Signals and Systems 03

7/29/2019 Signals and Systems 03

1/8

19

EEE303:SignalsandLinearSystems

Convolve ( )x t with ( )h t asshownbelowwhereT=3sec.

( ) ( ) ( ) ( ) ( )n

y t h t x t h t t nT

=

= = ( ) ( ) ( )n n

h t t nT h t nT

= =

= = .

TheresultingoutputisshowninFigure(b)above.

PeriodicConvolution

If

1( )x t and

2 ( )x t areboth

periodic

signals

with

common

period

0T ,

the

convolution

of

1( )x t and

2 ( )x t does

notconverge.Inthiscasewedefineperiodicconvolutionof 1( )x t and 2 ( )x t as,

0

1 2 1 20

( ) ( ) ( ) ( ) ( )T

y t x t x t x x t d = = .

FrequencyResponseFunctionofaLinearSystem

Let, ( )j t

x t e = .Theconvolutionintegralgivestheoutputas, ( )( ) ( ) j ty t h e d

= .

Or, ( ) ( ) ( )j t j jy t e h e d H j e

= = ,

Wherewedefine,( )( ) ( ) ( )j j H jH j h e d H j e

= = .

Theoutput ( )y t isthenexpressedas,( ( ))( ) ( ) j t H jy t H j e += .

Theoutputisacomplexexponentialofthesamefrequencyasinputmultipliedbythecomplexconstant

( )H j .This ( )H j iscalledthefrequencyresponseofthesystem.

Similarthingswillhappenforsin t and cos t input.Thefunctions ,sin and cosj t

e t t arecalledEigen

function,aswegetthesamefunctionintheoutputasintheinput.


2/8

20

Example

Theimpulseresponseofasystemisgivenas,/1( ) ( )t RCh t e u t

RC

= .Findtheexpressionforthefrequency

response.Whatwillbetheoutputofthesystemwith, ( ) cos2000 cos20000x t t t = + ?

/ ( 1/ )

0

1 1 1/( ) ( ).

1/

RC j j RC RCH j e u e d e d

RC RC j RC

+

= = =

+

Or,

( )

1tan ( )

22

1/( )

1/

j RCRCH j e

RC

=+

.45 84.29

1 2( ) 0.707 ; ( ) 0.995 .j jH j e H j e = =

o o

Thus, ( ) 0.707cos(2000 45 ) 0.995cos(2000 84.29 )y t t t = + o o .Ans

BlockDiagramRepresentation

Theimpulseresponseandthedifferentialequationdescriptionsrepresentonlytheinput=outputbehaviorof

asystem.Ontheotherhandblockdiagramrepresentationdescribesadifferentsetofinternalcomputations

usedtodeterminethesystemoutput.

Ablock

diagram

is

an

interconnection

of

elementary

operations

that

act

on

the

input

signal.

It

is

amore

detailedrepresentationofthesystemas itdescribeshowthesystems internalcomputationsareordered.

Blockdiagramrepresentationsconsistofaninterconnectionofthreeelementaryoperationsonsignals:

1. Scalarmultiplication: ( ) ( )y t cx t= ,2. Addition: ( ) ( ) ( )y t x t w t= + ,and3. Integration: ( ) ( )ty t x d

= .

AnNthordersystem is representedby theequation,0 0

( ) ( )k kN M

k kk kk k

d y t d x t a b

dt dt = == . Inorder todepict the

system

in

terms

of

integration

operation,

let

us

assumeM N=

,

1Na=

and

multiply

both

side

of

the

above

equationbyND .Afterrearrangement,

1 2 ( 1)

1 1 2 2 1 1 0 0( ) ( ) ( ) ( ) ( ) ( )N N

N N N N Ny t b x t D b x a y D b x a y D b x a y D b x a y

= + + + + + LL

(1)

Usingtheaboveequationwecandrawthesimulationdiagramas,

Thisformofrealizationiscalledcanonicalrealization.Differentiatorsarenotusedtosimulateasystem,asa

differentiatorenhancesnoise.Ontheotherhandintegratorssmoothorsuppressnoisepresentinaninput.

cx(t) y(t)

w(t)

y(t)x(t)

t

x(t) y(t)


3/8

21

Equation(1)inthepreviouspagecanbewritteninanotherformforN=2as,1 2 1 2

2 1 0 1 0( ) ( ) ( ) ( ) ( ) ( )y t b x t b D x t b D x t a D y t a D y t = + +

(2)

DirectformIanddirectformIIimplementationofthesystemisshowninfigures(a)and(b)below.

