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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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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How can I find inverse Z transform using synthetic division method when Region of convergence is bounded rather than right sided or left sided?

Explanation of synthetic division method Above example has region of convergence of type mod(z)>mod(a) I want to solve problems when region of convergence of type mod(e)>mod(z)>mod(d)
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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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36 views

z-transform and DTFT properties

I actually do not understand what to do with the third property of the impulse response g[n] and how it has to be determined. Thanks in advance!
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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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38 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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36 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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22 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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30 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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137 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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39 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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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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86 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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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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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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58 views

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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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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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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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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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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217 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 ...