# 0.1 Lecture 2: introduction to systems  (Page 8/8)

${\lambda }_{1,2}=\left(1+\frac{\text{rg}}{2}\right)±\sqrt{\left(1+\frac{\text{rg}}{2}{\right)}^{2}-1}$

The following plot shows the locus of natural frequencies as rg increases.

Both natural frequencies lie along the positive, real λ-axis. When rg = 0, ${\lambda }_{1,2}$ = 1. As rg increases, one natural frequency decreases toward λ = 0 and the other natural frequency increases. What is the physical significance of this pattern?

Note that the product of the two natural frequencies is 1,

${\lambda }_{1}{\lambda }_{2}=\left(1+\frac{\text{rg}}{2}+\sqrt{\left(1+\frac{\text{rg}}{2}{\right)}^{2}-1}\right)×\left(1+\frac{\text{rg}}{2}-\sqrt{\left(1+\frac{\text{rg}}{2}{\right)}^{2}-1}\right)=1$

Therefore, ${\lambda }_{1}=\frac{1}{{\lambda }_{2}}$ and the form of the homogeneous solution is

${v}_{o}\left[n\right]={A}_{1}{\lambda }_{{1}^{n}}+{A}_{2}{\lambda }_{{1}^{-n}}$

The two terms of the homogeneous solution are shown below.

Thus, the homogeneous solution consists of a linear combination of decaying and growing geometric functions whose rates of decrease and increase are the same. Increasing the quantity rg increases the magnitude of the rate of change of voltage.

2/ Particular solution

Next we find a particular solution to the difference equation

$\sum _{k=0}^{K}{a}_{k}{y}_{p}\left[n+k\right]=\sum _{l=0}^{L}{b}_{l}x\left[n+1\right]$

for n>0 and

$x\left[n\right]={\text{Xz}}^{n}$

where z does not equal one of the natural frequencies. We assume that

${y}_{p}\left[n\right]={\text{Yz}}^{n}$

and we solve for Y . Substitution for both x[n] and y[n]in the difference equation yields, after factoring,

$\left(\sum _{k=0}^{K}{a}_{k}{z}^{k}\right){\text{Yz}}^{n}=\left(\sum _{l=0}^{L}{b}_{l}{x}^{l}\right){\text{Xz}}^{n}$

After dividing both sides of the equation by ${z}^{n}$ we can solve for Y which has the form

$Y=\stackrel{˜}{H}\left(z\right)X$

where

$\stackrel{˜}{H}\left(z\right)=\frac{\sum _{l=0}^{L}{b}_{l}{z}^{l}}{\sum _{k=0}^{K}{a}_{k}{z}^{k}}$

3/ System function

$\stackrel{˜}{H}\left(z\right)=\frac{\sum _{l=0}^{L}{b}_{l}{z}^{l}}{\sum _{k=0}^{K}{a}_{k}{z}^{k}}\frac{⇐\text{zeros}}{⇐\text{poles}}$

• is called the system function
• is a rational function in z that has poles and zeros
• has poles that are the natural frequencies of the system
• is a skeleton of the difference equation
• characterizes the relation between x[n] and y[n]

a/ Example—reconstruction of difference equation from $\stackrel{˜}{H}\left(z\right)$

Suppose

$\stackrel{˜}{H}\left(z\right)=\frac{Y}{X}=\frac{z}{z+1}$

what is the difference equation that relates y[n] to x[n]? Crossmultiply the equation and multiply both sides by ${z}^{n}$ to obtain

$\left(z+1\right)Y{z}^{n}=zX{z}^{n}$

which yields

$\left({z}^{n+1}Y+{z}^{n}Y={z}^{n+1}X$

from which we can obtain the difference equation

y[n+1] + y[n]= x[n+1]

b/ Pole-zero diagram

$\stackrel{\text{~}}{H}$ characterizes the difference equation and $\stackrel{\text{~}}{H}$ is characterized by K + L + 1 numbers: K poles, L zeros, and one gain constant. Except for the gain constant, $\stackrel{\text{~}}{H}$ is characterized by a pole-zero diagram which is a plot of the locations of poles and zeros in the complex-z plane.

4/ Total solution

The general solution is

$y\left[n\right]=\sum _{k=1}^{K}{A}_{k}{\lambda }_{{k}^{n}}+X\stackrel{˜}{H}\left(z\right){z}^{n}\text{for}n>0$

and

y[n] = 0 for n<0.

