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These are determined from N initial conditions which must be specified. These conditions result in a set of N algebraic equations that need to be solved to obtain the initial conditions. We shall find another, and simpler, method to determine these coefficients later.

2/ Steady-state

Assume the particular solution is not zero. Then the particular solution dominates after some time, if the homogeneous solution decays more rapidly than does the particular solution. When this occurs, we call the resulting particular solution the steady-state response. Steady-state occurs if each term in the homogeneous solution decays more rapidly than the particular solution. Thus, steady-state occurs if

lim t e λ n t e st 0 for all λ n size 12{ {"lim"} cSub { size 8{t rightarrow infinity } } lline { {e rSup { size 8{λ rSub { size 6{n} } t} } } over {e rSup { ital "st"} } } rline size 12{ rightarrow 0`` ital "for"`` ital "all"``λ rSub {n} }} {}

Thus

lim t e ( λ n s ) t 0 for all λ n size 12{ {"lim"} cSub { size 8{t rightarrow infinity } } lline e rSup { size 8{ \( λ rSub { size 6{n} } - s \) t} } rline rightarrow 0`` ital "for"`` ital "all"``λ rSub {n} } {}

which implies that

R { λ n s } 0 for all λ n size 12{R lbrace λ rSub { size 8{n} } - s rbrace 0`` ital "for"`` ital "all"``λ rSub { size 8{n} } } {}

Thus, steady-state occurs when

R { λ n } < R { s } size 12{R lbrace λ rSub { size 8{n} } rbrace<R lbrace s rbrace } {} for all λ n size 12{λ rSub { size 8{n} } } {}

provided the particular solution is not zero. The conditions for steady state are depicted in the complex s plane below.

The particular solution dominates for s in the shaded region, and the total solution equals the steady-state solution i.e.,

y ( t ) = XH ( s ) e st size 12{y \( t \) = ital "XH" \( s \) e rSup { size 8{ ital "st"} } } {}

For which conditions is the particular solution zero? Suppose

H ( s ) = s 1 ( s + 1 ) ( s + 2 ) size 12{H \( s \) = { {s - 1} over { \( s+1 \) \( s+2 \) } } } {}

The particular solution dominates for s in the shaded region, and the total solution equals the steady-state solution except when s = 1 because at this value i.e.,

y ( t ) = XH ( 1 ) e s = 0 size 12{y \( t \) = ital "XH" \( 1 \) e rSup { size 8{s} } =0} {}

so that the particular solution is zero and steady -state does not occur.

IX. LINEAR DIFFERENCE EQUATIONS ARISE IN MANY DIFFERENT CONTEXTS

1/ Electric ladder network

r is the series resistance and g is the shunt conductance.

KCL at the central node yields

v o [ n 1 ] v o [ n ] r + v o [ n + 1 ] v o [ n ] r + g ( v i [ n ] v o [ n ] = 0 size 12{ { {v rSub { size 8{o} } \[ n - 1 \] - v rSub { size 8{o} } \[ n \]} over {r} } + { {v rSub { size 8{o} } \[ n+1 \] - v rSub { size 8{o} } \[ n \]} over {r} } +g \( v rSub { size 8{i} } \[ n \] - v rSub { size 8{o} } \[ n \]=0} {}

which yields the linear difference equation

v o [ n + 1 ] + ( 2 + rg ) v o [ n ] v o [ n 1 ] = rgv i [ n ] size 12{ - v rSub { size 8{o} } \[ n+1 \] + \( 2+ ital "rg" \) v rSub { size 8{o} } \[ n \]- v rSub { size 8{o} } \[ n - 1 \] = ital "rgv" rSub { size 8{i} } \[ n \]} {}

2/ Interest and accumulation

Let us consider a simple model of the accumulation of wealth through savings. At the end of year n you deposit x[n] dollars in the bank which pays an annual interest of r. Your accumulation at the end of year n is y[n]dollars. Therefore,

y [ n + 1 ] = y [ n ] + ry [ n ] + x [ n ] size 12{y \[ n+1 \] =y \[ n \]+ ital "ry" \[ n \] +x \[ n \]} {}

We can rewrite this equation as

y[n+1] − (1+r)y[n]= x[n]

This difference equation can be realized in a block diagram as shown below.

D is a unit delay unit.

3/ Discretized CT system

An important application of DT systems is a numerical simulation of a CT system. For example, consider the CT lowpass filter shown below.

The differential equation is

dv o ( t ) dt = 1 RC v o ( t ) + 1 RC v i ( t ) size 12{ { { ital "dv" rSub { size 8{o} } \( t \) } over { ital "dt"} } = - { {1} over { ital "RC"} } v rSub { size 8{o} } \( t \) + { {1} over { ital "RC"} } v rSub { size 8{i} } \( t \) } {}

To solve this equation numerically in a computer, the CT signals are discretized and the derivative is approximated.

To discretize the signals, we can define DT signals as samples of CT signals, i.e.,

v i [ n ] = v i ( t ) t = nT = v i ( nT ) size 12{v rSub { size 8{i} } \[ n \] =v rSub { size 8{i} } \( t \) \lline rSub { size 8{t= ital "nT"} } =v rSub { size 8{i} } \( ital "nT" \) } {} and v o [ n ] = v o ( t ) t = nT = v o ( nT ) size 12{v rSub { size 8{o} } \[ n \] =v rSub { size 8{o} } \( t \) \lline rSub { size 8{t= ital "nT"} } =v rSub { size 8{o} } \( ital "nT" \) } {}

The derivative can be approximated as follows

dv o ( t ) dt t = nT v o ( ( n + 1 ) T ) = v o ( nT ) T size 12{ { { ital "dv" rSub { size 8{o} } \( t \) } over { ital "dt"} } \lline rSub { size 8{t= ital "nT"} } approx { {v rSub { size 8{o} } \( \( n+1 \) T \) =v rSub { size 8{o} } \( ital "nT" \) } over {T} } } {}

Therefore, we can approximate the differential equation as

v o ( ( n + 1 ) T ) = v o ( nT ) T = 1 RC v o ( nT ) + 1 RC v i ( nT ) size 12{ { {v rSub { size 8{o} } \( \( n+1 \) T \) =v rSub { size 8{o} } \( ital "nT" \) } over {T} } = - { {1} over { ital "RC"} } v rSub { size 8{o} } \( ital "nT" \) + { {1} over { ital "RC"} } v rSub { size 8{i} } \( ital "nT" \) } {}

which can be written as

v o [ n + 1 ] = ( 1 T RC ) v o [ n ] + ( T RC ) v i [ n ] size 12{v rSub { size 8{o} } \[ n+1 \] = \( 1 - { {T} over { ital "RC"} } \) v rSub { size 8{o} } \[ n \]+ \( { {T} over { ital "RC"} } \) v rSub { size 8{i} } \[ n \] } {}

Let α size 12{α} {} = T/(RC). Then the difference equation is

v o [ n + 1 ] = ( 1 α ) v o [ n ] + αv i [ n ] size 12{v rSub { size 8{o} } \[ n+1 \] = \( 1 - α \) v rSub { size 8{o} } \[ n \]+αv rSub { size 8{i} } \[ n \] } {}

This equation can be solved iteratively for a given input and initial condition. Assume that v i [ n ] = u [ n ] size 12{v rSub { size 8{i} } \[ n \] =u \[ n \]} {} and that v o [ 0 ] = 0 size 12{v rSub { size 8{o} } \[ 0 \] =0} {} then

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
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