-
Home
- Discrete-time signals and systems
- The z-transform
- Z-transform properties
As with other signal transforms, the z-transform has a number of significant properties, including ways in which changes to a signal in one domain impacts its representation in another domain. To examine these properties, we will again use the notation of a z-transform "pair." If we have
$X(z)=\sum\limits_{n=-\infty}^{\infty}x[n]z^{-n}$,
then we express the relationship between $x[n]$ and its z-transform $X(z)$ as
$x[n]\leftrightarrow X(z)$
The DTFT is a special case of the z-transform. If we have $x[n]\leftrightarrow X(z)$ and the ROC of $X(z)$ contains the unit circle, then the DTFT of $x[n]$, $X(e^{j\omega})$, is simply the z-transform $X(z)$ evaluated on the unit circle $z=e^{j\omega}$:
$X(e^{j\omega})=X(z)|_{z=e^{j\omega}}$
Given that the z-transform is simply an infinite sum, if follows then that it will be a linear operator. If we have $x_1[n]\leftrightarrow X_1(z)$ and $x_2[n]\leftrightarrow X_2(z)$, and $\alpha_1,\alpha_2\in C$, then:
$\alpha_1 x_1[n]+\alpha_2 x_2[n]\leftrightarrow \alpha_1 X_1(z)+\alpha_2 X_2(z)$However, the ROC of the new transform is the intersection of those of $X_1(x)$ and $X_2(z)$: $ROC_{\alpha_1 X_1+\alpha_2 X_2}=ROC_{X_1}\bigcap ROC_{X_2}$
If $x[n]$ and $X(z)$ are a z-transform pair, then:$x[n-m]\leftrightarrow z^{-m}X(z)$and the ROC remains the same as for $X(z)$ (with the possible addition/removal of a pole at zero and/or zero at infinity):
$\begin{align*}\sum_{n=-\infty}^{\infty} x[n-m]\, z^{-n}&= \sum_{r=-\infty}^{\infty} x[r]\, z^{-(r+m)}\\&=\sum_{r=-\infty}^{\infty} x[r]\, z^{-r} \, z^{-m} \\&=z^{-m} \sum_{r=-\infty}^{\infty} x[r]\, z^{-r}\\&=z^{-m} X(z)
\end{align*}$
If $x[n]$ and $X(z)$ are a z-transform pair, then:$z^n_0 x[n]\leftrightarrow X(\frac{z}{z_0})$To determine the ROC of the new z-transform, substitute $\frac{z}{z_0}$ for $z$ in the original ROC definition, and simplify. For example, if the ROC of $X(z)$ is $|z|\gt 1$, then the new ROC will be $|\frac{z}{z_0}|\gt 1 \rightarrow |z|\gt |z_0|$
Proof:$\begin{align*}
\sum_{n=-\infty}^{\infty} (z_0^n x[n]) z^{-n}&=\sum_{n=-\infty}^{\infty} x[n](z/z_0)^{-n}\\&=X\left(\frac{z}{z_0}\right)\end{align*}$
If $x[n]$ and $X(z)$ are a z-transform pair, then:$x^*[n] \leftrightarrow} X^*(z^*)$Proof:
$\begin{align*}\sum_{n=-\infty}^{\infty} x^*[n] z^{-n}&=\left(\sum_{n=-\infty}^{\infty} x[n] (z^*)^{-n}\right)^*\\&=X^*(z^*)
\end{align*}$
If $x[n]$ and $X(z)$ are a z-transform pair, then:$x[-n]\leftrightarrow}X(z^{-1})$With the ROC inverted: to find the ROC, substitute $\frac{1}{z}$ for $z$ in the original ROC expression and simplify.
Proof:$\begin{align*}
\sum_{n=-\infty}^{\infty} x[-n]z^{-n}&=\sum_{m=-\infty}^{\infty} x[m]z^{m}\\&=\sum_{m=-\infty}^{\infty} x[m] (z^{-1})^{-m}\\&=X(z^{-1})
\end{align*}$
If $x[n]\leftrightarrow X(z)$, $h[n]\leftrightarrow H(z)$, and $y[n]\leftrightarrow Y(z)$ all all z-transform pairs, and $y[n]=x[n]\ast h[n]$, then:
$Y(z)=X(z)H(z)$,with the ROC of $Y(z)$ being the intersection of the ROCs of $X(z)$ and $H(z)$.
$\begin{array}{c|c|c}
x[n]&X(z)&ROC \\
\hline \\\delta[n]&1&all z\\
u[n]&\frac{1}{1-z^{-1}}&|z|\gt 1 \\
\alpha^n u[n]&\frac{1}{1-\alpha z^{-1}}&|z|\gt|\alpha| \\
-\alpha^n u[-n-1]&\frac{1}{1-\alpha z^{-1}}&|z|\lt|\alpha|
\end{array}$
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Omeprazole
Cimetidine / Tagament
For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
how does the stomach protect itself from the damaging effects of HCl
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
function of digestive
Ali
what is the normal body temperature
please why 37 degree selcius normal temperature
Mark
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
how does Lysin attack pathogens
Diya
I information on anatomy position and digestive system and there enzyme
anatomy of the female external genitalia
Got questions? Join the online conversation and get instant answers!
Source:
OpenStax, Discrete-time signals and systems. OpenStax CNX. Oct 07, 2015 Download for free at https://legacy.cnx.org/content/col11868/1.2
Google Play and the Google Play logo are trademarks of Google Inc.