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.
1
vote
1answer
147 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
$$y[n] = \alpha_1 y[n-1] + \alpha_2 y[n-2] + \beta_0 ...
1
vote
1answer
59 views
Steady state value of a complex convolution
I am trying few problems on the introductory part of DSP. One of the problem asks to calculate the steady state response of a system with impulse response $h[n] = (\frac{j}{2})^{n} u[n] $ to an input ...
1
vote
1answer
47 views
Evaluate the Z Transform
Evaluate the Z transform of $x[n] = n^3$ where the signal is two sided.
I have tried using the basic definition of the Z transform ie.,
$$X(z) \triangleq \sum_\limits{n=-\infty}^{+\infty} x[n] \, z^...
0
votes
2answers
345 views
Z transform stability
What is the causality & stability status for three cases shown (aso in attached photo) ?
$$H(z) = \frac{z(z-1)}{(z+1)(z+\frac{1}{3})} $$
for three possible regions of convergence as:
a-) |z| > ...
1
vote
1answer
186 views
How to determine poles and zeros of the z-transform?
This is a simple question, but I just don't understand how we determine the poles and zeros of a rational system function.
For example, for the LTI system described by this constant coefficient ...
0
votes
1answer
53 views
How does an IIR system affect magnitude and phase of a sinusoidal signal
Consider an IIR system with impulse response $h[n]=\left( \frac{1}{\sqrt{3}} \right)^n u[n]$. If I apply $x[n]=\cos(n \frac{\pi}{2} + \varphi)$ at the input, how can I determine the change in ...
0
votes
1answer
76 views
Transfer function of resonant filter with 2 poles, peak at $f_0 = 500\text{ Hz}$, and $\Delta f = 32\text{ Hz}$
This a contest question. I'd like some help because I can't find any materials related to this topic.
https://www.qconcursos.com/questoes-de-concursos/questao/ecd6c966-51
My english translation:
...
0
votes
1answer
68 views
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 ...
1
vote
2answers
132 views
How to compute impulse and frequency response of Flanger?
I have to the implementation of a Flanger effect on Matlab, buy previously I have to plot its frequency response and impulse response.
The difference equation is $y[n]=x[n]+a\cdot x[n-d[n]]$
where $a$...
1
vote
1answer
64 views
Poles of the transfer function in Z Transform
Given that:
$H(z)$ has 4 poles maximum.
$H(z)$ has a pole at $z_1=a+bi$
Given that the impulse response $h[n]$ is:
Symmetric: $h[n] = h[-n]$
Real: $\forall$$n$ , $h[n]$$\in$$\mathbb{R}$
How we can ...
0
votes
1answer
62 views
What happens when the poles of this z-transform function are outside the ROC for a signal?
I am given a z-transform function for a signal $h[n]$. It's $H(z)=\frac{2z^2-0.75z}{(z-0.25)(z-0.5)}$. I am supposed to find $h[n]$ and check the stability of the system for these cases:
a)ROC: $|...
1
vote
1answer
255 views
How to perform this spectral decomposition in MATLAB?
Given a filter $X(z)$ I want to find $G(z)$ such that it is stable, causal and minimum-phase, and it accomplishes that $$X(z)=K_0G(z)G^*(1/z^*)$$
where $K_0\in\mathbb{R}$. Of course, $G^*(1/z^*)$ ...
0
votes
2answers
71 views
Discrete filter $y[n] = \frac{1}{3} x[n] + \frac{1}{3} x[n-1] + \frac{1}{3} x[n-2]$
Consider the filter which equation can be represented by $y[n] = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]$, in $x[n]$ and $y[n]$ are sequence of input and output of the system ...
2
votes
1answer
57 views
Find a stable transfer function $G(z)$ such that $|G(z)| = |H(z)|$
Consider the following causal IIR transfer function:
$$ H(z) = \frac{2z^3 - 4z^2 + 9}{(z-3)(z^2+z+0.5)} $$
Is $H(z)$ a stable function? If it is not stable, find a stable transfer function $G(z)...
0
votes
1answer
1k views
Z-domain transfer function to difference equation
So I have a transfer function $ H(Z) = \frac{Y(z)}{X(z)} = \frac{1 + z^{-1}}{2(1-z^{-1})}$. I need to write the difference equation of this transfer function so I can implement the filter in terms of ...
1
vote
1answer
246 views
Impulse response of a causal system from transfer function in z-domain
The transfer function is $$H(z)=(z+1)/(z^2+z+0.5)$$
I need to find the impulse response h[n] of a causal system with x[n] as unit impulse.
