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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Multirate Control System Transfer Functions

I'm interested in oversampling the inputs to a digital controller to increase the SNR of the input process variable signal. I've read on this site and in articles like the one below that it is not ...
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Different PI controller implementations and their respective discrete transfer functions

So I need to implement a PI-controller and I found an Implementation of an PID-controller with some background explanation. I adapted the implementation to an PI-controller, implemented it and got the ...
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Z-Transform of a Complex Number

I have a more general question about $z$-transforms that I am having difficulty finding an answer to. Suppose we have a two-sided sequence something along the lines of: $$x(n)=a^{|n|}$$ where $a \in \...
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What is causal inverse of a system?

Let's say that I have a system $H(z)$. What is causal inverse and how do I compute the causal inverse of $H(z)$?
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Sampling with impulse train

There are many times when the textbooks used by all university students like me make an introduction to the notion of sampling of a continuous-time signal with an impulse train as shown below. Why do ...
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Calculating transfer function for a comb filter with low-passed feedback

I'm creating a comb filter bank as part of a project I'm working on, and I need a way to visualize its magnitude response. The project already has FFT-based analyzers that can generate this response, ...
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Problem with the existence of inverse DTFT

I am having trouble on the following exercise and I can't figure out where I am doing something wrong: Given an LTI system described by the following difference equation: $$y(n)=x(n)+2x(n-2)+y(n-1)$$ ...
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Validity of taking an inverse $\mathcal{Z}-$ transform instead of taking an inverse DTFT

I have the following problem: I am using the Convolution Theorem and have got an expression of $H(z)X(z)$ and now I need to take $\text{DTFT}^{-1}(H(z)X(z))$, namely I have to take the inverse DTFT ...
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A problem on $\mathcal{Z}-$transforms and signal stability

I am doing the following problem for my DSP exam and I am doubting my answers while being stuck on the last part: Given a causal LTI system and difference equations $y(n)=x(n)+20y(n-1)-100y(n-2),$ ...
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If $y(n)-y(n-1)=bx(n)$ is an LTI system, find $b$ such that $\left|H(e^{i\omega})\right|=1$ for $\omega=0$

I am stuck on the following problem and would like your help: Given an LTI system described by the difference equation: $$y(n)-\frac{y(n-1)}{10}=bx(n),$$ find $b$ such that $\left|H(e^{i\omega})\...
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Finding and displaying Laplace or Z transform ROC(region of convergence) using MATLAB

Is there any way, we can use MATLAB for finding and displaying Laplace or Z transform Region of convergence?
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Poles and zeros form of a transfer function

I know that a transfer function for a discrete-time LTI system can be written in the form $$ H(z) = \frac{Y(z)}{X(z)} = \frac { \displaystyle\sum_{m=0}^M {b_m z^{-m}}} {1 + \displaystyle\sum_{n=1}^...
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What kind of filter is z / (z + 0.5)?

z / (z - e^-aT) is a low pass filter. z / (z - 1) is an integrator. What is z / (z + 0.5) ? In the time domain, it is Yn = Xn - 0.5Yn-1. It changes sign every iteration. Is this useful or should it ...
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What is the difference between delay and difference properties of z-transform?

I'm working on a discrete updating algorithm as follows: $x[n+1]=Kx[n]$ Here $K$ is a constant. The continuous counterpart of this algorithm translates to: $\dot{x(t)}=Kx(t)$ While the Laplace ...
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How to compute the inverse Z-transform

How to compute the inverse Z-transform of the form $$ G(z)=\frac{z^{2n}}{a(z^{2n})+b(z^n)+c} $$ I started by taking $$ F(z^n)=G(z) $$ so $$ \frac{F(z)}{z}=\frac{z}{az^2+bz+c}$$ This can be solved ...
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Discrete signals - Z Transforms

If a system has transfer function $H(z) = 1 + z^{–1} + z^{–2}$, then for an input signal of the sequence = {1,2,3,1,4,-2, 4} calculate the output sequence of the system. DO i put transfer function ...
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Confusion implementing inverse z transform in MATLAB

I am trying to use MATLAB commands ztrans and iztrans , but i am not getting proper results My code is below, why i am not getting H1=H2??Keeping in view that "H2" is "inverse Z ...
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Convolution between bit stream sampled by N number of samples per bit with a channel modeled as low pass filter

I generated a bit stream in MATLAB, from the transmitter to be convoluted by the channel I use MATLAB function "randi" between 0 & 1, so it generates random vectors of 0s & 1s and ...
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IIR Filters H(z)=Y(z)/X(z) Why do we insert the coefficents of the input for Y(z)?

