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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 perform this spectral decomposition in MATLAB?

Given a filter $X(z)$ I want to find $G(z)$ such that it is stable, causal and minimum-phase, and it accomplishes that $$X(z)=K_0G(z)G^*(1/z^*)$$ where $K_0\in\mathbb{R}$. Of course, $G^*(1/z^*)$ ...
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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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Find a stable transfer function $G(z)$ such that $|G(z)| = |H(z)|$

Consider the following causal IIR transfer function: $$ H(z) = \frac{2z^3 - 4z^2 + 9}{(z-3)(z^2+z+0.5)} $$ Is $H(z)$ a stable function? If it is not stable, find a stable transfer function $G(z)...
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584 views

Z-domain transfer function to difference equation

So I have a transfer function $ H(Z) = \frac{Y(z)}{X(z)} = \frac{1 + z^{-1}}{2(1-z^{-1})}$. I need to write the difference equation of this transfer function so I can implement the filter in terms of ...
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184 views

Impulse response of a causal system from transfer function in z-domain

The transfer function is $$H(z)=(z+1)/(z^2+z+0.5)$$ I need to find the impulse response h[n] of a causal system with x[n] as unit impulse. I have tried to find the impulse response by the following ...
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47 views

Explicit a succession using z inverse transform

Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?: $ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $ I can't write ...
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Finding Z transform of a signal: Intermediate steps

Find the Z transform of $y(n)=x(n+2)u(n)$ I have solved the problem. I have doubt whether it is correct or not. It would be very helpful if someone could check whether the steps that I have ...
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50 views

Sampling and ideal reconstruction of signal

Two time discrete signals $x_1(n)$ and $x_2(n)$ are produced by sampling the continuous signal $$x_a(t) = \cos(2\pi300t) + \cos(2\pi600t) $$ with the sample frequency $F_s = 1000\ \rm Hz$. For the ...
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160 views

Help needed with finding ROC of transfer function

I've been doing some practice with the $\mathcal Z$-transform for an exam, and I'm not sure if my approach is correct to this problem: My approach: I wrote $y[n]$ as follows: $$ y[n] = 2\cdot\...
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193 views

Confusion over impulse invariance, matched z-transform, and bilinear transformation methods

In the DSP course that I am taking in my university as an undergraduate student, three methods are presented for mapping analog filters to digital filters - namely, impulse invariance, matched z-...
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63 views

Question regarding ROC of transfer function

I've been trying to understand how to determine the Region of Convergence (ROC) of $H(z)$ given $X(z)$ and $Y(Z)$ for some time, and just can't wrap my head around it. I know that $Y(Z) = X(z)H(z) \...
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66 views

How do I convert a two-pole two-zero transfer function from the s-domain to the z-domain?

I'm trying to convert the following IIR transfer function from the s-domain to the z-domain: $$ H(s) = \displaystyle\frac{\frac{s^2}{\omega_z^2} + 2\zeta_z\frac s\omega_z + 1}{\frac{s^2}{\omega_p^2} + ...
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108 views

Design a filter which passes all frequencies except $\omega=\pm\frac{\pi}{2}$ and plot its pole-zero diagram

Also draw its normalized frequency response. What is the ROC? This has to be done in z-plane so there must be two poles at $+i$ and $-i$ since they cannot be included in region of convergence. Is my ...
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192 views

Autocorrelation of a Shifted Sequence

Suppose I have a sequence $x[k]$ with $\mathcal{Z}$-transform $$ X(z) = x_{0} + x_{1}z^{-1} + x_{2}z^{-2} + \ldots + x_{N-1}z^{N-1}$$ I know that for real-valued $x[k]$ the $\mathcal{Z}$-transform ...
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51 views

Inverse $\mathcal Z$-transform when region of convergence goes outwards from the inner pole?

