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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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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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35 views

Absolute Value z Transform

just a quick question on how to perform z transform and provide the ROC when the sequence is: $$x[n] = 5|n|$$ Thanks!
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37 views

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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32 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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21 views

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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28 views

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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125 views

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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35 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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Linear Combiner Based LMS Transfer Function

I am currently working on narrowband beamforming and looking to compute the transfer function for a linear combiner based LMS. While doing a quick search I have found the following article which ...
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24 views

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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74 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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39 views

Please explain Multiplication property in Z transform?

I am having problem visualizing contour any example will be great help. As far as as where I need it I was trying to find Z transform of a contracted signal by first multiplication by impulse train.
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46 views

Region of convergence for discrete system

I am confused by two questions on the region of convergence (ROC) when applying the $Z$-transform. Consider a discrete-time linear system $$x[n+1] = Ax[n]+Bu[n], \qquad y[n]=Cx[n]+D$$ whose transfer ...
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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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Determine filter type using recurrence relation

Given the recurrence relation: $y[n] = x[n] + 0.5y[n-1]$ I want to determine the filter type (i.e. LPF, HPF etc.) I try to use Z transform, and get that the the transfer function is $H(z) = \frac{2z}{...
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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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Why is the Z-transform so important in digital filters analysis and design? [closed]

Please elaborate on why this mathematical transform can help analyzing as well as designing any type of digital filter.
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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 symmetry does system function H(z) have if h[n] is real?

assuming h[n] is real... If frequency response $H(e^{j\omega})$ is Conjugate Symmetric: $$ H(e^{-j\omega}) = H^*(e^{j\omega}) $$ $$ H(e^{j\omega}) = H^*(e^{-j\omega}) $$ Then, what symmetry does ...
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31 views

Squared magnitude of System Function H(z)

If: $$ |\alpha|^2 = \alpha \alpha^* $$ Then, why does: $$ |H(z)|^2 = H(z) H(z^{-1}) $$ instead of: $$ |H(z)|^2 = H(z) H^*(z) $$
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RoC of Z transform of signal consisting of 3 values

Let's consider the signal $x[n]=\{x[1],x[2],x[3]\}$ It's $\mathcal{Z}$ transform is $\frac {x[1]}{z}+\frac {x[2]}{z^2}+\frac {x[3]}{z^3}$ My textbook says it converges for all values of $z$ but ...
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Hidden zero in system equation $H(z)$

An FIR linear phase filter has unit sample response $h[n]$ that is real with $h[n]=0$ for $n < 0$ and $n > 7$. If $h[0]=1$ and the system function has a zero at $z=0.4e^{j\pi/3}$ and a zero at $...
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graph of poles at the same location

Just wanted to make sure: This cases would have three poles at the exact same location of (z=1) on the complex plane? $H(z)=\frac{1}{(z-1)^3}$ But this case, would have three poles spread out on a ...
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system function $H(\omega)$ relationship to odd and even components of h[n]

What qualities of $h[n]$ are necessary for: $$ H(e^{j\omega}) = DTFT\{h_{even}[n]\} + j\ DTFT\{h_{odd}[n]\} $$ Do all real / causal h[n] have the property that: $$ H(e^{j\omega}) = DTFT\{h_{even}[n]...
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103 views

Is the inverse of a causal system also causal?

If I have a causal system H(z) and I find the inverse of this system: $$ G(z) = \frac{1}{H(z)} $$ Is G(z) also causal?
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ROC of inverse system function

If the region of convergence (ROC) for system function $H(z)$ is $R_h$, what is the ROC of the inverse function $G(z)=\frac{1}{H(z)}$? $$$$
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z-transform causality properties: negative coefficents are zero ($x[-1]z^1=0$, $x[-2]z^2=0$, …)

Let's suppose I have a system: $$Y(z)=X(z)H(z)$$ If the system is causal, does that mean that all the negative coefficients (example: x[-1]) of the transform for $Y(z)$, $X(z)$, and $H(z)$ are zero?...
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chirp z-transform for different output sizes

I am attempting to use the chirp z-transform for an application that requires arbitrary FFT output sizes less than or equal to the length of the input signal. However, I've encountered an issue where ...
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Is this calculation of inverse z-transform proper

I wonder whether my calculation of inverse z transform are correct. My IIR system is described as follows in Z-domain $H(z) = \frac{z^{-2}}{1-0.5z^{-2}}$ After using partial fraction decomposition I ...
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Is “initial value theorem” sufficient to prove causality of a system?

Textbook states Initial value theorem as follows: if $x[n]$ is equal to zero for $n < 0$, the initial value, $x[0]$, may be found from $X(z)$ as follows: $$ x[0] = \lim_{\ z \rightarrow \...
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25 views

What is the sequence type if ROC is an annulus in the z-plane

if: $|z|>|\alpha|$ is causal and right-sided and: $|z|<|\alpha|$ is anti-causal and left-sided then, what is the annulus/donut-ring ROC? $|\beta| < |z| < |\alpha|$ non-casual? ...
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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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2answers
70 views

Trouble with inverse Z-transform and calculating of samples

I have a little problem. I have to solve this task but I can't. Z-transform of sequence $\{x(k)\}$ describe by the formula: $$X(z) = \frac{2.5 -3.15z^{-1} + 1.2 z^{-2}}{1-2.3z^{-1} + 1.2z^{-2}...
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172 views

z-Transform Methods: Definition vs. Integration Rule

The definition of the z-transform is defined as $z = e^{sT}$ where "s" is complex frequency for continuous-time systems and "T" is the sample period. Why are rules such as the forward rectangular ...
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How do I examine if the signal can be z-transformed?

It's given signal $x[n]=\sin(\frac{2 \pi}{N} m n) u[n]$ where $u[n]$ is the unit step function. Can I calculate Z-Transform? $\mathcal{Z}$ transform exists when $$ \sum_{n=-\infty}^{\infty} x[n]z^{...
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Inverse z-transform of a modulus square

Suppose we have the z-transform of $x[n]$ is $X(z)$ and that of $y[n]$ is $Y(z)$. Then we know that the inverse z-transform of $G(z)=X(z)Y(z)$ is $g[n]=x[n]*y[n]$, where $*$ is convolution. What will ...
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71 views

Determining asymptotic stability using transfer function?

In an exam task, I am asked to determine the transfer function of the following direct-time system and decide whether it's stable. I think this system is canonical and the amplifiers 'on top' ...
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52 views

Calculating an output of a system (Z- transform question)

I have a following question to answer: An LTI system is described by its impulse response h[n]. For input x[n] it gives output y[n]. $$h[n] = u(n) - u(n-N) $$ $$x[n] = u(n) - u(n-M)$$ I want to ...
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100 views

Trivial and non-trivial zeros

I am new to DSP, and I'm self studying. Could someone please explain to me what do we mean by trivial and non-trivial zeros?
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26 views

Transfer function of a nonhomogeneous difference equation

Consider the following difference equation: $y_k=\alpha y_{k-1}+\beta x_k$ The transfer function for this is given by: $\displaystyle\frac{Y(z)}{X(z)}=\frac{\beta}{1-\alpha z^{-1}} = \frac{\beta}{...
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118 views

partial fractions expansion inverse Z-transform, help

I have the correct solution from teacher's solution guide, but I was slightly confused by some algebra about the partial fractions expansion evidently difference equation is as follows $ y[n] = \...
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30 views

solving recurrence relation with Z-transform (LTI and causal)

I just wanted to doublecheck answers for my sanity's sake (exam next week) problem statement recurrence relation, solve it $y[n+1]= 35 + y[n]*0.5$ according to my teacher it will be such that the ...