The System Function of IIR Filters is defined as $$H(z)=\frac{Y(z)}{X(z)}$$ The Output Signal as $$y(n)=\frac{1}{a_0}\left(\sum^{P}_{i=0}b_ix[n-i]-\sum^{Q}_{j=1}a_jy[n-j]\right)$$ I do not understand ...
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Find a band-pass filter

The question is: how can I define $h_1[n]$ in such a way that $h [ n ] = \delta [n - 1 ] + 2 \delta [n -2 ] + h_ 1 [n]$ is a band-pass filter. My thought was the following. Firstable, I wrote the $Z$-...
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Phase of DTFT transform of impulse response

I am approaching to the study of FIR systems. In particular, I was analyzing the graphs of amplitude and phase of the transfer function, when I had some trouble understanding how the phase behaves. ...
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Find moments of a discrete process using $\mathcal{Z}$-Transform method

For a linear process relating a variable $Y_i$ to random, independent variables $X_i$ using the equation: \begin{equation} Y_i = aY_{i-1} + (1-a)X_i \end{equation} which has the solution: \begin{...
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Find Transfer Function and Appropriate Coefficients of the Transfer Functions from Pole Zero Plot

I got a Transfer function problem and I am confused in finding a solid solution step. Below is the problem description: 1st and 2nd order discrete-time filters with different pole-zero locations shall ...
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Calculating tranfer function, poles, zeros and impulse response given input and outpul signals in matlab

I have been given an input and output signal. input: x(n)=(0.3^n)*u(n) + (5^n)*u(-n-1) output: y(n)=(3^n)*u(-n-1) - ((2^(n+2))/(3^n))*u(n). or x=(0.3.^n).*(n>=0) + (5.^n). *(n<=-1) --> in ...
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Filter filters out more than needed

I am currently coding a school assignment. I have a 1.5s recording of someone's speech that has 4 rogue cosines mixed in. With sampling rate 16000Hz, I divided the recording into frames of 1024 ...
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Recursive equation Of Euler's Backward PID With Derivative Filter

I'm doing the implementation of the PID with derivative filter using the Euler's backward method, but I got stuck in this part since I have z in the denominator of most terms. I didn't realize yet how ...
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Butterworth filter poles

Hi, I'm looking at this textbook question and trying to get a better idea of exactly what its asking. For the processing to be real valued each pole would have to have a complex conjugate right? So ...
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How would one solve this question if no initial conditions are given? Which assumption can I make?

This was a question on our test, I know it can be easily solved by Z-transforms but there are no initial conditions specified. In this case, what would be the right approach? Assume all initial ...
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Z Transform of M-Fold Decimation

I know this is probably a common question, but after some searching I think that my version of the question is slightly different -- apologies if this is a repeat. I have seen that it is inaccurate to ...
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Poles and Zeros of a DiscreteTimeModelFunction with delay in Wolfram Mathematica

help me please, I have a problem with this TransferFunctionModel, When I want to obtain the poles from the Discrete Model Wolfram gives me an error, what am I doing wrong? Could it be the delay of the ...
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Find Fourier Transform of Unit Step using the $z$-Transform [duplicate]

Since the unit step $u[t]$ is not absolutely summable, it has no Fourier Transform. In the DSP book (Proakis), the Fourier Transform of the unit step is formed by evaluating its $z$-Transform on the ...
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Value of $\sum\limits_{n=-\infty}^{\infty}(x*x)[n]$

If $x[n]=(0.5)^nu[n]$ and $y[n]=(x*x)[n]$ then what is the value of $\sum\limits_{n=-\infty}^{\infty}y[n]$ ? I calculated the $\mathcal{Z}$-transform of $x[n]$ and then applied the accumulation ...
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Can I Apply Multiple Properties of the Z Transform Simultaneously?

Using the time shifting, time reversal, and scaling, I want to derive the form of the Z Transform of $$x[n]=-a^n u[-(n+1)]$$ $u[n]$ is the discrete-time unit step function: $$ u[n] \triangleq \begin{...
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ROC of $\mathcal{Z}$-Transform and zeros

Theorem: Let $$f(z) = \sum_{n=0}^{+\infty}a_nz^n$$ where $z\in\mathbb{C}$. If $f(z_0)$ exists for some $z_0\in\mathbb{C}$ then it converges for all $z\in\mathbb{C}$ such that $|z|\lt|z_0|$. Proof: It ...
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How do I proceed to calculate this Z Transform?

