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& ) " ) ' % & ! ' % ! V " H 0 U ! R W ! ' " ) ' " W ( ! & 0 R T & ) " 0 U ! R T W & ! H I R H ' & S

yZSR% S 0 T I ( 0 R ( ! 3 ( 3 U ! " 0 ! ) ' % & ! ' T % I H W % ) I U ! ) I # 4 # X g # I

yZIR% S 0 % R V ! 3 ( !

0

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andny(t)

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bM= 0

Q V !

& ! ( W T ! H 0 ! ! U " W S ! " ( ! & S % P & W

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aNndny(t)

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dtm.(2.1.2)

( 2.1.1)T U Q ! " % ' W R ! R 0 Q ! " % ! " ' ! ) T U 0 ( ! " T ) Q 0 U Q ! " V

y(k)(t).

= dky(t)dtk

Q " % R

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{y(t0), y(1)(t0), . . . , y(N1)(t0)} {y(t) : t t0} $ % ( 2.1.1) $ # t0 % &

d R ! S & ! W % ! R " ( " Q ! " %

x(t) & W ( 0

tW 3 & ( T ' " & U 0 ' R ! S ( 0

xd Q " % T 3 ! R P & W

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x() 0 ! U " ( U T W " R ! ' ' R ! S ( 0

x(t)d Q " % R ! R I T W ) S ' R 0 T ( W T

x1

( 2.1.1)

y1

2.1.2` D 9 b

{y1(t0), y(1)1 (t0), . . . , y(N1)1 (t0)},

x2

( 2.1.1)

y2'

{y2(t0), y

(1)2 (t0), . . . , y

(N1)2 (t0)

},

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( 2.1.1)

y1 + y2 &

{y1(t0) + y2(t0), y(1)1 (t0) + y(1)2 (t0), . . . , y(N1)1 (t0) + y(N1)2 (t0)}.

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( 2.1.1)d R P ` 7

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! 3 ( & ) 3 " T I ( 0 R ( W ) I & ) "

x

% R V ( ! 0 & ! W P 0 Q ! " ( V & W " & ) 3 " & ) "

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y0 ! " ! 0

T I ( 0 R ( T % I ( T ) V

Q ! ) R U R ! U 0 & U & ) " T Q ! & ( S ( R ( ! U T W & ! H I R ( ! V & W " ( ! 3 " V & ) " Q ! ) R U R S T

R 3 ! " ! Q ! & ( S ! & S Q ! & ( S ! U I 0

(

t

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( 2.1.1) ! %

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"

( 2.1.1) $ % %

2.1.4` D 9 b

#

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( ! V & W " & h ) "

"

yh + yp

T U "

2.1.2 R W " W ! R P

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% S 0 % R V Q ! & ( S T ! % S 0 T I ( 0 R ( Q ! & ( S T ' ! T I ) T W 0 Q ! & ( S ( R T R U U

#

x 0 % ( 2.1.1) % 2.1.5 ` 8 F B `

yZIR' & % %

#

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( 2.1.1) %

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( 2.1.1) $ %

y.

= yZIR + yZSR 2.1.6` D 9 b

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yZIRQ ! & ( S R 3 ! " ! Q ! & ( S 0 !

! & V W ! R T W T S U " " Q V ! " V % S 0 T I ( 0 R ( & ! W ' & & S Q ! & ( S 0 !

yZSR( 0 H ( " ! W T

& S Q ! & ( S 0 ! R 3 ! " ! Q ! & ( S ! & S Q ! & ( S T U ! V % ! R 3 ! " ! Q ! & ( S 0 ! R 3 ! " ! ( ! R ! & ( S T U & 0 R T

( 2.1.1) $ " #

D & & ! %

2.1.7` 8 F B `

Nn=0

anDny(t) =

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bmDmx(t)(2.1.3)

( $

0

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0

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ann(2.1.4)

( 2.1.4) "

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! % & $

1, . . . , N' %

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% # ( !

0

! %

fi(t)

fi(t) = eit & $ i

& &

i = i+1 = = i+k # k + 1 i

0 j k fi+j(t) = tjeit

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September 2006


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( ! V & W " & h ) "

& ) " Q ! & ( S V ! R 0 & " 3 ! ) V ( ! U " U " & ) " & U 0 V ( ! V & W " T & ( ! ( T T V & ! % I ( R H W %

) T % R V ( W S U H T U H V & ( R T I ( 0 R ( 3 ! % R V 3 ! T (

0 ! % R V Q & U ' ( ! & ) 3 " ( ! V & W " Q R ! ( R

(

0

% #

' % #

Input Output MapIOM '

2.2.1` 8 F B `

Y' % # (

0

% #

X' % #

! & U & ) & U ' ( 0 Q " % R

H( V & W " T 0 & ' R P 0 Q ! " % U " U ( R ! R 0 U I " & ! T

HEx Ey

0 % R V ! S " ( V & W " P 1 # & ! 0

P

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y = (x)d V 3 ( 0 H Q " % R !

