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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Difference equation when transfer function expressed as poles and zeros

The transfer function $H(z)$: $$ H(z) = \frac {b_0 + b_1 z^{-1} + b_2 z^{-2}} {1 + a_1 z^{-1} + a_2 z^{-2}} \tag{1} $$ Has difference equation: $$ y(n) = b_0 x(n) + b_1 x(n - 1) + b_2 x(n - 2) - ...
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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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1answer
37 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
48 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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32 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
20 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 ...
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24 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} \\ &= ...
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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)\\ ...
4
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2answers
202 views

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
37 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 ...
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1answer
41 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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29 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 $(1.1)$ $$x[n]=s[n]-e^{-8\alpha}s[n-8]$$ $$\alpha>0$$ ...
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1answer
42 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
33 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 ...
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1answer
69 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
34 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
35 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 ...
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1answer
79 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) = ...
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2answers
55 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
53 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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177 views

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
33 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 ...
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1answer
50 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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90 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 ...
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1answer
73 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
109 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
59 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
39 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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73 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
70 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 ...
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62 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 ...
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2answers
116 views

What does 'z' in Z-transform represent ? Is it frequency or something else?

my question is about the Z- transform. My first question is what the title says. What does 'Z' in Z-transform represent ? Say in Fourier transform, 'w' (omega) represents frequency ? From Fourier ...
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1answer
31 views

What is the right way to calculate the inverse Z-transform of $zX(z^{-1})$

say the signal $x(n)$ has the z transform $X(z)$ and there is signal $x_1(n)$ that $X_1(z)=zX(z^{-1})$ I tried 2 different approach to get the relationship between $x(n)$ and $x_1(n)$ and the ...
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1answer
76 views

Z-Transform of $x(n) = 3^n$

First of all, thank you all for your answers. I know the z transform for $$ x(n)=3^n \space ; \space n\geqslant 3 $$ or rather $$ x(n)= 3^n u(n-3) ...
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1answer
29 views

Phase response for conjugate zeros

If a second order system has 2 poles/zeros that are conjugate symmetric, how does this affect the phase response? I know that if there are 4 zeros/poles that are conjugate reciprocals, then it is a ...
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2answers
32 views

the ROC of a Z-transform for shifted signal

I have got two different answers for the ROC of the signal. In that PIC, I have solved it in 2 methods, but I'm getting different answer. Which one is correct? Also please explain how to find the ...
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3answers
77 views

The right way to approach z transform?

I am a student learning dsp. I like the subject. I could understand the discrete time signals. When I move into z transform. I could not understand it. Z transform is the mapping from discrete ...
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91 views

What does z-transform imply?

As z tranform is the transformation of discrete time signals into complex frequency domian. What do you get out of complex Stuff. As wikipedia calls it complex frequency domain. Why do you need it ? ...
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39 views

System Stability: Can we derive stability of a discrete system (Frequency domain, Z-transform) by applying analogous methods?

So given some analogue system function in the complex s-domain. Can we perform a stability analysis in the $s$-domain, before actually transfer it into the $z$-domain? So in other words analysis in ...
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31 views

Non-causal z transform in MATLAB?

In MATLAB, a causal z transfer function can be specified with filt(). Is it possible to specify a transfer function with acausal terms (positive exponent on z)?
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1answer
54 views

Z-transform of an FIR filter

QUESTION Compute the Z-transform of $y[n] = x[n] + 2x[n-1]$. and find the poles and zeros. I just bombed an interview where I couldn't do this (because I have no grounding in fundamentals and have ...
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Inverse z transform - Pair of complex conjugate poles

How can I perform the inverse z-transform on the following $H(z)$ to be able to calculate a real-valued impulse response $h[n]$? $$ H(z)=\frac{z^2}{z^2+0.8\sqrt{2}z+0.64} $$ My idea was to find an ...
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116 views

pole/zero locations for real and imaginary signal

In Z-Transform, For a real signal, $x(n)$ =$x^*(n)$ . Taking Z-transform on both sides, $X(z)$=$X^*(z^*)$ , which gives certain pole/zero condition similarly for a purely imaginary signal ...
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1answer
59 views

Find the Fourier transform of $g(k)$ from $G(z)$ for frequency $=1/2$

$$G(z)=\displaystyle \frac{\frac{1}{z}}{1+\frac{5}{6z}+\frac{1}{6}z^{-2}}$$ I found: $$g(k)=\displaystyle \left(\frac{-1}{3}\right)^k - \left(\frac{-1}{2}\right)^k$$ I don't understand how I can ...
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66 views

Finding fourier transform of a discrete signal from the z transform

Is it possible to find the fourier transform of a discrete signal if you know the $\mathcal{Z}$-transform of it?
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88 views

What is the ROC for this discrete signal:

$$ x(k)=4[u(k-2)-u(k)*δ(k-3)]$$ I found that the $\mathcal{Z}$ transform of the signal is $X(z)=4/(z^2)$. What would the ROC be?
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1answer
70 views

How can the order of a transfer function be derived from its equivalent state space representation?

Suppose I have a discrete state space model: $\begin{eqnarray*} \theta[k+1] &=& A \theta[k] + B u[k]\\ y[k] &=& C \theta[k] \end{eqnarray*}$ I know that the equivalent transfer ...
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1answer
47 views

What is the inverse Z transform of this:

$X(z) = \displaystyle \frac{1}{z}{\left(1-\frac{z^2}{4}\right)\left(1+\frac{1}{z}\right)\left(1-z\right)}$ Using partial fractions expansion i came up to this: $\displaystyle \frac{1}{X(z)} = ...
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1answer
42 views

Bilateral z-tranform of exponential

We all know that $a^nu(n)$ has unilateral $Z$-transform. But what is the $Z$-transform of $a^n$? (bilateral) When i tried to solve, i got answer as 'zero'. But bilateral Laplace transform of $e^t$ ...
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1answer
57 views

FIR filter design for response inside unit circle

I would like to design an FIR filter such that its Z-transform has a certain profile in certain regions. For example, if I'd like to have an FIR filter that nulls the decaying exponential ...