ExamplesConstructtheblockdiagramrepresentationofthesystembelow.

Or,2 1 1 1( ) ( )D D v t Di t

RC LC C

+ + =

; here,1 0 2 1 0

1 1 1, ; 0, , 0a a b b b

RC LC C= = = = = .

Thecanonical

block

diagram

representation

of

the

system

is

shown

below:

Thesystemcanbesimulatedusingmultiplier,summerand

integratorusingOPAMP.

State-variableRepresentation

The statevariable description of an LTI system consists of a series of coupled firstorder differential

equationsthatdescribeshowthestateofthesystemevolvesandanequationthatrelatestheoutputofthe

system to the current state variables and the input. These equations arewritten inmatrix form. State

variableanalysistransformsanNthorderdifferentialequation intoNfirstorderdifferentialequationsofa

setofstatevariables.

Thestateofasystemisdefinedasaminimalsetofsignalsthatrepresentsthesystemsentirememoryof

thepast.Givenonlythevalueofthestateatapointintime 0t ,andtheinputforthetimes 0t t wecan


4/8

22

determinetheoutputforalltimes(or, 0t t ).Thestateof

asystemisnotuniqueandtherearemanypossiblestate

variabledescriptionscorrespondingtoasystemwitha

giveninputoutputcharacteristic.

Considertheelectricalcircuitshowninfigureright.Derive

thestatevariabledescriptionforthissystemiftheinputis

( )x t andtheoutputisthecurrentthroughtheresistor ( )y t .Choosethestatevariablesasthevoltageacross

eachcapacitor.

1 1

1

1 1

( ) ( ) ( )

1 1, ( ) ( ) ( )

x t R y t q t

or y t q t x t R R

= +

= +

Thisequationexpressestheoutputasafunctionofthestatevariablesandtheinput.

Let, 2 ( )i t bethecurrentthrough 2R . Weget, 2 1 22 2

1 1( ) ( ) ( )i t q t q t

R R= .Also, 2 2 2( ) ( )i t C q t = & .

Thus,2 1 2

2 2 2 2

1 1( ) ( ) ( )q t q t q t

C R C R= &

Again,currentthrough 1C is, 1 1 1 2 1 22 2

1 1( ) ( ) ( ) ( ) ( ) ( ) ( )i t C q t y t i t y t q t q t

R R= = = +&

Thus,

1 2 1 2 1 1 2

1 1 2 1 2 1 1 1 1 1 2 1 2

1 2

1 1 1 2 1 2 1 1

1 1 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1( ) ( ) ( )

q t y t i t y t q t q t q t x t q t q t C C R C R C R C R C R C R

q t q t x t C R C R C R C R

= = + = + +

= + + +

&

1 1

1 11 1 1 2 1 2

2

22 2 2 2

( ) q (t) 11 1 1

( )1 1

( )0q (t)

q t

C RC R C R C Rx t

q tC R C R

+

= +

&

&

; Stateequation

1

1

2

( )1

( ) ( )

0 ( )

1

1

q t

Ry t x t

q tR

= +

; Outputequation

Or,q(t) = Aq(t) + Bx(t)

y(t) = Cq(t) + Dx(t)

&,wherethematricesA,B,CandDdescribetheinternalstructureofthesystem.

Here,Aiscalledthesystemmatrix,Biscalledtheinputmatrix,CiscalledtheoutputmatrixandDiscalled

thetransmission

matrix.

Findthestatevariabledescriptionofthecircuitdepictedinfigurebelow.

(Ans)


5/8

23

Inblockdiagramrepresentation,thestatevariablecorrespondstotheoutputsoftheintegrators.Consider

thefollowingexample:

Theblockdiagramaboveindicatesthat,

ForanNthordersystemwithminputandpoutput,thedimensionsofA,B,CandDareasfollows:

( ), ( ), ( ) and ( )A N N B N m C p N D p m .

State-spacerepresentationofadifferentialequation

Thegeneral

form

of

an

Nth

order

differential

equation

is,

LetusdefineNstatevariables 1 1( ), ( ), , ( )Nq t q t q t L as,

1

2

1

( ) ( )

( ) ( )

( ) ( )NN

q t y t

q t y t

q t y t

=

=

=

&

M Then,

1 2

2 3

1 1 2 1

( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )NN N N N

q t q t

q t q t

q t y t a q t a q t a q t x t

=

=

= = +

&

&

M

& L

And,1

( ) ( )y t q t= .