& W ( 0 ( W ) T & R & U 0 V ! T ! V ( ! 0 T W

! S " T W ! S T I ( P V Q T U H Q ! " %

y(t) = [x](t) ( ! 0 ' R ! S " W 0 U

! S " ( 0 ! (

t( 0 R

tQ ! ( R W 3 &

y 0 ! ( T U

0 U 0 I R P U & ! S " ! H ) ! W Q R 0 T Q V T ! 0 % R V ! S " ( ! V & W " & ' W ' ! % W R U "

0 % R V ! S " 3 ! % " 0 R ! ) R ( V & W " ( & I 0 & " 0 R 0 T

% &

Y'

X $ %

2.2.2` B F

& # % & # # $ (

0' $ !

0

$ &

& # ! & # $ ! # &

Q " H & 3 ! % 0 ! ( V & W " ! ( ! 0 T U R S 0 " ) I 0

n% &

x(n)' & %

$ $ % & $ $ % &

2.2.3` 8 F B `

% & # % & $ $

$ $ % & % $ $ % &

& $

% & % % &

% & $ $ % & $ $

Version 1.1 15 September 2006


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( ! V & W " & h ) "

S ! % Q " H & T W ) ) Q " H ( ! 0 0 !

x(n0), x(n0 + 1), . . . , x(n1)P & ) % U W " T 0 ! ) ) Q " H ( ! 0

S ! % Q 0 ! 0 ) W n0 d " T U " T S ! % d Q 0 I 3 ( ! T T ! V Q " H &

% & $ %

( 2.1.1) $ $ !

2.2.4` B F

y !

x ! $ $ ! $

y = Ax(2.2.1)

x, y' #

A

$ $ !

y(n) = x(n) + 2x(n 1) 1 1 1

( 2.2.2)

0

$ $ % & ! % $ $ % &

! $ % & (

10 $ #

% &

0

$ ! ( % &

0

%

$

( $ $

) I 0

y ! 3 ( 3 ! V ! & " T ! 0 & T ' % & S % " 0 !

x(t) ( 2.1.1) ) T W ( & ) 3 ! " ( V & W " & ! W

( ) " "

$ ' $ % ! %

SISO: Single Input Single Output

2.2.5` 8 F B `

%

MIMO: Multiple Input Multiple Output

MISO'

SIMO $ ! $

t% & " " $

t% & % & % &

2.2.6

` 8 F B `

( ! " R ) ( ! V & W " T ' & Q ! " V ( ! W " U " ( T W 0 ! H & ) 3

" $ % & %

( 2.1.1) $ $ !

2.2.7` B F

! ! " ( & ! $

0

!

y(t) = 2x(t) y(t) = x2(t) + (t + 1)2 1 1 4

% & " # $ %

y(t) = x(t 1) , y(t) = x(t + 1) 1 1 X

$

0

! " $

t ! ! " % % & %

( #

% & % %

( 2.2.2) $

( 2.1.1) $ $ !



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( ! V & W " & h ) "

! H & ) 3 ! ! ! & W % W R V V & W ' & ! T (

y(t) ! 3 (

tW 3 & T V 0 ( ( % ( 0 & ' R ( V & W "

! R T W & H 3 & ! T Q V U ( ( % R 0 ! H V & ) 3 S T & H ! 3 0 U ( V & W " T U " T P ( " ( " R I " ( ( W

& ( ! ( V ! % " & ) 3 & ! U Q V T ( & H 3 R T U & ) 3 ! & V H ! ! T & W " W " % R V V & W ( 0 ( W ) T

% &

y(t) " #

t < t0'

t0

Causal, Non anticipative

2.2.8` 8 F B `

{x(s), s < t0} ' # " $ t % & "

%

( 2.2.2)

2.2.9` B F

& # ' !

0

$

( $

" # % %

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y(t) = x(t + 1) 1 1

$ " ! " %

y =dx

dt

1 1 f

! $ $ ) $ ! # $ !

!

t ! " #

t

x(s) $

t ! & ! " $ $

&

$ ! # & ! & # & !

! % % & " # !

( 2.2.6)

t ! # ! "

& U '

( %

R 0 U

T ) ! " ( ! R T I ! R

( W T

T ! 0 ( ! ( % Q ! " V Q ( ! T ' H S ( ! V & W " ! &

Q ! " Q 0

t R ( U " T 0 & I 0 ( ! ( ! 0 3 0 R ! ! T & ! R 0 ( ! ( % 3 U ! " U Q ! " V ( ! R ! " ( ) ! W T U

( ( % R 0 U ( T ' H S ( V & W " Q V ( ( H 0 Q " H T U T ' H S

,

x1, x2 ! &

2.2.10` 8 F B `

(x1 + x2) = (x1) + (x2).

