<< Chapter < Page Chapter >> Page >

Y=H(s)X

where

H ( s ) = m = 0 M b m s m n = 0 N a n s n size 12{H \( s \) = { { Sum cSub { size 8{m=0} } cSup { size 8{M} } {b rSub { size 8{m} } s rSup { size 8{m} } } } over { Sum cSub { size 8{n=0} } cSup { size 8{N} } {a rSub { size 8{n} } s rSup { size 8{n} } } } } } {}

b/ System function definition

H ( s ) = m = 0 M b m s m n = 0 N a n s n size 12{H \( s \) = { { Sum cSub { size 8{m=0} } cSup { size 8{M} } {b rSub { size 8{m} } s rSup { size 8{m} } } } over { Sum cSub { size 8{n=0} } cSup { size 8{N} } {a rSub { size 8{n} } s rSup { size 8{n} } } } } } {}

  • is called the system function
  • is a rational function in s
  • is a skeleton of the differential equation
  • characterizes the relation between x(t) and y(t)

Example — reconstruction of differential equation from H(s)

Suppose

H ( s ) = Y X = s s + 1 size 12{H \( s \) = { {Y} over {X} } = { {s} over {s+1} } } {}

what is the differential equation that relates y(t) to x(t)? Cross-multiply the equation and multiply both sides by e st size 12{e rSup { size 8{ ital "st"} } } {} to obtain

( s + 1 ) Ye st = sXe st size 12{ \( s+1 \) ital "Ye" rSup { size 8{ ital "st"} } = ital "sXe" rSup { size 8{ ital "st"} } } {}

which yields

sYe st + Ye st = sXe st size 12{ ital "sYe" rSup { size 8{ ital "st"} } + ital "Ye" rSup { size 8{ ital "st"} } = ital "sXe" rSup { size 8{ ital "st"} } } {}

from which we can obtain the differential equation

dy ( t ) dt + y ( t ) = dx ( t ) dt size 12{ { { ital "dy" \( t \) } over { ital "dt"} } +y \( t \) = { { ital "dx" \( t \) } over { ital "dt"} } } {}

2/ Poles and zeros

H(s) can be expressed in factored form as follows

H ( s ) = K m = 1 M ( s z m ) n 1 N ( s p n ) size 12{H \( s \) =K { { Prod cSub { size 8{m=1} } cSup { size 8{M} } { \( s - z rSub { size 8{m} } \) } } over { Prod cSub { size 8{n - 1} } cSup { size 8{N} } { \( s - p rSub { size 8{n} } \) } } } } {}

where K = b M a M size 12{K= { {b rSub { size 8{M} } } over {a rSub { size 8{M} } } } } {}

  • { z 1 , z 2 , . . . , z M } size 12{ lbrace z rSub { size 8{1} } ,z rSub { size 8{2} } , "." "." "." ,z rSub { size 8{M} } rbrace } {} are the roots of the numerator polynomial and are called zeros of H(s) because these are the values of s for which H(s) = 0.
  • { p 1 , p 2 , . . . , p M } size 12{ lbrace p rSub { size 8{1} } ,p rSub { size 8{2} } , "." "." "." ,p rSub { size 8{M} } rbrace } {} are the roots of the denominator polynomial and are called poles of H(s) because these are the values of s for which H ( s ) = size 12{H \( s \) = infinity } {} .

a/ Poles are the natural frequencies

Note that poles of H(s) are the natural frequencies of the system. Recall that natural frequencies are given by the roots of the characteristic polynomial

( n = 0 N a n λ n ) = 0 size 12{ \( Sum cSub { size 8{n=0} } cSup { size 8{N} } {a rSub { size 8{n} } λ rSup { size 8{n} } \) =0} } {}

and the poles are the roots of denominator polynomial of H(s)

( n = 0 N a n s n ) = 0 size 12{ \( Sum cSub { size 8{n=0} } cSup { size 8{N} } {a rSub { size 8{n} } s rSup { size 8{n} } \) =0} } {}

