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a X ( z ) + b [ z - 1 X ( z ) + x ( - 1 ) ] + c [ z - 2 X ( z ) + z - 1 x ( - 1 ) + x ( - 2 ) ] = Y ( z )

solving for X ( z ) gives

X ( z ) = z 2 [ Y ( z ) - b x ( - 1 ) - x ( - 2 ) ] - z c x ( - 1 ) a z 2 + b z + c

and inversion of this transform gives the solution x ( n ) . Notice that two initial values were required to give a unique solution just as theclassical method needed two values.

These are very general methods. To solve an n th order DE requires only factoring an n th order polynomial and performing a partial fraction expansion, jobs that computers are well suited to. There are problemsthat crop up if the denominator polynomial has repeated roots or if the transform of y ( n ) has a root that is the same as the homogeneous equation, but those can be handled with slight modifications givingsolutions with terms of the from n λ n just as similar problems gave solutions for differential equations of the form t e s t .

The original DE could be rewritten in a different form by shifting the index to give

a x ( n + 2 ) + b x ( n + 1 ) + c x ( n ) = f ( n + 2 )

which can be solved using the second form of the unilateral z-transform shift property.

Region of convergence for the z-transform

Since the inversion integral must be taken in the ROC of the transform, it is necessary to understand how this region is determined and what it meanseven if the inversion is done by partial fraction expansion or long division. Since all signals created by linear constant coefficientdifference equations are sums of geometric sequences (or samples of exponentials), an analysis of these cases will cover most practicalsituations. Consider a geometric sequence starting at zero.

f ( n ) = u ( n ) a n

with a z-transform

F ( z ) = 1 + a z - 1 + a 2 z - 2 + a 3 z - 3 + + a M z - M .

Multiplying by a z - 1 gives

a z - 1 F ( z ) = a z - 1 + a 2 z - 2 + a 3 z - 3 + a 4 z - 4 + + a M + 1 z - M - 1

and subtracting from [link] gives

( 1 - a z - 1 ) F ( z ) = 1 - a M + 1 z - M - 1

Solving for F ( z ) results in

F ( z ) = 1 - a M + 1 z - M - 1 1 - a z - 1 = z - a ( a z ) M z - a

The limit of this sum as M is

F ( z ) = z z - a

for | z | > | a | . This not only establishes the z-transform of f ( n ) but gives the region in the z plane where the sum converges.

If a similar set of operations is performed on the sequence that exists for negative n

f ( n ) = u ( - n - 1 ) a n = a n n < 0 0 n 0

the result is

F ( z ) = - z z - a

for | z | < | a | . Here we have exactly the same z-transform for a different sequence f ( n ) but with a different ROC. The pole in F ( z ) divides the z-plane into two regions that give two different f ( n ) . This is a general result that can be applied to a general rational F ( z ) with several poles and zeros. The z-plane will be divided into concentricannular regions separated by the poles. The contour integral is evaluated in one of these regions and the poles inside the contour give the part ofthe solution existing for negative n with the poles outside the contour giving the part of the solution existing for positive n .

Notice that any finite length signal has a z-transform that converges for all z . The ROC is the entire z-plane except perhaps zero and/or infinity.

Relation of the z-transform to the dtft and the dft

The FS coefficients are weights on the delta functions in a FT of the periodically extended signal. The FT is the LT evaluated on the imaginaryaxis: s = j ω .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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