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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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How does the ROC (Region of Convergence) related to a real world application?

In class, we are often given exercises to find the impulse response, output, and Z-transform of a system. In addition, we are often asked to define the Region of Convergence (ROC) depending on where ...
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1answer
367 views

Find the difference equation and draw the simulation diagram

Calculate the difference equation and then draw the simulation diagram of the below transfer function. $$ H(z) = \frac{Y(z)}{X(z)} = \frac{0.4142 + 0.4142z^{-1}}{1.4142 - 0.5858z^{-1}} $$ I ...
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321 views

Why correlation property of $\mathcal Z$-transform contains a time reversal operation

I'm reading through Digital Signal Processing, Proakis and Manolakis, third edition. I've reached section 3.2: Properties of $\mathcal Z$-transform. One property is the convolution: $$x(n) = x_1(n)\...
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203 views

For a discrete LTI system, does “bounded memory” imply “rational transfer function?”

Every LTI system with a $\mathcal Z$-domain transfer function that is a rational function - aka a quotient of two polynomials in $z$ - can be implemented using a bounded amount of memory. Is the ...
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135 views

$\mathcal Z$-transform of auto-correlation

Assume signal $x(n), d(n)$, filter $\mathbf{w}(n)$, define error signal as $$e(n)=d(n)-\mathbf{w}^T(n)\mathbf{x}(n)$$ Auto-power spectrum is defined as $$ S_{ee}(z)= \mathcal{Z}\left[r_{ee}(k)\right]...
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142 views

$\mathcal{Z}$-transform of $\frac{1}{n^2}$

This is a Question asked in IISC ( Indian Institute of Science,Bangalore,India) interview for MS admission. What is the $\mathcal{Z}$-transform of $\dfrac 1{n^2}$ ?
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1answer
110 views

Converting transfer function that is a sum of unusual rational polynomials to finite difference equation

I have the following rather exotic transfer function: $$ H(z) = cz^{-m} + \frac{b_0 z^{-1} + b_1z^{-2} + \dots + b_{2m}z^{-2m}}{1 + a z^{-1}} + \frac{q_0 z^{-1} + q_1z^{-2} + \dots + q_{2m}z^{-2m}}{1 ...
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87 views

Learning about inverse-z-transform and how to apply it to a rational transfer function

I have been studying IIR filters and know that a rational transfer function: $$ H(z) = \frac {b_0 + b_1 z^{-1} + ... + b_N z^{-N}}{1 + a_1 z^{-1} + ... + a_N z^{-N}} $$ has a finite difference ...
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1answer
42 views

Inverse $\mathcal Z$-transform problem

$B(z)+B(-z) = 2c$, explain the structure of $b[n]$ and find the constraint of its length given that $c$ cannot be $0$. This is a homework problem. "Explain the structure" means that $b[n]$ is zero ...
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75 views

$\mathcal Z$-transform, ROC of a system from dividing to others

TRUE / FALSE Given three systems with rational $\mathcal Z$-transform. Systems A and B are not stable with $\mathcal Z$-transform $H(z)$, $G(z)$ respectively. A and B have no common poles. System's C ...
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How/why are the $\mathcal Z$-transform and unit delays related?

The $\mathcal Z$-transform uses the same notation as the unit delay $z^{-1}$, but in $\mathcal Z$-transform $z$ is a complex number. What's the relation between the $\mathcal Z$-transform and the ...
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49 views

$\mathcal Z$-transform if the output is given

A impulse response for a LTI system is given by: $$h[n]=\left(\frac{2}{3}\right)^n u[n]+2 \left(\frac{1}{5}\right)^n u [n]$$ and if the putput for the system is given by: $$y[n]= \left(\...
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1answer
417 views

Converting poles/zeros to differential/difference equation solutions

Does anyone have a reference handy on how to convert poles and zeros of a system to differential/difference equations. Here is a quick draft of math, but I am not sure if it's at all correct. First, ...
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1answer
153 views

LTI system input upsampling

Let's assume that a linear and time-invariant system is sampled at 2 different frequencies $F_{s}$ and $2F_{s}$ (e.g. 5Hz and 10 Hz). It gives $$Y_{F_{s}}(z) = H_{F_{s}}(z)X_{F_{s}}(z)$$ $$Y_{2F_{s}}(...
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266 views

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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5k views

What are the advantages and disadvantages of Laplace transform over Z transform?

