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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An example of a signal that neither have a $z$-transform nor a DTFT? [closed]

What is an example of a signal that do not have a $z$-transform or a DTFT? With explanation.
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How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?

Let $H(z)$ be the rational system function of an LTI system. How can we show that $|H(e^{j\omega})|=|H(e^{-j\omega})|$?
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Calculating the output of a pole eigen signal in a difference equation

Let an IAR system be defined by the following difference equation: $$y[n]-\frac{1}{4} y[n-2]=x[n]+3x[n-1]$$ and an input signal $x[n]=(-0.5)^n$. The transfer function is defined as $H^z(z)=\frac{1+3z^{...
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confusion related to finding inverse Z-Transform using Complex Integral Method

I am facing problems related to evaluation of inverse Z-Transform using Complex Integral Method; Consider $X(z)=\frac{z}{z-2} $ and $ROC: |z|>2$ then, $$x(n)= \frac{1}{2\pi j}\oint_c X(z) z^{n-1} \...
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Can we choose a sampling frequency to remove unwanted noise at a specific frequency?

I am studying for my exam in signal processing. In one of the old exam papers I am told to find a sampling frequency, which will remove 80 Hz noise. The filter the exam question is based around has an ...
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Why not use the same “standard” exponentials for both continuous and discrete time

In continuous time the standard exponential signal is usually defined as $$ e^{st}, \quad\text{with}\quad s = \sigma+j \omega $$ In discrete time the standard exponential signal is usually defined as ...
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Simple Filter representation in Matlab

Suppose I have an IIR filter in the $z$-domain in the following form: $$ H\left(z\right)=\frac{1}{1-0.2z^{-1}-0.1z^{-2}} $$ How do I represent this in MATLAB? I am pretty sure if I just listed the ...
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38 views

Magnitude response of mirrored (with respect to unit circle) poles and zeros

I just want to check that my understanding about the following paragraph from Optical Filter Design and Analysis by Christi K. Madsen, Jian H. Zhao is correct: A filter’s magnitude response is equal ...
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1answer
54 views

Squared magnitude of the Z-transform

I am basically new to the $z$-transform and there are some points regarding its square magnitude that I do not understand. Basically I do not understand how in slide 4 of PDF, they arrive at the ...
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Uncountable Set of Poles?

It is easy to define an (ideal) LTI system that would have an infinite number of poles - for instance, if the transfer function is $$ H(z)=\frac{1}{\cos(z)-1} $$ However, this would only define a ...
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Why can't this system be adequately represented using a z-domain transfer function?

According to this question and answer the following system cannot be adequately captured by a z-transform transfer function. $$y[n] = y[n-1] + F_{\psi}(y[n-1)) + F_{\phi}(x[n-1])$$ where $F_{\alpha}(z)...
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How can initial conditions be taken into account to calculate a system's terminal value using the final value theorem or some other technique

I wish to calculate the Final Value of the following system given initial conditions not at rest, and assuming the X will not change from its initial condition value. The transfer function is $$H(z) = ...
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Discrete time Final Value Theorem applied to feedback system

I wish to calculate the Final Value of systems in which a high pass filter of the output feeds back into the input. A simple example would be: where is a 1st order high pass filter with transfer ...
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1answer
55 views

Single-sided Z transform with difference equations and the system function

I am working on this problem: Given an impulse response, find the system function, find the difference equation representation, find pole-zero plot, find output $y[n]$ if the input is $x[n] = 0.25^n *...
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Disjoint ROCs of input and system function

I am trying to understand how the output behaves when the input and the system function do not have a common region of convergence (ROC) for an LSI system. Consider an LSI system with $x[n]$, $h[n]$, $...
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Inverting a sampled system

I'm doing some self-study for an upcoming exam and came across the following question: My first idea to solve it was using the bilinear transform to get some approximation of $H(Z)$ (or just using ...
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Why is Fourier space not adequate for (theoretical or digital) filters?