The general solution can be written compactly as follows

$y\left[n\right]=\left(\sum _{k=1}^{K}{A}_{k}{\lambda }_{{k}^{n}}+X\stackrel{˜}{H}\left(z\right){z}^{n}\right)u\left[n\right]$

5/ Initial conditions

To completely determine the total solution we need to determine the K coefficients { ${A}_{1},{A}_{2},\text{.}\text{.}\text{.}{A}_{K}$ }. These are determined from K initial conditions which must be specified. These conditions result in a set of K algebraic equations that need to be solved to obtain the initial conditions so that the total solution can be specified. We shall find another, and simpler, method to determine the total solution later.

Example — discretized CT system

We have previously considered the discretized approximation to a lowpass filter.

The equilibrium equation is

$\frac{{\text{dv}}_{o}\left(t\right)}{\text{dt}}=-\frac{1}{\text{RC}}{v}_{o}\left(t\right)+\frac{1}{\text{RC}}{v}_{i}\left(t\right)$

We know that the unit step response of this network starting from initial rest is

${v}_{o}\left(t\right)=\left(1-{e}^{-t/\text{RC}}\right)u\left(t\right)$

We showed that a discretized approximation to this system yields the difference equation

${v}_{o}\left[n+1\right]=\left(1-\beta \right){v}_{o}+{\mathrm{\alpha v}}_{i}\left[n\right]$

where α = T/(RC). We will determine the solution by finding the homogeneous and particular solution. But all the information we need is contained in the system function which we can obtain by substituting ${v}_{i}\left[n\right]={V}_{i}{z}^{n}$ and ${v}_{o}\left[n\right]={V}_{o}{z}^{n}$ into the difference equation to obtain

so that

$\stackrel{˜}{H}\left(z\right)=\frac{{\stackrel{˜}{V}}_{o}\left(z\right)}{{\stackrel{˜}{V}}_{i}\left(z\right)}=\frac{\alpha }{z-\left(1-\alpha \right)}$

The natural frequency is $\lambda =1-\alpha$ so that the solution has the form

${v}_{o}\left[n\right]=\left(A\left(1-\alpha {\right)}^{n}+\stackrel{˜}{H}\left(1\right)\left(1{\right)}^{n}\right)u\left[n\right]$

where we have made use of the fact that $u\left[n\right]={\text{1}}^{n}u\left[n\right]$ . Since $\stackrel{\text{~}}{H}$ (1) = 1,

${v}_{o}\left[n\right]=\left(A\left(1-\alpha {\right)}^{n}+1\right)u\left[n\right]$

Finally, the initial condition, ${v}_{o}\left[0\right]=\text{0}$ , implies that A = −1 so that the solution is

${v}_{o}\left[n\right]=\left(1-1-\alpha {\right)}^{n}\right)u\left[n\right]$

which is the same result we obtained earlier by solving the difference equation iteratively.

We compare the step response of the CT system with the DT approximation

${v}_{o}\left(t\right)=\left(1-{e}^{-t/\text{RC}}\right)u\left(t\right)$ and ${v}_{o}\left[n\right]=\left(1-\left(1-\frac{T}{\text{RC}}{\right)}^{n}\right)u\left(t\right)$

The solutions are shown below for RC = 1 and T/(RC) = 0.1.

XI. CONCLUSIONS

• Systems are typically described by an arrangement of subsystems each of which is defined by a functional relation. Systems are classified according to such properties as: memory, causality, stability, linearity, and time-invariance. Linear, time-invariant systems (LTI) are special systems for which powerful mathematical methods of description are available.

Logic for an analysis method for LTI systems

• H(s) characterizes system $⇒$ compute H(s) efficiently.
• In steady state, response to ${\text{Xe}}^{\text{st}}$ is $H\left(s\right){\text{Xe}}^{\text{st}}$
• Represent arbitrary x(t) as superpositions of ${\text{Xe}}^{\text{st}}$ on s.
• Compute response y(t) by superposing $H\left(s\right){\text{Xe}}^{\text{st}}$ on s.

1/ Structural versus functional descriptions

Just as with CT systems, DT systems can be described either structurally, with a block diagram or a network diagram, or functionally by a system function.

$\stackrel{\text{~}}{H}\left(z\right)$ characterizes system

2/ Steady-state response to zn is particularly simple

Since the steady-state response to a complex geometric (exponential) is so simple, it is desirable to represent arbitrary DT functions as sums (integrals) of building-block complex geometric (exponential) functions chosen so that steady-state dominates. The steady-state response to each complex geometric (exponential) is readily computed. For a DT LTI system, the response to an arbitrary input can be computed by superposition.

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
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
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
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Sebastian Sieczko... By OpenStax By OpenStax By Rhodes By Madison Christian By OpenStax By Madison Christian By OpenStax By OpenStax By Cameron Casey