I have tried to find the impulse response by the following ...
0
votes
1answer
49 views
Explicit a succession using z inverse transform
Is it possible to explicit $y(n)$ of this mathematical succession in recursive form using z inverse transform?:
$ y(0) = 1 \\ y(n+1) = 2y(n) + 3 $
I can't write ...
1
vote
0answers
38 views
Finding Z transform of a signal: Intermediate steps
Find the Z transform of
$y(n)=x(n+2)u(n)$
I have solved the problem. I have doubt whether it is correct or not. It would be very helpful if someone could check whether the steps that I have ...
0
votes
0answers
74 views
Question regarding ROC of transfer function
I've been trying to understand how to determine the Region of Convergence (ROC) of $H(z)$ given $X(z)$ and $Y(Z)$ for some time, and just can't wrap my head around it.
I know that $Y(Z) = X(z)H(z) \...
0
votes
0answers
72 views
How do I convert a two-pole two-zero transfer function from the s-domain to the z-domain?
I'm trying to convert the following IIR transfer function from the s-domain to the z-domain:
$$
H(s) = \displaystyle\frac{\frac{s^2}{\omega_z^2} + 2\zeta_z\frac s\omega_z + 1}{\frac{s^2}{\omega_p^2} + ...
1
vote
1answer
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 ...
1
vote
2answers
246 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 ...
0
votes
1answer
52 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 $\...
1
vote
2answers
310 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 ...
1
vote
1answer
138 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 ...
0
votes
1answer
1k 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 ...
0
votes
1answer
151 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 ...
1
vote
1answer
115 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 ...
0
votes
1answer
40 views
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.
$$\frac{U(z)}{\bar{U}(z)} = G(z) = \frac{1-p}{z-p} \quad \leftrightarrow \quad u(k) = \bar{...
2
votes
1answer
84 views
Transient response of system with single pole $0 \le p < 1$
$G(z) = \frac{1-p}{z-p}$
If the value of p satisfies $ 0 \leq p < 1$ there are no oscillations
in the transient response.
Question: Why is that $\uparrow$ true? I know roughly what a ...
0
votes
2answers
108 views
How to find $h[n]$ system response of this equation?
$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response.
I transformed this system via using Z transform:
$$\frac{Y(z)}{X(z)}=H(z)=\frac{z^4 - 10z^...
-1
votes
1answer
53 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^...
2
votes
0answers
81 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 ...
1
vote
0answers
48 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 ...
3
votes
2answers
105 views
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 $$X(z)=\frac{1}{G(z)...
-3
votes
1answer
53 views
$\mathcal Z$-Transformation in Discrete Time [closed]
I want to find the inverse $\mathcal Z$-transform of this, in discrete time:
$$X(z) = \frac{1}{1+3z^{-1}+2z^{-2}}$$
0
votes
3answers
101 views
LTI system phase response given $z$-transform
I have been given this question
\begin{equation}
H\left(z\right)\:=\:\frac{1}{6}\left(1+z^{-2}\right)^6
\end{equation}
(a) Compute and plot the phase response of the system.
(b) Determine ...
0
votes
3answers
1k views
How to compute the Laplace transform of a discrete signal?
Assume I have a discrete random signal, $f(t)$ for which I want to calculate the laplace transform.
How can I do it in matlab without using sym variables, for ...
1
vote
1answer
469 views
Bilinear transformation confusion
Wikipedia says in bilinear transformation from \$s\$ domain to $z$ domain relation is
$$\boxed{s \longleftarrow \frac{2}{T}\frac{z-1}{z+1}}$$
But here this relation is given like this
$$\boxed{w=\...
1
vote
2answers
193 views
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.
...
2
votes
1answer
142 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 ...
5
votes
1answer
235 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 ...
4
votes
1answer
8k views
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 $\mathcal Z$-...
3
votes
2answers
2k views
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.
0
votes
1answer
3k 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. ...
0
votes
1answer
347 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
\...
1
vote
1answer
752 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 ...
0
votes
1answer
167 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
$$E(z)=\frac{1}{z+...
0
votes
1answer
532 views
ROC of transfer function
given:
$$ H(z) = \frac{4z(z-1)}{z-0.5} $$
I would say, when all poles are in the unit circles, the impulse response is right sided and causal.
-1
votes
0answers
148 views
Inverse Z-Transform of a Complex Filter
What is the inverse z-transform of
$$
1/(1-az)
$$
where a is complex and |a| < 1
