# 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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### Can use of Fourier transform be minimized completely with the help of Laplace and Z transform?

Fourier transform has different types like continuous Fourier transform (CFT), Discrete time Fourier transform (DTFT) and Discrete Fourier transform ( DFT). CFT can be used in case of continuous ...
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### Why is $\int^\infty _{0^-}\delta(t-nT)e^{-st}dt = e^{-nsT}$?

I'm currently in the process of going over the $\mathcal Z$-transform and more specifically its derivation. I understand and I am able to follow it up until the final step whereby involving the ...
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### Applying a time delay with Z-Transform

I am trying to take a signal $F(t)$ that has been sampled at some time DelT, I then wish to pass this signal through a channel $H(s)$. To do this I am sampling my signal $H(s)$ at DelT time intervals ...
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### Inverse $\mathcal Z$-transform of rational functions

What will be inverse $\mathcal Z$-transform for this function: $$H(z) = \frac{\left(1+\beta z^{-1}\right)\left(1+\beta z\right)}{\left(1+\alpha z^{-1}\right)\left(1+\alpha z\right)}$$
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### Problems calculating Z-transform

I am trying to solve a class exercise in which I am given the following, in Laplace domain: $$G(s)=\dfrac{e^{-Ts}}{s+3}$$ $$H(s)=\dfrac{1}{s}$$ And I need to calculate $\dfrac{C(z)}{R(z)}$, which is ...
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### Z-transform of difference equations and stability of a process

According to this paper: $y(t)$ is stationary if all of the roots (of characteristic equation) lie outside the unit circle Here, $y(t)$ is causal. To me it seems the case is exactly the opposite, ...
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### Is $y[k] = y[k-1] + x[k]$ an integrator?

It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$ Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \...
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### What is the difference between natural response and zero input response?

I am new to DSP and was going through different responses of a system subjected to an input. My understanding of zero input response is: it is the response/output of the system when the input signal ...
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### Transfer function of an Exponential system in Z domain

Hi, I am really confused with the system in the diagram. The input-output relation of the system is given by $y[n]=\exp(x^2[n])$ I need to find the transfer function of this system $Y(z)$ in $z$-...
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### Inverse $\mathcal Z$-transform of system with an 8th order pole

Can I find the inverse $\mathcal Z$-transform of this transfer function: $$H(z)=\frac{1}{1-\alpha z^{-8}}$$ in a way other than contour integration and finding the residues of the 8 poles? If so, how?
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### Fourier Transform of triangle function $x(t)=\Delta\left(\frac{t-1}{2}\right)$

Can you please tell me if my working is right for the Fourier Transform of this function: $$x(t)=\Delta\left(\frac{t-1}{2}\right)$$ My workings are: I have used the fourier transform standard ...
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### IDFT of H(z) sampled in N values

If a have a causal IIR filter described by $H(z)$ and I sample it in $N$ equispaced values around the unit circle, I get a DFT of $N$ points. That DFT corresponds to $h[n]$ truncated in $n=N-1$ or to ...
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### Analysis of a LTI system using DFT

Consider an LTI system $$H(z)=1-\frac{1}{2}z^{-1}+\frac{3}{4}z^{-2}$$ Let $x[n]=(\frac{1}{3})^n\cdot u[n]$ be the input signal. It is desired to determine the output for $n=0,1...,N_a$. To achieve ...
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### ROC of the product of two Z-Transforms

Suppose I have an LTI system $$H(z)=\frac{z}{(z-2)(z-\frac{1}{2})}$$ and I want to know its response to the step function $u[n]$. The LTI system $H(z)$ has three possible ROCs: $$|z|<\frac{1}{2}$$ ...
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### Relationship between z-transform and DFT

I'm studying for a Signals Processing exam and came across an exercise that I'm finding pretty difficult to solve. It says: Asume there is a signal $x[n]$ of length $N$. Its $\mathcal{Z}$-Transform ...
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### Z-transformation

Hi everybody i'm a student. Yesterday i had a test about my Engineering subject about signal processing and there was this problem: You have the sequence $x(n) = N+1 - |n|$. With $|n|\leq N.$ ...
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### DFT/FFT Transfer function

I want play and record a sine sweep. When i have both signals the recorded one and the send one i can create a Transferfunction. That is what i know so far. $$H_0 = \frac{OUT}{IN} = \frac{Y}{X}$$ ...
i try to calculate the $\mathcal z$-transform of $x[a-n]$ (where $n$ is my variable) i can't find any transform. the best suited transform is $x[-n] \longleftrightarrow X(z^{-1})$ i took the sum ...