# Second order system impulse response generation

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This module describes the recursive generation of the impulse response of a second order system.

## Introduction

This module examines the recursive generation of the impulse response of a second order system. This analysis can be used to determine how to generate a cosine or sine function recursively.

## Second order systems

The transfer function of a general second order system is given as:

$H\left(z\right)=\frac{{b}_{0}{z}^{2}+{b}_{1}z+{b}_{2}}{{z}^{2}+{a}_{1}z+{a}_{2}}$

The Direct Form II structure for the implementation of the second order system is shown in the following figure.

The difference equation for the second order system is

$y\left(n\right)+{a}_{1}y\left(n-1\right)+{a}_{2}y\left(n-2\right)={b}_{0}x\left(n\right)+{b}_{1}x\left(n-1\right)+{b}_{2}x\left(n-2\right)$
$y\left(n\right)=-{a}_{1}y\left(n-1\right)-{a}_{2}y\left(n-2\right)={b}_{0}x\left(n\right)+{b}_{1}x\left(n-1\right)+{b}_{2}x\left(n-2\right)$

To recursively determine the impulse response of the system start with the following:

$h\left(-1\right)=h\left(-2\right)=0,x\left(n\right)=\delta \left(n\right)$

Then using Equation 3 the impulse response can be found recursively by:

$h\left(0\right)=-{a}_{1}h\left(-1\right)-{a}_{2}h\left(-2\right)+{b}_{0}\delta \left(0\right)+{b}_{1}\delta \left(-1\right)+{b}_{2}\delta \left(-2\right)={b}_{0}$
$h\left(1\right)=-{a}_{1}h\left(0\right)-{a}_{2}h\left(-1\right)+{b}_{0}\delta \left(1\right)+{b}_{1}\delta \left(0\right)+{b}_{2}\delta \left(-1\right)$
$h\left(1\right)=-{a}_{1}{b}_{0}+{b}_{1}$
$h\left(2\right)=-{a}_{1}h\left(1\right)-{a}_{2}h\left(0\right)+{b}_{0}\delta \left(2\right)+{b}_{1}\delta \left(1\right)+{b}_{2}\delta \left(0\right)$
$h\left(2\right)=-{a}_{1}\left(-{a}_{1}+{b}_{1}\right)-{a}_{2}{b}_{0}+{b}_{2}$

For the values of n >2 the recursive equation reduces to

$h\left(n\right)=-{a}_{1}h\left(n-1\right)-{a}_{2}h\left(n-2\right)$

because the values of $x\left(n\right),x\left(n-1\right)$ and $x\left(n-2\right)$ will all be zero for n >2.

So, if you know the values of $h\left(0\right)$ and $h\left(1\right)$ , then the Equation 10 can be used to find future values of $h\left(n\right)$ .

#### Questions & Answers

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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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
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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?
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Renato
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
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