I am looking for the inverse $\mathcal Z$-transform of the following: $$ \frac{1}{1-\frac 12 z^{-1}}+\frac{1}{1+\frac 13 z^{-1}} $$ When the region of convergence is $z > 1/3$. I have found the $\...
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282 views

Determine the stability of a system without using the $\mathcal Z$-transform (described by a difference equation)

For example, let's say a causal LTI System is described by the following equation: $$y[n] - ay[n-1] = x[n] - bx[n-1],\quad n \in Z$$ Is there a way to determine (in this case) the stability of the ...
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116 views

Where does this star came from? [EDIT - Detailed Question]

First of all, it's a rare topic in google, so I can't find intuitive explanation about step-invariant, so I don't understand actually what it is, except to solve zero order hold transfer function. In ...
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889 views

Marginal Stability based on Poles

We know that a discrete-time system with a (Z-transform) transfer function that has a pole of magnitude 1 (i.e. $|z|=1$ is a pole of the transfer function) is marginally stable if the pole at $z=1$ is ...
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123 views

How do I find the ROC of a system if it has no poles

The output of a system of discrete time $y[n]$ is corellated with the input $x[n]$ through the equation $y[n]$. $$y[n] = \frac 13\big(x[n-1]+x[n]+x[n+1]\big)$$ It then asks me to find the system ...
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99 views

Relation between time domain, DTFT domain and frequency domain

Problem The sampling frequency of a continuous-time signal is $S$ kHz, what does $\frac{\pi}{4}$ radians/sample in DTFT domain represent in Hz in frequency domain? Prove the relationship. Doubts I ...
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Unclear inverse Z-transform of $G(z) = \frac{1-p}{z-p}$

In this paper on page 5 equation (10) is supposed to be the reverse z-transform of equation (5) on page 4. $$\frac{U(z)}{\bar{U}(z)} = G(z) = \frac{1-p}{z-p} \quad \leftrightarrow \quad u(k) = \bar{...
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75 views

Transient response of system with single pole $0 \le p < 1$

$G(z) = \frac{1-p}{z-p}$ If the value of p satisfies $ 0 \leq p < 1$ there are no oscillations in the transient response. Question: Why is that $\uparrow$ true? I know roughly what a ...
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99 views

How to find $h[n]$ system response of this equation?

$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response. I transformed this system via using Z transform: $$\frac{Y(z)}{X(z)}=H(z)=\frac{z^4 - 10z^...
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49 views

Is $\mathcal{Z}\{4\delta[n-8]\delta[n-8]\} = 4z^{-16}$?

When I try to calculate the $\mathcal{Z}$-transform of $4\delta[n-8]\delta[n-8]$, I put the statement into the formula of $\mathcal{Z}$-transform from $-\infty$ to $+\infty$, and I get the result $4z^...
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75 views

How to transform a Fractional Order Laplace Transfer Function into a digital filter?

I'm working with loudspeaker impedance analysis. Electrical behavior of loudspeakers can be modeled with RLC networks. But real loudspeakers have components, that exhibit some non-linear and frequency ...
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47 views

periodicity, minimum phase, maximum phase, interpretation

I have a finite linear difference equation $$y(n)=ax(n-1)+bx(n-2)+cx(n-3)+\ldots+fx(n-m)\text,$$ relating an input $x(n)$ to an output $y(n)$. If I assume periodicity of type $x(n-2)=x(n)$, the ...
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When inverting a transfer function, solving for the input using the output does the causality status change

suppose $y(n)=ax(n-1)+bx(n-2)+\dots$ ($y$ is the output and $x$ the input). What happens if I want to solve $x(n)$ from $y(n)$? Z transform: $$Y(z)=G(z)X(z)\tag{1}$$ then $$X(z)=\frac{1}{G(z)...
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$\mathcal Z$-Transformation in Discrete Time [closed]

I want to find the inverse $\mathcal Z$-transform of this, in discrete time: $$X(z) = \frac{1}{1+3z^{-1}+2z^{-2}}$$
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91 views

LTI system phase response given $z$-transform

I have been given this question \begin{equation} H\left(z\right)\:=\:\frac{1}{6}\left(1+z^{-2}\right)^6 \end{equation} (a) Compute and plot the phase response of the system. (b) Determine ...
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1k views

How to compute the Laplace transform of a discrete signal?

Assume I have a discrete random signal, $f(t)$ for which I want to calculate the laplace transform. How can I do it in matlab without using sym variables, for ...
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405 views

Bilinear transformation confusion

Wikipedia says in bilinear transformation from \$s\$ domain to $z$ domain relation is $$\boxed{s \longleftarrow \frac{2}{T}\frac{z-1}{z+1}}$$ But here this relation is given like this $$\boxed{w=\...
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Invertibility of Room Impulse Response: Reproducing Research Paper

I have been trying to reproduce this paper¹. Few things which are unclear to me. The paper talks about finding whether a given Room Impulse Response(RIR) is invertible or not based on Nyquist plot. ...
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121 views

Can a unit delay in discrete time be represented by exponential functions?

