# 2.6 Z-transform analysis of discrete-time filters

 Page 1 / 1

## $z$ -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.

what is variations in raman spectra for nanomaterials
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
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!