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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

A flowchart of a system.

We can analyze this system via the equations

v [ n ] = b 0 x [ n ] + b 1 x [ n - 1 ] + b 2 x [ n - 2 ]


y [ n ] = v [ n ] + a 1 y [ n - 1 ] + a 2 y [ n - 2 ] .

More generally,

v [ n ] = N k = 0 b k x [ n - k ]


y [ n ] = M k = 1 a k y [ n - k ] + v [ n ]

or equivalently

N k = 0 b k x [ n - k ] = y [ n ] - M k = 1 a k y [ n - k ] .

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

M k = 0 a k y [ n - k ] = N k = 0 b k x [ n - k ] .

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

Z k = 0 M a k y [ n - k ] = Z k = 0 M b k x [ n - k ] k = 0 M a k Z { y [ n - k ] } = k = 0 N b k Z { x [ n - k ] } .

Note that

Z y [ n - k ] = n = - y [ n - k ] z - n = m = - y [ m ] z - m · z - k = Y ( z ) z - k .

Thus the relationship above reduces to

k = 0 M a k Y ( z ) z - k = k = 0 N b k X ( z ) z - k Y ( z ) k = 0 M a k z - k = X ( z ) k = 0 N b k z - k Y ( z ) X ( z ) = k = 0 N b k z - k k = 0 M a k z - k

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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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