pole and b) the number of zeros must not be greater than the ...
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Contour Integral and Residue Theory for Inverse $z$-Transform

I'm aware that the inverse $z$-transform can be evaluated using contour integration which leads to the use of Residue Theory as a corollary and I do know of the two definitions. My question is how ...
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The range of r can be r<1 and r>1

I have to find the range of r which makes H(z) stable. There is no restriction of left sided or right sided. Then, both r<1 (z>r) and r > 1 can be the answer?
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Cepstrum Calculation of Rational Function H(z)

I am trying to solve my first problems at cepstrum calculation. I want to calculate the complex cepstrum $\hat{h}[n]$ of a signal $h[n]$ with Z-Transform: $$H(z)=\frac{(1-0.5z^{-1})(1+4z^{-2})}{(1-0....
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273 views

What is the type number of a discrete time system given $H(z)$?

Given a continuous time impulse response $h(t)$, if I take the Laplace transform and count the no. of poles at origin, that gives the type number of the system. For e.g., $$H(s) = \frac{2}{s(s+2)}$$ ...
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Plane Settings of the Matched $z$-transform Method

I've come across that the matched $z$-transform maps poles of the $s$-plane design to locations in the $z$-plane. My question is, what is the $s$-plane and what does this mean? I'm aware that the ...
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Time Setting of $z$ and Laplace Transforms

I'm aware that the z-transform and the Laplace Transform have an analogous relationship but I want to be doubly-sure that the z-transform only works in discrete-time and that the Laplace transform ...
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598 views

Z domain transfer function to difference equation

I want to convert this transfer function: $$\ \frac{2\cdot(z-0.5)\cdot(z-0.6)}{z-1}$$ to a difference equation, but I have no idea how. Can someone help me? Thanks, Arjon
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Region of Convergence

In attached image why does the ROC have these values for $$ X(z) = \frac{1}{1-\frac{1}{3}z^{-1}} - \frac{1}{1-2z^{-1}} ~~~~~,~~~~~ 1/3 < |z| < 2 $$ and for $$ Y(z) = \frac{5}{1-\frac{1}{3}z^...
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“Dirac Comb” vs “Ones Comb”

While learning sampling theory - I noticed that examples of continuous signal sampling always achieved the goal via multiplying the signal with a "Dirac Comb". I was intrigued by the requirement to ...
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How to determine if a filter is bandpass/stopband from its pole-zero diagram in z-domain

How can we determine if a filter is bandpass or stopband, just by looking at its pole-zero diagram in z-domain? For exmaple, if we have a system with third-order pole at the origin and a zero on the ...
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Causal unstable system turn into stable anticausal?

I would appreciate it very much if someone would be able to provide some clarity, help or comment on this problem. I have been reading several papers on time series identification such as https://www....
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Double check z-transform ROC of $a^nu[n]$

For the z-transform ROC of signal $a^nu[n]$, it has been computed to be $|z|>a$. For example (as I have found on Wikipedia), the signal $(\frac{1}{2})^nu[n]$'s ROC will be $|z|>\frac{1}{2}$, as ...
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z-transform help

I'm trying to solve this exercise: And the solutions manual states that the resolution is this one: but I can not understand the last step, which is indicated with an arrow. Also, how do you find ...

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