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 ( z ) = b 0 z 2 + b 1 z + b 2 z 2 + a 1 z + a 2 size 12{H \( z \) = { {b rSub { size 8{0} } z rSup { size 8{2} } +b rSub { size 8{1} } z+b rSub { size 8{2} } } over {z rSup { size 8{2} } +a rSub { size 8{1} } z+a rSub { size 8{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 ( n ) + a 1 y ( n − 1 ) + a 2 y ( n − 2 ) = b 0 x ( n ) + b 1 x ( n − 1 ) + b 2 x ( n − 2 ) size 12{y \( n \) +a rSub { size 8{1} } y \( n - 1 \) +a rSub { size 8{2} } y \( n - 2 \) =b rSub { size 8{0} } x \( n \) +b rSub { size 8{1} } x \( n - 1 \) +b rSub { size 8{2} } x \( n - 2 \) } {}
y ( n ) = − a 1 y ( n − 1 ) − a 2 y ( n − 2 ) = b 0 x ( n ) + b 1 x ( n − 1 ) + b 2 x ( n − 2 ) size 12{y \( n \) = - a rSub { size 8{1} } y \( n - 1 \) - a rSub { size 8{2} } y \( n - 2 \) =b rSub { size 8{0} } x \( n \) +b rSub { size 8{1} } x \( n - 1 \) +b rSub { size 8{2} } x \( n - 2 \) } {}

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

h ( − 1 ) = h ( − 2 ) = 0, x ( n ) = δ ( n ) size 12{h \( - 1 \) =h \( - 2 \) =0,x \( n \) =δ \( n \) } {}

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

h ( 0 ) = − a 1 h ( − 1 ) − a 2 h ( − 2 ) + b 0 δ ( 0 ) + b 1 δ ( − 1 ) + b 2 δ ( − 2 ) = b 0 size 12{h \( 0 \) = - a rSub { size 8{1} } h \( - 1 \) - a rSub { size 8{2} } h \( - 2 \) +b rSub { size 8{0} } δ \( 0 \) +b rSub { size 8{1} } δ \( - 1 \) +b rSub { size 8{2} } δ \( - 2 \) =b rSub { size 8{0} } } {}
h ( 1 ) = − a 1 h ( 0 ) − a 2 h ( − 1 ) + b 0 δ ( 1 ) + b 1 δ ( 0 ) + b 2 δ ( − 1 ) size 12{h \( 1 \) = - a rSub { size 8{1} } h \( 0 \) - a rSub { size 8{2} } h \( - 1 \) +b rSub { size 8{0} } δ \( 1 \) +b rSub { size 8{1} } δ \( 0 \) +b rSub { size 8{2} } δ \( - 1 \) } {}
h ( 1 ) = − a 1 b 0 + b 1 size 12{h \( 1 \) = - a rSub { size 8{1} } b rSub { size 8{0} } +b rSub { size 8{1} } } {}
h ( 2 ) = − a 1 h ( 1 ) − a 2 h ( 0 ) + b 0 δ ( 2 ) + b 1 δ ( 1 ) + b 2 δ ( 0 ) size 12{h \( 2 \) = - a rSub { size 8{1} } h \( 1 \) - a rSub { size 8{2} } h \( 0 \) +b rSub { size 8{0} } δ \( 2 \) +b rSub { size 8{1} } δ \( 1 \) +b rSub { size 8{2} } δ \( 0 \) } {}
h ( 2 ) = − a 1 ( − a 1 + b 1 ) − a 2 b 0 + b 2 size 12{h \( 2 \) = - a rSub { size 8{1} } \( - a rSub { size 8{1} } +b rSub { size 8{1} } \) - a rSub { size 8{2} } b rSub { size 8{0} } +b rSub { size 8{2} } } {}

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

h ( n ) = − a 1 h ( n − 1 ) − a 2 h ( n − 2 ) size 12{h \( n \) = - a rSub { size 8{1} } h \( n - 1 \) - a rSub { size 8{2} } h \( n - 2 \) } {}

because the values of x ( n ) , x ( n − 1 ) size 12{x \( n \) ,x \( n - 1 \) } {} and x ( n − 2 ) size 12{x \( n - 2 \) } {} will all be zero for n >2.

So, if you know the values of h ( 0 ) size 12{h \( 0 \) } {} and h ( 1 ) size 12{h \( 1 \) } {} , then the Equation 10 can be used to find future values of h ( n ) size 12{h \( n \) } {} .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
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Jyoti Reply
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
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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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Brian Reply
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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
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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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
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Damian Reply
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Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
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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, Dsp lab with ti c6x dsp and c6713 dsk. OpenStax CNX. Feb 18, 2013 Download for free at http://cnx.org/content/col11264/1.6
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