Inmatrix

form

the

above

two

equations

can

be

expressed

as,


6/8

24

Where,

Solutionofstateequations

Letusdeterminethesolutionofthestateequation, ; 0q(t) = Aq(t) + Bx(t) x(0) = x& .

Rewritingtheequation, =q(t) Aq(t) Bx(t)&

Multiplyingbothsidesbyt

e A

weget, [ ] [ ]t t td

e e edt

= =A A Aq(t) Aq(t) q(t) Bx(t)&

Integratingbothsidesbetweenthelimits 0 andt,weget

0 0

ttte e d

= A A

q(t) Bx( )

Or,0

0t

te e d =

A Aq(t) q( ) Bx( ) or,

( )

00 0

tt t t t e e d e e t

= + = + A A A A

q(t) q( ) Bx( ) q( ) Bx( )

Iftheinitialstateisknownat 0t t= ,theaboveequationbecomes,

0

0

( ) ( )

0

tt t t

te t e d

= +

A Aq(t) q( ) Bx( ) .

ThematrixfunctionteA isknownasthestatetransitionmatrixofthesystem.

Theoutputequationisgivenby,

0

( )0t

t t

te e d = + +

A Ay(t) C q( ) C Bx( ) Dx(t) .

Thezerostateresponseofthesystemis(when 0=q(0) ),t te t e t = + = + A Ay(t) C B x( ) Dx(t) C B x( ) D(t) x(t)

Or, [ ]te= + = Ay(t) C B D(t) x(t) h(t) x(t)

Thematrix

h(t) is

known

as

impulse

response

matrix.

Evaluationof teA

Let,Abean ( )N N matrixand I bean ( )N N identitymatrix.TheEigenvalues , 1,2, ,i i N = L ofA

aretherootsoftheNthorderpolynomial, det[ ] 0 =A I .

Evaluationofthestatetransitionmatrixt

eA isbasedontheCayleyHamiltontheorem.ThistheoremstatesthatthematrixAsatisfiesitsowncharacteristicequation,i.e.,

If,1

1 1 0( ) 0N N

NQ a a a

= + + + + =L

Then,1

1 1 0( ) 0N NNQ a a a

= + + + + =A A A A IL

1

1 1 0

N N

Na a a

= A A A IL (1)

Now,

2 2

2! 2!

N Nt t t

e t= + + + + +AA A

I A L L

Applyingequation(1),teA maybeexpressedas,

2 1

0 1 2 1

t N

Ne

= + + + +A

I A A AL (2)

Ifall i saredistinctwemaywrite,2 1

0 1 2 1itN

i i N ie

+ + + + =L fromwherewecanfound s

as,

We canalso calculateteA using Laplace

transform.


7/8

25

Example

Findastatespacerepresentationofthesystemasshowninfigurebelow.

The system is of order 3.We assume three state variables 1 2 3( ), ( ) and ( )q t q t q t . From figure abovewe

obtain,

(Ans)

Example

Finda statespace representationof thecircuit shown in figurebelowassuming that theoutputsare the

currentsflowingin 1 2andR R .

Wechoose

the

state

variables

1( ) ( )Lq t i t = and2 ( ) ( )Cq t v t = . Let,

1 1( ) ( )x t v t= and2 2( ) ( )x t v t= . Also,

1 1( ) ( )y t i t= and 2 2( ) ( )y t i t= .

ApplyingKVLtothetwoloopsweget,


8/8

26

Rearrangingandwritinginmatrixformweget,

(Ans)

Example: Find teA for0 1

6 5

=

A usingCayleyHamiltontheorem.

ThecharacteristicpolynomialofAis,

21

( ) 5 6 ( 2)( 3)6 5

q

= = = + + = + +

+I A .

ThustheEigenvaluesofAare, 1 2 = and 2 3 = .

Hencewehave,0 1

0 1

1 0 16 5

te

= + =

AI A ,wheresareobtainedas,

2

0 1

3

0 1

2

3

t

t

e

e

=

= solvingweget,

2 3

03 2t te e = and 2 31

t te e = .

(Ans)

Documents

Signals and Systems 03