Both originate from the left-hand side of the differential equation

n = 0 N a n d n y ( t ) dt n size 12{ Sum cSub { size 8{n=0} } cSup { size 8{N} } {a rSub { size 8{n} } { {d rSup { size 8{n} } y \( t \) } over { ital "dt" rSup { size 8{n} } } } } } {}

b/ Pole-zero diagram

H(s) characterizes the differential equation and H(s) is characterized by N + M + 1 numbers: N poles, M zeros, and the constant K. Except for the multiplication factor K, H(s) is characterized by a pole-zero diagram which is a plot of the locations of poles and zeros in the complex-s plane. The ordinate is jI { s } = size 12{ ital "jI" lbrace s rbrace =jϖ} {} and the abscissa is R { s } = σ size 12{R lbrace s rbrace =σ} {} where

s = σ + size 12{s=σ+jϖ} {}

Example — system function of a network

The differential equation relating v(t) to i(t) is

di ( t ) dt = C ( d 2 v ( t ) dt 2 + 1 RC dv ( t ) dt + v ( t ) LC ) size 12{ { { ital "di" \( t \) } over { ital "dt"} } =C \( { {d rSup { size 8{2} } v \( t \) } over { ital "dt" rSup { size 8{2} } } } + { {1} over { ital "RC"} } { { ital "dv" \( t \) } over { ital "dt"} } + { {v \( t \) } over { ital "LC"} } \) } {}

The particular solution is obtained from

sI = C ( s 2 + 1 RC s + 1 LC ) V size 12{ ital "sI"=C \( s rSup { size 8{2} } + { {1} over { ital "RC"} } s+ { {1} over { ital "LC"} } \) V} {}

With v(t) as the output and i(t) as the input, the system function of the RLC network is

H ( s ) = V I = 1 C ( s s 2 + 1 RC s + 1 LC ) size 12{H \( s \) = { {V} over {I} } = { {1} over {C} } \( { {s} over {s rSup { size 8{2} } + { {1} over { ital "RC"} } s+ { {1} over { ital "LC"} } } } \) } {}

Two-minute miniquiz problem

Problem 3-1

Given the system function

H ( s ) = s + 2 ( s + 3 ) ( s + 4 ) size 12{H \( s \) = { {s+2} over { \( s+3 \) \( s+4 \) } } } {}

  • Determine the natural frequencies of the system.
  • Determine a differential equation that relates x(t) and y(t).

Solution

  • The natural frequencies of the system are the poles of the system function and are −3 and −4.
  • The differential equation can be obtained by cross multiplying and multiplying by e st size 12{e rSup { size 8{ ital "st"} } } {} to obtain

( s + 3 ) ( s + 4 ) Ye st = ( s + 2 ) Xe st size 12{ \( s+3 \) \( s+4 \) ital "Ye" rSup { size 8{ ital "st"} } = \( s+2 \) ital "Xe" rSup { size 8{ ital "st"} } } {}

( s 2 + 7s + 12 ) Ye st = ( s + 2 ) Xe st size 12{ \( s rSup { size 8{2} } +7s+"12" \) ital "Ye" rSup { size 8{ ital "st"} } = \( s+2 \) ital "Xe" rSup { size 8{ ital "st"} } } {}

so that

d 2 y ( t ) dt 2 + 7 dy ( t ) dt + 12 y ( t ) = dx ( t ) dt + 2x ( t ) size 12{ { {d rSup { size 8{2} } y \( t \) } over { ital "dt" rSup { size 8{2} } } } +7 { { ital "dy" \( t \) } over { ital "dt"} } +"12"y \( t \) = { { ital "dx" \( t \) } over { ital "dt"} } +2x \( t \) } {}

VIII. TOTAL SOLUTION

The general solution is

y ( t ) = n = 1 N A n e λ n t + XH ( s ) e st for t > 0 size 12{y \( t \) = Sum cSub {n=1} cSup {N} {A rSub { size 8{n} } e rSup { size 8{λ rSub { size 6{n} } t} } + ital "XH" \( s \) e rSup { ital "st"} size 12{ ital "for"`````t>0}} } {}

and

y(t)=0 for t<0

Hence, provided there are no singularity functions (e.g., impulses) at t = 0, the general solution can be written compactly as follows

y ( t ) = ( n = 1 N A n e λ n t + XH ( s ) e st ) u ( t ) size 12{y \( t \) = \( Sum cSub {n=1} cSup {N} {A rSub { size 8{n} } e rSup { size 8{λ rSub { size 6{n} } t} } + ital "XH" \( s \) e rSup { ital "st"} size 12{ \) u \( t \) }} } {}

As we shall see later, no singularity functions occur in the response provided the order of the numerator polynomial of H(s) does not exceed that of the denominator.

