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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357 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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33 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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447 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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193 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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37 views

How is the ROC of a transform function determined?

Suppose I have $x[n]$ and $y[n]$, and I calculate their respective Z-transforms $X(z)$ and $Y(z)$ as well as their respective ROCs. Calculating $H(z)$ is as simple as calculating the quotient of $\...
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317 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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758 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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378 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
189 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: ...
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224 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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659 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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834 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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252 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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238 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 ...
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1answer
97 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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1answer
110 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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974 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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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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73 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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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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111 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 ...
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2answers
168 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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108 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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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: $|...
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628 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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90 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 ...
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73 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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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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524 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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60 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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113 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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117 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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483 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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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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424 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
300 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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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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320 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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187 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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106 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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161 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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1answer
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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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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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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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2answers
381 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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$\mathcal Z$-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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3answers
144 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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3answers
2k 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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