What is Z-transform: Definition and 39 Discussions
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation.
It can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
Hello,
I am reading a course on signal processing involving the Z-transform, and I just read something that leaves me confused.
Let ##F(z)## be the given Z-transform of a numerical function ##f[n]## (discrete amplitudes, discrete variable), which has a positive semi-finite support and finite...
Hello everyone !
I am working on ultrasound scan and the processing of the signal received by the probe. I made the model I wanted and as I do not have an ultrasound scan machine I want to simulate the signal processing.
I will do that with the Python package Scipy and the function...
Hi,
I am trying to work out how I could obtain an expression for a z-transform for the following expression:
x_{n + 1} = r x_{n} \left( 1 - x_{n} \right)
I am hoping to derive X(z) and then use the final value theorem to show agreement with numerically calculated steady state values.
I...
Homework Statement
Hi,
I'm asked to find the transfer function for the following digital filter described by a difference equation
Homework EquationsThe Attempt at a Solution
[/B]
Usually if there's no constant term (0.7) here, I can just rearrange in the form of Y(z)/X(z) to give me H(z)...
Hello everyone.
Iam just learning the z-transform for discrete signals and I can't get my head around the Region of covergence (ROC).
As far as I have understood describes the ROC if the z-transform excists or not ?
But how to I actually calculate it? Is there any kind of formula?
I all...
Homework Statement
Hello,
I am currently doing some holiday pre-study for signals analysis coming up next semester. I'm mainly using MIT OCW 6.003 from 2011 with some other web resources (youtube, etc).
The initial stuff is heavy on the old infinite series stuff, that seems often skimmed...
Hi
I am currently learning Digital Signal Processing from Applied Digital Signal Processing by Manolakis and Ingle. However, my background is in mathematics, so I like being shown why an assertion is true. So, in the chapter about the z-transform is where I start seeing this becoming an issue...
This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property
\lim_{|z|\to\infty} \frac{X(z)}{z} = 0
Is that sufficient to guarantee that
\frac{1}{2\pi...
Has there been any applications in engineering to the following generelization to Z-transform?
##X(z) = \sum_{n=-\infty}^\infty x[n] e^{-zn}##
We get the ##z##-trasnform by using the substitution ##z \mapsto Log(z)##.
Thanks in advance.
Hi guys,
I am trying to find the z-tranform of the following equation: x(n) =-2n u(-n-1)
Using the Z-transform definition,summation and geometric series
I am getting 1/(1-2z-1)
But according to my lecturer the answer is suppose to be 0.5/(1-2z-1)
I have tried simplification/factorization and...
Homework Statement
In Figure Q1b (on the next page), two plots display samples of the continuous-time signal uct(t) = sin(2πt) and two plots display samples of the continuous-time signal vct(t) = e −t sin(2πt). For each signal, the samples in the corresponding plots are obtained with two...
Homework Statement
Homework Equations
H(z) = Y(z)/X(z)
The Attempt at a Solution
I realized this wasn't in partial fraction form because the 1+z-1+0.5z-2 has non-real roots. I multiplied the 1st fraction part by z1 and the 2nd fraction by z2, then I combined them into one fraction and I...
Homework Statement
H(z)=1/(1-2/3*z-1)
h(n)=?
Homework Equations
I tried to use the residue method to solve this problem but it doesn't give me the good result and i am not sure if i don't know how to use the method or this problem can't be solved using this method.
Also, its pretty...
Homework Statement
"Let x[n] be a real-valued DT signal. Using the Pole-Zero plot of X(z), argue as to why:
a. |X(e^jω)| is even in ω.
b. ∠X(e^jω) is odd in ω.
Hint: if x[n] is real, what can we say about the poles and zeros of X(z)?Homework Equations
None.
The Attempt at a...
Hello,
Homework Statement
I would like to find the inverse Z transform of the following:
F(z)=1-1.25z-1+0.25z-2/[1-(5/6)z-1+(1/6)z-2]
using (a) partial fractions, and (b) residue theorem
I have obtained different results and hence would appreciate some insight on the discrepancy and how...
I have a question about the z-transform:
In the (Oppenheim's Signals and Systems) book it is written:
If the z-transform of x[n] is X(z) with ROC=R, then the z-transform of nx[n] is -zdX(z)/dz with the same ROC.
I don't understand why the ROC remains unchanged. Some books say "it...