As far as I have seen, almost all theoretical filter design occurs in Laplace or Z-space. Also, there is a pervasive connection to real life analog filters in the design. If one is just thinking in a ...
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How to find inverse z transform

Suppose $$Y(z) = \frac{\frac 12 z + 1}{z+\frac 12} \cdot \frac{z}{z-\frac12}\text.$$ According to Wolfram Alpha the inverse transform is, $2^{-n - 2} \cdot(5 - 3 \cdot (-1)^n)$. However, I cannot show ...
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finding power spectral density from a vector

I have been given a vector: \begin{equation} v= \:\begin{pmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{pmatrix} \end{equation} my job is to find the power spectral density from this vector \...
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Stability of $x(n) = A x(n-1)+b$

I am looking at the following system: $x(n) = A x(n-1) + b$ where x and b are vectors and A is a matrix. How can I derive the stability and causality conditions for such a system using Z transform? If ...
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ROC of Z transform Doesnt include a pole on the boundary?

I cannot figure out what is going on here. I have an example problem in my book that says the ROC of a certain function is $$ 0.5 < |z| < \infty$$ The function's denominator is $$ 1 - z^{-1} + 0....
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Find the length of the impulse response for the given output and input

Homework Question: Consider a signal $x[n]=\alpha e^{j \omega_{0} n}+\beta e^{j \omega_{1} n}+\gamma e^{j \omega_{2} n} .$ What is the length of impulse response $h[n]$ of a system (non-trivial) such ...
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Calculation of an impulse response of h[n]

I am currently looking at the z-transform and am using a great youtube reference to help me, however I am struggling on some basic step. How do I get the impulse response array of h[n] = [ ... ] shown ...
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Confusion in property of Z transform of ideal sampled waveform

I was reading about z transfom of ideal sampled signals and one of the properties of Z transform of sampled signal that surprised me,here it is (image) So here this property of Z transform is quite ...
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Alternative function for MatLab iztrans to Octave?

I am trying to compute the reverse Z transform on Ocatve and I get the following error: error: 'iztrans' undefined near line 1, column 1 The code I am running is ...
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1answer
50 views

Summations in Z-Transform

I'm currently working on a problem that involves a Z-Transform. Basically, the essence of the problem is that if: \begin{equation} H\left(z\right) = \sum_{n=0}^{N-1}h\left(n\right)z^{-n} \end{equation}...
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What is the Z-transform of $0.8^{n+2}u(n-1)$?

I have 2 signals. One is $x(n)=(-0.5)^nu(n)$ and the other one is $y(n)=0.8^{n+2}u(n-1)$. I know that for the first one it is $X(z)= 1/(1+0.5z^{-1})$, but what about the other one? I know $y(n)$ is ...
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Calculating IIR Filter gain at given frequency

Let's consider an IIR filter with transfer function: $H(z)$. Given the sampling frequency $F_s$ how can I calculate gain at say $F$ ? When I was dealing with analog systems when I wanted to calculate ...
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66 views

Can time-invariance be determined from a given a transfer function?

I've the following function. $$ G(z) = 2 + \frac{-1+5z^{-1}}{(1-0.5z^{-1})(1-z^{-1})}$$ Calculating it's inverse using $\mathcal Z$-Transform, I get the following function: $$g[n] = 2\delta[n] + 8u[n] ...
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Determine causality and stability from given filter structure

I have the diagram above. I found the transfer function below from it; The question asks me to find out if the system is causal and stable, but didn't it have to indicate whether it was left-sided or ...
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Is there a simpler way to calculate the amplitude response of the following filter

I would like to calculate the amplitude response $|H(z)|$, $z=e^{j\omega}$, of the following filter: $$H(z)=\frac{\frac{b}{2}+z^{-2}}{2+bz^{-2}}$$ and I would like to avoid using Euler's formula and ...
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Convolution that outputs a unit impulse

Im thinking whether any convolutional operation can output a unit impulse, an example to further explain: where a convolution between system $h[n]$ and unknown system $g[n]$ would output $\delta[n]$. ...
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Z-transform of a cosine without a unit step [duplicate]