1/ Initial conditions

To completely determine the total solution we need to determine the N coefficients

{ A 1 , A 2 , . . . , A N } size 12{ lbrace A rSub { size 8{1} } ,A rSub { size 8{2} } , "." "." "." ,A rSub { size 8{N} } rbrace } {}

Questions & Answers

what are the money value
Wisdom Reply
Nothing more than a purchase power, in other words, $100 now, must have the same value after 1 year.
Carlos
what is Monopoly
Rebecca Reply
what is money
Lawal Reply
It can be define as a big transaction that can control any business for one place to another base.
Akinlo
money is recognisable note to accept both parties selling and buying
Hassan
i don still understan
Rene
hey
Abdul
hi
Rene
money is anything generally accepted as a medium of exchange
Awwal
Money is anything generally accepted as a medium of exchange and for the settlement of goods and services .
Korda
hi good ppl, pls help out
Tumi
discuss human and natural resources as develop strategies ro improving living condition of citizens in developing countries.
Tumi
I don't understand the question.
Naomi
it's a form of currency used for 2 or more individuals or parties in order to reach their amicable personal or business attainment. one must understand that money itself can manifest in multiple fashions for which the individuals or parties adheres.
are u trying to say we shld discuss ways in which human natural resources help in improving living condition of citizens in developing countries?
Naomi
money is a legal thunder generally accepted as a medium of exchange for the payment of debt ,goods and services
Naomi
money is a way of payment.
Carlos
money is any thing that is generally accepted as a medium of exchange good for good and settlement of debt and means of payment
Yillah
money is nothing but a object which is used for exchange of goods and services.
Harshita
money is anything that is generally accepted as payment of goods and services and settlement of debt
Rebecca
what is demand
Melissa Reply
demand is where the customer is willing and able to buy goods and services during a given period of time
idk
demand is the ability and willingness of an individual to buy goods and services at a given price in a particular period of time
Alpha
demand is the ability to buy a specific quantities of goods and services at a given price and at a specific period of time
rosemary
what are the rules of demand
rosemary
Rosemary Nsebon, Do you mean laws of demand?
Alpha
what are the rules of demand
Rene
the rule of demand is the higher the price the lower the quantity demanded and the lower the price the higher the quantity demanded
mbi
thank
Rene
what is unemployment
Rebecca
unemployment is a scenario or a phenomenon in an economy whereby people are willing are able to work but cannot a job
mbi
Suppose you have a team of two workers: one is a baker and one is a chef. Explain why the kitchen can produce more meals in a given period of time if each worker specializes in what they do best than if each worker tries to do everything from appetizer to dessert. please I need a urgent answer
Oladosu Reply
Enables individuals and countries to consume a variety of goods and services
Iddrisu
what is the meaning of competency
Oladosu Reply
competency is an ability and courage to do something perfectly
Abdullahi
ability to perform some task
Segun
rival
Ray
thanks 🙏 it is also the same with the core competency
Oladosu
A sufficient supply
Ebenezer
Ebenezer you mean the (core competency) right?
Oladosu
what is mean,median and mode
Ikeh Reply
mean is the average number of a given data
Gallant
median is the middle number of a given data
Gallant
in a given data sorry
Mitchel
hi
Sajib
Pls am new here
Physcal
what are development bank in Nigeria
Adedigba Reply
.hi
Physcal
hi
Adedigba
hw
Physcal
and cool
Rosie
nice to meet everyone
Rosie
hi how are dears
Mumtaz
how can we development economic in our country
Mumtaz
hi
Charm
Payroll and​ 4p
Wasuroj
Agriculture
Wasuroj
Export
Wasuroj
Transport
Wasuroj
Change management​ and​ cerrancy
Wasuroj
Empoyee
Wasuroj
Lawyer army and​ Lawyer​
Wasuroj
animal husbandry essay
Rakuane Reply
what's the primary location of capital and money market respectively
ALIMI
what is bank
Nyakeh Reply
A bank is an institution set up purposely for the save keeping of money and other valuables
Alpha
A bank is a financial institution which helps people to save their money
Cyprian
pls define the HRM and HRP
Mumtaz
we have no money in bank....the bank owes us
Ray
When a supply curve start from the origin price elasticity of supply is unitory. Provide a simple proof
Felix Reply
Oui
Bobbo
please help someone should help me this question
Felix
ok
Anita
what is price
Divine Reply
the perchesing amount of something is called price
Nasir
OK pls tell me about economic elasticity of supply and demand
Mumtaz Reply
elasticity in economics is a measurement of the ratio of percentage change in quantity of a particular commodity to the percentage change in a factor that influence demand-price, consumer's income and price of another good
Epie
same with supply. How ever economics focus only on price elasticity of supply(PES)
Epie
using diagrams defferentiate between price ceiling and price floors
VIDELIS Reply
price ceiling lies below the equilibrium price and vice versa
Freeman
who is a broker
ALIMI Reply
a broker is a middle person between two other parties who makes all the arrangements required to conduct the the transaction.
rkesh
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10803/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask