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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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372 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 ...
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479 views

finding inverse Z transform with usage of tables (LTI and causal sequences)

problem is as follows try to find the inverse Z $Z^{-1} (\frac{3}{z+2}) = ???$ with the usage of z-transform tables Ok, so in order to find something from the table, I thought that we expand with ...
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957 views

BIBO Stability in Z-domain

I'd really appreciate it if someone could please explain to me the condition for a LTI system to be BIBO stable, in z-domain. I have a background in control, and in linear control for example, if we ...
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587 views

Partial Fraction Expansion for Inverse Fourier Transform

In many textbooks, I've seen the application of Partial Fraction Expansion (PFE) to find an inverse Fourier Transform. Let's stick to the discrete time case, and let me give you an example. Let's say ...
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1answer
1k 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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1answer
237 views

How can I implement a triangular filter in MatLab, given it's Z-transform representation?

I have to implement an anti-aliasing filter for a certain processing step in MatLab. While searching in literature for inspiration, I came accross a paper in which the authors wrote the following: We ...
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2answers
250 views

Block diagram for a complex impulse response

I have this question regarding digital systems which might not handle sinusoidal signals in general. So let us say that I have a system with impulse response $h(n) = \{1+j, -1, 2j \}$ with first ...
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3answers
888 views

How to find the difference equation directly from Direct Form II signal flow graph

I am trying to solve for the difference equation of the following signal flow graph: I am aware that Direct Form II can be converted to Direct Form I, which finding the difference equation directly ...
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2answers
1k views

Stability of system with poles inside unit circle - conflict with differential equation

I am trying to understand why a system with a single pole inside the unit circle is stable. For example, take a system with one pole at $z=\frac{1}{2}$. The literature says the system is stable. As a ...
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2answers
271 views

Z Transform - Do i always need 2 poles for every “peak”?

I am quite new to digital signal processing (and also Z transforms). I am reading about frequency response modelling and have some questions do we always need a pair of poles for each "peak" in the ...
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1answer
137 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^...
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1answer
121 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 ...
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1k 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| > ...
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1answer
1k 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 ...
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2answers
287 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. ...
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81 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 ...
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219 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: ...
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1answer
665 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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78 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)...
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185 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 $a$...
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1answer
115 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 ...
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148 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: $|z|&...
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1answer
823 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, $G^*(1/z^*)$ ...
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206 views

Can the Z-Transform be used to create smoothed 3D surfaces from point clouds?

According to Dr. Math the Z-transform can create closed-form solutions for 1D series defined by difference equations (e.g. the Fibonacci series). My 3D surface $z=LC(x,y)$ is defined by difference ...
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1answer
64 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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1answer
1k views

How to find out the transfer function of a FIR filter?

$$h[n]=\begin{cases}a^n & \text{if } 0 \le n < N \\ 0 & \text{otherwise}\end{cases}$$ And for which values of $a$ the filter is stable I know that the transfer function will be $$H(z)=\...
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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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2answers
2k views

Minimum phase systems with pole at infinity

If a system is given by a transfer function in the $z$ domain that has all poles and zeros inside the unit circle except for a factor of $z^{-1}$ in the denominator (pole at infinity), can it still be ...
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148 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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1answer
120 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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2answers
587 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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58 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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2answers
452 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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1answer
386 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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1answer
3k 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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451 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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1answer
204 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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64 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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1answer
41 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{...
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1answer
110 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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215 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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53 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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135 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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3answers
152 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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1answer
396 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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2answers
619 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)...
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1answer
14k views

Z-Transform of a^|n|

I am wanting to compute the Z-transform of $f(n) = a^{|n|}$ . 'a' is a positive constant. Looking at the transform table, I found that Z-transform for $a^n u(n)$ is available from the tables and is $...
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3answers
49k views

How to compute magnitude and phase response from transfer function in Z-domain?

I have a transfer function $$H(z)=\frac{1+1.2z^{-1}+0.8z^{z^-2}}{1-0.9z^{-1}}$$ from which I'm supposed to sketch the magnitude and phase response. I know that you can transform $z=e^{j\omega}$ to get ...
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1answer
168 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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55 views

$\mathcal Z$-Transformation in Discrete Time [closed]

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

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