Laplace transform for continuous signal $x(t)$ is given by $$ X(s) = \int\limits_{-\infty}^{+\infty} x(t) e^{-s t} dt. \quad (1) $$ Z-transform for discrete signal $x(n)$ is given by $$ X(z) = \...
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167 views

Efficient computation of Chirp Z Transform

Chirp Z Transform (1, 2, 3) is more powerful than zooming techniques (I use it to actually trace non-stationary chirp signals) and very usable in signal processing, but it's flexibility comes at price ...
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122 views

Solving a linear ODE in the z-Domain

I'm stuck with an excercise or rather unsure how the solution came to be. the problem is the following: The following ODE is given: $$ x[k] + x[k-3] = 0 $$ Calculate the non trivial solution $...
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1answer
126 views

$\mathcal Z$-transform ROC

Let's say I have a $\mathcal Z$-transform that represents some transfer function and its has some ROC. My question is how do I know if this system is causal? I know that if the ROC contains the ...
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206 views

Difference equation when transfer function expressed as poles and zeros

The transfer function $H(z)$: $$ H(z) = \frac{Y(z)}{X(z)} = \frac {b_0 + b_1 z^{-1} + b_2 z^{-2}} {1 + a_1 z^{-1} + a_2 z^{-2}} \tag{1} $$ Has difference equation: $$ y[n] = b_0 x[n] + b_1 x[n-1] + ...
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177 views

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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1answer
667 views

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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1answer
70 views

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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42 views

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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1answer
94 views

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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188 views

Z transform of $\sum_{k=0}^{n}3^{k}$

My task is to calculate z transform of signal $x[n]=\sum\limits_{k=0}^{n}3^{k}$ ? By definition, $$ \begin{align} X(z) &= \sum\limits_{n=-\infty}^{n=\infty}x[n]z^{-n} \\ &= \sum\limits_{n=-\...
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3answers
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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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2answers
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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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1answer
175 views

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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53 views

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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70 views

Multi-Channel LTI Inverse system problem

A sequence $x[n]$ is the output of a linear time-invariant system whose input is $s[n]$. This system is described by the difference equation $(1.1)$ $$x[n]=s[n]-e^{-8\alpha}s[n-8]$$ $$\alpha>0$$ a)...
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1answer
155 views

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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1answer
52 views

Differentials - Differences: Non causality in the system

I'm still learning DSP and referring to Oppenheim video lectures. In that lectures, differential difference equation is obtained for IIR filter design, in Lecture 14. $$\mathcal{L}[\frac{\mathrm d}{\...
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1answer
354 views

Closed form of $\mathcal Z$-transform : decomposition signal $x(n)$

The text of my exercise ask : Determine the closed form of the $\mathcal{Z}$-transform for this $x(n)$ $$ x(n) = \begin{cases} |n-N| & \text{if 0<$n$<2N} \\ 0 & \text{elsewhere} \...
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1answer
44 views

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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1answer
161 views

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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704 views

Downsampling and Then Upsampling

Given this system: I need to show the $\mathcal Z$-transform of $y[n]$ as a function of the $\mathcal Z$-transform of $x[n]$. Now I know that for downsampling alone: $$Y(z) = \frac1M\sum_{m=0}^{M-1}...
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2answers
87 views

Sampling H(z) to get DFT

Suppose that I have a $H(z)$ and I sample it to get a DFT of 15 values. Let's call this DFT $H_{1}[k]$. Then, suppose I antitransform $H(z)$ and grab the first 10 values of the sequence, and then I ...
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1answer
127 views

Problem designing a specific FIR filter

Consider an LTI system whose impulse response is $$h[n]=\frac{1}{2^n}u[n]+\frac{1}{3^n}u[n]$$ The input signal to this system is $x[n]$ and is null for $n<0$ but may or may not be null for $n=0$. ...
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1answer
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Identifying the magnitude and impulse response from pole zero plot quickly

I have an exam next week and it's verty certain that a task of this kind will be there. Are there some good tips how to match the right pole zero plot to the right responses? No proof is needed in ...
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1answer
105 views

Is this system LTI?

Assuming the system $h[n]$ is LTI (and has an associated $H(z)$ transform), is the whole system below LTI? I found the impulse response of the system and I got that it is $$h_{0}[n]=\alpha ^{-n}\cdot ...
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1answer
314 views

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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2answers
5k views

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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1answer
81 views

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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2answers
1k views

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. $$ H_0 = \frac{OUT}{IN} = \frac{Y}{X} $$ ...
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1answer
61 views

Z-transform of x[a -n]…where a is int

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 ...
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1answer
93 views

doubt of intersection of ROC Z Transform

dear friends of StackExchange. I have a doubt of the intersection of two ROC. I have H(Z), X(Z) and and i have to determine: $$ \begin{align} Y(Z)= H(Z)X(Z)\end{align}$$ $$ \displaystyle $$ $ ...
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2answers
106 views

Is the below filter linear phase

$$h(t) = \frac{1}{1 + t^2}$$ and is it IIR or FIR filter. I tried finding the Laplace transform of this filter to get the data flow diagram with 5 taps and T=2s, however, I am unable to solve this. ...
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1answer
4k views

Inverse Z-transform with complex conjugate poles

I was computing an inverse z-transform here, and I am facing some problems. So, the z-transform is: $$ X(z) = \frac{2+3z^{-1}}{1 - z^{-1} + 0.81z^{-2}} , |z| > 0.9 $$ I found the following poles:...
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2answers
299 views

Question about z transform

After studying z transform from different books and literature on internet I want to ask few which makes me confuse. a) From the Discrete Time Fourier Transform we have drive equation for z ...