What is the $\mathcal Z$-transform of a cosine without a unit step, i.e. $x[n] = \cos(\omega_0 n)$ and not $x[n] = \cos(\omega_0n)u[n]$?
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Z transform of finite signals

I was trying to solve the Z-transform for u[n] - u[n-N], where u[n] means discrete unit step function, and N is some finite integer. I solved this using 2 methods. ...
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Frequency response of filter <-> signal mix

This might be a weird question but here's the setup: I have a few biquads that filter a signal $x[n]$ and output the filtered signal $y[n]$. I can calculate the frequency response of those biquads ...
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Determine the Z-Transform for the following sequence: $ |n|(\frac{1}{2})^{|n|} $

Determine the Z-Transform for the following sequence: $$ |n|(\frac{1}{2})^{|n|} $$ I have tried to solve the above problem. However, the answer that I got is the negative of what is given in the ...
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Finding inverse Z Transform

Find the inverse Z transform: I have done the solution but my answer does not match with the one given in the textbook. What I may have done wrong?
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How to determine whether a filter is high/low or band pass from the Z transform?

How to solve questions of these kind? I have tried by replacing $z=re^{jw}$ and taking the limits from $0$ to $\infty$. But I am not sure what $e^{j\infty}$ is.
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Average image over $N$ frames with one frame buffer

Can you give some filter that is averaging an image over $N$ samples in a recursive way that only needs one frame buffer? $\frac{1}{N} \sum_{k=1}^{N} y_{k}(m,n)$ I can only imagine filters that need ...
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Unilateral Z transform

I tried to calculate the unilateral Z transform of x[n-2], is it right?
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On the stability and causality of a discrete system

On MIT's open course a simple exercise with two questions is given. On the first part, they question about the properties of the following discrete system: $$ y[n]=x[n]+0.5y[n−1]−2y[n−2] $$ The ...
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How to find H(z) and H(k) from a given causal function

Consider the causal function, $y[k] = 2x[k] - 40x[k - 1] + 10y[k - 1]$ 􀀀 $16y[k - 2]$; where $y[k]$ is the output and $x[k]$ is the input. Assume that the system is initially at rest. Please someone ...
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How can I plot the frequency response on a bode diagram with Fast Fourier Transform?

Assume that we have an unknow dynamical system and we only want to estimate its parameters. The system can be discribed as: Continous time: $$G(s) = \frac{3s + 5} {5s^2 + 3s + 2}$$ Discrete time ...
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Z-Transform vs. Fourier Transform convergence

Are there signals for which the Fourier transform is known to exist (perhaps including singularities) and for which the z-transform does not converge?
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Impulse response of an LTI system given the input and output signals

I have been given the input and output signals of an LTI system as: $x[n] = (\frac{1}{2})^nu[n] + 2^nu[-n-1]$ $y[n] = 6(\frac{1}{2})^nu[n] - 6(\frac{3}{4})^nu[n]$ I have found the system function $...
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Do all systems in z domain are filters?

Normally a system transfer function is represented by H(z) I want to know wether all those system transfer functions are only representing a digital filter? Or their any other thing/entity in z ...
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116 views

Using ROC to find stability of system in specific example

I've started learning about finding the ROC from the transfer function, but I'm confused about an example. $$H(Z) = \frac{2Z + 1}{Z^2 + Z - \frac{5}{16}}$$ I understand the poles lie at $z = \frac{-...
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66 views

Transfer Function Block Diagram Confirmation

Can someone confirm if this is the correct block diagram for the following transfer function? The original equation provided was: y[n+1] = y[n] + 0.01x[n] Which I rearranged into H(z) = Y(z)/X(z) = ...
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When to use Fourier, Laplace and Z transforms?

If we have an LTI system, with an input signal $x(t)$, impulse response $h(t)$ and output $y(t)$, I was under the assumption that if the input and impulse response were continuous in time, then you ...
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In z-transform, if z means “delay”, why do we talk about the value of z?

I've been watching MIT's signals course and trying to understand $z$-transform. The course introduces $z^{-1}$ as an operator that delays the signal by $1$ time unit (which works very well with the ...

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