Z-transform of a conjugated sequence ("a straightforward" exercise)
Homework Statement
The conjugation property is expressed as x^*[n] \stackrel{Z}{\leftrightarrow} X^*(z^*)
This property follows in a straightforward maner from the definition of the z-transform, the details of which are left...
Homework Statement
I am attempting to perform a z-transformation on the terms 1/(z-3) and 1/(z-5) but I cannot find a sequence which fits this form in any of my tables. Is there a transform for these terms or do you just leave them as they are?
Homework Equations
The Attempt at a...
Homework Statement
An LTI digital system with impulse response h[n] = 2(1/4)^n u[n-1] produces an output y[n] = (-3)^n u[n-1]. Determine the corresponding input x[n].
Homework Equations
I believe that Y[z] = X[z]H[z]
The Attempt at a Solution
So I started by finding the...
Greetings,
We just had 1.5 classes about this topic (due to some unavoidable circumstances in our country...) so our teacher just scribbled and roughly explained them on the board...So after i got home i couldn't get some steps he did for convolution in z-transform:
find inverse z-transform...
1. Homework Statement [/b]
You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|.
a) x is a finite-length signal
b) \hat{X} has...
Homework Statement
You are given the following pieces of information about a real, stable, discrete-time signal x and its DTFT X, which can be written in the form X(\omega)=A(\omega)e^{i\theta_x(\omega)} where A(\omega)=\pm|X(\omega|.
a) x is a finite-length signal
b) \hat{X} has exactly two...
Hi,
Suppose we have these two functions and their z-transforms are
P(r,z)=\sum_{t=0}^{\infty}P(r,t)z^t
and
F(r,z)=\sum_{t=0}^{\infty}F(r,t)z^t.
Now we are going to transform the following convolution of P and F:
\sum_{t'\le{t}}F(r,t')P(0,t-t').
The result is said to be
F(r,z)P(0,z).
But I don't...
Homework Statement
H(z) = \frac{6-z^{-1}}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}} = k + \frac{A}{1+0.5z^{-1}} + \frac{2}{1-0.4z^{-1}}
where A = (6-z^{-1}) is evaluated at z^{-1}=-2.
Homework Equations
Partial fraction expansion.
The Attempt at a Solution
Why is z^{-1} set equal to -2? I...
Homework Statement
I need to find a transfer function using the given functions which i have no problem with. But i don't know how to exactly define the Region Of Convergence of the resulting transfer function.
I have h1[n] and h2[n] as to be convoluted to find the transfer function. Homework...
Hi,
I'm currently studying some introductory discrete control theory and I've run into a problem with the Z-transform, although I could pose the same question regarding the Laplace transform. I know I'm completely off with this question but I've just stared myself blind to it. Here goes:
I...
Homework Statement
Find the impulse response of the following: Assume the system is initially at rest.
y[n] - (1/2) y[n-1] = x[n] + 2 x[n-1] + x[n-2]
The Attempt at a Solution
To find the impulse response y[n]: we know that y[n] = homogenous solution + particular solution
so...
lamda -...
Homework Statement
Given a ramp function x(t) = t*heaviside(t) with the known z-transform ( z / (z-1)^2 ).
If the time-domain signal has been delayed by 2 seconds, then sampled with an unknown sampling rate (T). How could I get the z-transform (one-sided) for the discrete signal, for any values...
Homework Statement
I'm trying to take the (two-sided, aka defined for all n) Z-transform of
x(n)=sin(Bn)Homework EquationsThe Attempt at a Solution
What I tried to do was split x(n) into
x(n)=sin(Bn)u(n)+sin(Bn)u(-n-1)
The Z transform of the first part of x(n) was found from a table. The...
Homework Statement
Compute the Z-transform of
x[n] = (0.5)^{-n} * u[n-1]
And find the ROC (Region of Convergence)
Homework Equations
Z-Transform for discrete time
x(z) = \sum_{n=0}^\infty x[n] * z^{-n}
The Attempt at a Solution
To solve this, I used the definition of...
I am having some trouble with this z-transform, and would like a little help.
Question:
Find the Z-transform, and sketch the pole/zero plot with the region of convergence.
x[n] = \cos( \Omega_0 n) u[n]
Where u[n] is the step function.
"Answer"
Directly from a table in my book, x[n]...
Hi,
I try to visualize a sine wave, z-transformed in a Bode plot. The frequency shows up correctly, but the amplitude is way off - why?
Thanks for your comments.
Bmalp
function zdemo
format compact
syms w Ts n z phi t
Ts = 1
w = 0.1
phi = 0
vz =...