If we have a signal $y[n]$ and its unit delayed version $y[n-1]$, can we write $y[n-1]$ in terms of $y[n]$ times some exponential? The reason I want to do this is to then take $y[n]$ common and ...
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213 views

Is there a z-transform like for variable sampling rate signals?

I'm working with signals with variable sampling rate (the time space between samples is not constant). I know the delay between samples but I don't wont to interpolate the signal. Is there a way to ...
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Position of poles and Stability in $z$ domain

We know in Laplace Transform, if the poles lie on the left of $j\omega$ axis, we can say the system is stable. Similarly can we comment on the stability based on poles position in $\mathcal Z$-...
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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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2k views

Determine whether the system is a FIR or IIR by looking the transfer function

I have the following system: $$ y[n]=\frac{1}{3}(x[n+1]+x[n]+x[n-1]) $$ After the Z-Transform we get $$ \frac{y[z]}{x[z]}=\frac{z^2+z+1}{3z} $$ which is of course the transfer function of the system. ...
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267 views

Z-transform of an impulse signal in discrete time

am trying to compute the Z-transform of the following signal \begin{equation*} x\left[n\right]\:=\:\sum_{k=-\infty \:}^{\infty \:}\:\delta \:\left[n-k\right] \end{equation*} so I thought it would be \...
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671 views

Minimum number of Poles and zero of transfer function H(z)?

Suppose $G(z)=H(z)(1-\frac{1}{2}z^{-1})$ now in question its saying ROC of G(Z) is entire Z plane except Z=0,so here we need not to add anything because G(Z) already a right sided signal with ROC ...
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152 views

Same z transformed function, but different answers of inverse z transform?

Given a $\mathcal{Z}$ transformed function $E(z)=\frac{1}{z+4}$. I know there are several ways to get the inverse $\mathcal{Z}$ transform of this function : Using partial fraction $$E(z)=\frac{1}{z+...
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448 views

ROC of transfer function

given: $$ H(z) = \frac{4z(z-1)}{z-0.5} $$ I would say, when all poles are in the unit circles, the impulse response is right sided and causal.
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138 views

Inverse Z-Transform of a Complex Filter

What is the inverse z-transform of $$ 1/(1-az) $$ where a is complex and |a| < 1
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1answer
399 views

Z transform of a function with delay?

i have this open loop system , and i've been asked to find out the response $C(kT)$ due to a unit step input. I am able to find the transfer function without the delay unit i.e $$\frac{C(z)}{R(z)}=\...
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1answer
46 views

which of the analog sinusoidal frequency can not pass through the filter?

First time I am encountering this type of question so i just tried but not getting whether my logic correct or not. First let the sinusoidal signal be $X(t)=\cos(2\pi ft)$. After sampling this ...
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1answer
317 views

How to find the inverse Z transform of this function in z domain?

i have a function $$F(z)=\frac{z-0.4}{z^2+z+2}$$ i need to find the inverse z transform of it , i have tried it with residues but the roots are too much ugly and it involves lots of messy ...
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96 views

Z-transform of alternating sequence

I'm having some difficulty in going through the z-transform of a sequence that is "on" every other sample. The sequence is $$x(n) = na^{|n|/2},$$ when $n$ is an even integer, and 0 otherwise. I have ...
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1answer
109 views

Analyzing a particular discrete-time LTI system for input signal $x[n]=(1/3)^n$ for *all* $n$

I'm considering the following problem from some course notes. Suppose the following is known about a discrete-time LTI system: Given the input $x[n]=(1/3)^n$ for all $n$, the system ...
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1k views

Transfer function and difference equations: why does $H(z)$ numerator polynomial not correspond to $Y(z)$?

For a discrete time LTI system, I understand that from a difference equation description of the system in the form $$ \sum\limits_{k=0}^N{a_k y[n-k]}=\sum\limits_{k=0}^M{b_k x[n-k]} $$ I can ...
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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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282 views

How does a digital filter $H(z)=1/(1-z^{-1})$ change a continuous rectangular pulse? [closed]

I inputted the following signal in this discrete time filter and got this output Could anyone explain how ? Also how does a discrete filter process a continuous time output ?