# 2.6 Z-transform analysis of discrete-time filters

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## $z$ -transform analysis of discrete-time filters

The $z$ -transform might seem slightly ugly. We have to worry about the region of convergence, and we haven't even talked about how to invert it yet (itisn't pretty). However, in the end it is worth it because it is extremely useful in analyzing digital filters with feedback. For example, consider the system illustrated below

We can analyze this system via the equations

$v\left[n\right]={b}_{0}x\left[n\right]+{b}_{1}x\left[n-1\right]+{b}_{2}x\left[n-2\right]$

and

$y\left[n\right]=v\left[n\right]+{a}_{1}y\left[n-1\right]+{a}_{2}y\left[n-2\right].$

More generally,

$v\left[n\right]=\underset{k=0}{\sum ^{N}}{b}_{k}x\left[n-k\right]$

and

$y\left[n\right]=\underset{k=1}{\sum ^{M}}{a}_{k}y\left[n-k\right]+v\left[n\right]$

or equivalently

$\underset{k=0}{\sum ^{N}}{b}_{k}x\left[n-k\right]=y\left[n\right]-\underset{k=1}{\sum ^{M}}{a}_{k}y\left[n-k\right].$

In general, many LSI systems satisfy linear difference equations of the form:

$\underset{k=0}{\sum ^{M}}{a}_{k}y\left[n-k\right]=\underset{k=0}{\sum ^{N}}{b}_{k}x\left[n-k\right].$

What does the $z$ -transform of this relationship look like?

$\begin{array}{cc}\hfill Z\left\{\stackrel{M}{\sum _{k=0}},{a}_{k},y,\left[n-k\right]\right\}& =Z\left\{\stackrel{M}{\sum _{k=0}},{b}_{k},x,\left[n-k\right]\right\}\hfill \\ \hfill \stackrel{M}{\sum _{k=0}}{a}_{k}Z\left\{y\left[n-k\right]\right\}& =\stackrel{N}{\sum _{k=0}}{b}_{k}Z\left\{x\left[n-k\right]\right\}.\hfill \end{array}$

Note that

$\begin{array}{cc}\hfill Z\left\{y,\left[,n,-,k,\right]\right\}& =\stackrel{\infty }{\sum _{n=-\infty }}y\left[n-k\right]{z}^{-n}\hfill \\ & =\stackrel{\infty }{\sum _{m=-\infty }}y\left[m\right]{z}^{-m}·{z}^{-k}\hfill \\ & =Y\left(z\right){z}^{-k}.\hfill \end{array}$

Thus the relationship above reduces to

$\begin{array}{cc}\hfill \stackrel{M}{\sum _{k=0}}{a}_{k}Y\left(z\right){z}^{-k}& =\stackrel{N}{\sum _{k=0}}{b}_{k}X\left(z\right){z}^{-k}\hfill \\ \hfill Y\left(z\right)\left(\stackrel{M}{\sum _{k=0}},{a}_{k},{z}^{-k}\right)& =X\left(z\right)\left(\stackrel{N}{\sum _{k=0}},{b}_{k},{z}^{-k}\right)\hfill \\ \hfill \frac{Y\left(z\right)}{X\left(z\right)}& =\frac{\left(\stackrel{N}{\sum _{k=0}},{b}_{k},{z}^{-k}\right)}{\left(\stackrel{M}{\sum _{k=0}},{a}_{k},{z}^{-k}\right)}\hfill \end{array}$

Hence, given a system like the one above, we can pretty much immediately write down the system's transfer function, and we end up with arational function, i.e., a ratio of two polynomials in $z$ . Similarly, given a rational function, it is easy to realize this function in a simple hardware architecture. We will focus exclusively on such rational functions in this course.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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