I would like to calculate the Z-Transform of following discrete signal: $$x[n] = 3^{-|n|}$$ Plugging it into the known formula, I got: $$X(z) = \sum_{n=-\infty}^\infty x[n]z^{-n} = \sum_{n=-\infty}^\...
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Z-transform of $\cos(\omega_0 n(n+1))u[n]$

I'm doing some research on Zadoff-Chu sequences and as a part of it I wanted to find the Z-transform of: $$\cos(\omega_0 n(n+1))u[n]$$ Wolfram Alpha / Mathematica couldn't help out. I couldn't find a ...
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Z domain Transfer function to magnitude

How would one go about calculating the Magnitude of... $\mathcal{Z}$-transform = $$\frac{1}{1-z^{-2}}$$ I understand that z can be replaced with $${exp}^{j\omega}$$ and I am aware of the identities $${...
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Z transforms doubt -(ROC and its purpose)!

i had this doubt previously queried in another forum, but unfortunately had no answer. Consider a signal 3^n u[n]. Take its Z transform, which is Z/(Z-3). Now i know that in real sense, Z is a delay ...
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Determine transfer function based on a diagram with ai coefficients given

From a diagram with input $x(n)$ a summer and three feedback delay taps I get the difference equation... $$y(n) = x(n)+a_1y(n-1)+a_2y(n-2)+a_3y(n-3)$$ Then I am given values for ai coefficients. Case1:...
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DSP: newbie not understanding z transform/complex sinusoidal frequency and phase

Im reading Will Pirkles Designing audio effect plugins book and I'm not sure if I'm understanding the z transform correctly. I got up to differential equations in college, but haven't done math in ...
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Polar Fourier Transform and its similarity to the Z-Transform

The DFT is given by $X_{\mathcal{F}}(k) = \sum_{n=0}^{N-1} x_n e^{-j 2 \pi k n / N}$ Knowing that complex numbers can also be represented using a polar form $A \angle\theta$, I was looking for a ...
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Having trouble interpreting z-transform description of a predictor from a codec paper

I've been looking at the opus paper (https://arxiv.org/pdf/1602.04845.pdf); in particular, in section 4.1, they describe the predictor for the current band energy based on energy from both the current ...
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how to calculate $H_{lp} = z(1- H_{hp}) $ , given coefficients for $H_{hp}$?

Given a high pass transfer fn of the form $H_{hp}=a_{1}*z^0 +a_{2}*z^{-1} + ... a_{n}*z^{-n}$ Is it possible to calculate a causal low pass filter using $H_{lp} = z*(1-H_{hp})$ ? attempting $H_{...
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Mixed - Discrete and Continuous system Laplace domain stability - Effect of Sampler and DAC

I have a system whose the plant transfer function is continuous and the compensation is discrete. I have an ADC which allows to measure the output of the system and a DAC which allows to control the ...
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Identify out signal from in signal and impulse response

Hi there I am studying a course in signal processing and systems. I was given an excecise which I am having a great deal of trouble solving. I am allowed to solved using matlab, but for top marks I ...
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Zeros and poles from transfer function

I have a transfer function $$ H(z) = \frac{Y(z)}{X(z)} = 1 - 0.5z^{-1} \text{.}$$ I'm interested in zeros and poles. I know I need to adjust the function to $$ H(z) = \frac{\prod_i(z-n_i)}{\prod_i(z-...
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Difference equation to FIR filter coefficients

I have a difference equation $$y[n] = y[n-1] + 0.5x[n] - 0.5x[n-2] \text{.}$$ According to this answer, the filter should have finite impulse response. I just can't figure out how to get the FIR ...
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$Z$-transform of a multilinear function/ consecutive multplication of $k$ signals $y_1(n), \ldots, y_k(n)$

How should one go about calculating the $Z$-transform of a signal that is the multiplication of $k$ signals (i.e. a multilinear function with regards to signals $y_1(n) \ldots y_k(n)$ ? Namely, $\...
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inverse z transform performed on 6th order IIR filter

we are told to find coefficients and impulse response of IIR filter of order of 6. There are 6 zeros and 6 poles in the design. Pole and zero pairs are conjugate and poles are within the unit circle ...
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If we take Z-transform of a signal & find its ROC. How to use this ROC? There are ∞ signals in ROC, suppose we choose any signal what to do with it?

If we take Z-transform of a signal & find its ROC. How to use this ROC? There are infinite signals in ROC, suppose we choose any signal from ROC, what to do with it?

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