$ ! ! #

0

3

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( 2.2.2)( V & W " V ) " W ! R & ) 3 "

( & 0 R T

P ( ! 0 ( ! R ! V ( ( U ( 0 ( " ' " 0 0 ' & ! 0 ( & 0 R T 0 ( V & W "

d !

(x1 + x2) = (x1) + (x2)P

(Additivity)( ! ) 0

(x) = (x)P

Homogeneity( ! R 3 ! " !

Version 1.1#

$

September 2006


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( ! V & W " & h ) "

( R ! V ( ( 0 ( " ' " % R V T U U " " ' T I 0 0 & U 0

y = Re[x]( V & W " " 3 ! ) T

) ! U " " & S % " 0 ! T 0 " U ) Q V U

Re[x] = Re[x] H 0 V ! & " & S % " 0 ! 0 T ! 0 ( ! ) 0 V ! & " " ) ' " & ( " ! R 0 U " ( & 0 R T ( V & W " R 0 ! H Q V T V ! & " & S % " Q "

& T ' % 3 T 3 0 T 0 ( V & W " 3 T W ) " ' & 0 T ( U & ! ) ( ! & 0 R T ( T 0 U V T ! " U P ( " ( " & W

( & I 0 & " 0 R 0 0 T 0 V ! & " & T ' % I R R ! R 0 ! V ! V ! & " U " " P & ( ! "

( ! ( ! 0 T ! 3 ( I ( R T & 0 U Q ! R T & U S 0 ( 0 & ( ! W " U " ( T W R ! V ( 0 ( ! & 0 R T ( R ! V (

& ( ! ! U S ( ! ( ! 0 T U % ! 0 T ' ! & S ) T W V ! % "

d U Q ! ! V

b = 0 0 ( & 0 R T R 0 y = a x + b ( V & W "

(x1 + x2) = a(x1 + x2) + b = ax1 + b + ax2 + b = (x1) + (x2).

[x] [z] = [x z] ' " Affine 2.2.11 ` 8 F B `

0 R ( W

( 2.1.1) ) T W ( & 0 ! ( " ( V & W "

(x) = axd d d ( R S 0 0 T W T

y = a x + b( V & W "

T I ( 0 R ( & U 0 V % R V ( ! ( ! 0 R U T ! 3 ( U I R 0 P ( R S 0 ( V & W " 0 R ! ( R ! W ! ' T I (

& " ! T V ( & 0 R T ( V & W " W ! ) V ! H ! d d d % S 0 T I ( 0 R ( & ! W

( 2.1.1)( 0 ' U & S H 0 H

) V T I ( 0 R ( ( W S U T T 3 % R V T % I ( & 0 R T ( V & W " ( & 0 ( " R 0 ( T 0 R & S ) 0 ! ! U "

% S 0 " R ! U T I ( 0 R ( !

mT V T

bm = 0& U 0 V

( 2.1.1)T W ! U I R ( 0 H ( ! 0 & T

& " ! T V % S 0 T I ( 0 R ( ( V & W " T ( ! % I ( ) T W 0 & ) " ( ! & 0 R T ( R ! V ( T ' T ( I 0 & )

T I ( 0 R ( & ) 3 ! " ! W ! ' Q " H & ! I T T 0 R ! R 0 P Q ! & % I U ! H & ) T ) T

yZSRQ ! & ( S T ( ! % I (

0 3 U ! " & ) V T W & 3 ( R

x(s) = 0' " (

0tx

% & $ $ $

x(t) $ $ $

2.2.12` 8 F B `

$ $ $ $ ! $

s < tx

u(t) =

0 t < 0

1 t > 0

1 1

$

(

3.3 $ #

t = 0' $ ! &

0

# ! $

x % $

u' % #

x(t) = x(t)

u(t

tx

)

x(t) = x(t) u(tx + t) $

% S 0 ( " ( ! 0 0 ! R " ) ) ) I ( ! 0 " 3 ! ) T

tx = 0d U 0 R ! ! V H 0

txQ " H T U & ! S " % I ( R 0 T 0

0d T T 0 " U "

!

(Initially At Rest) $ $

2.2.13` 8 F B `

s < tx

y(s) = 0 # $

s < tx

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2

September 2006


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( ! V & W " & h ) "

$

2.2.14` D 9 b

T I ( 0 R ( ( ( T I ( I ! R " ( V & W " ( & ) 3 " R " ( ! 0 0 ! R " ( ! ( ! 0 ! V % V T U R P 7

T U "

2.1.2 R W " ( W ! R ( ! & 0 R T Q V T ' S % " ) ' ! " Q " H & ! W % S 0

Q V 0 ! % R V ( W S U R S T ! 3 ( Q 0 P ( ! ( % ( & ) 3 ( 0 ! R T & V H " ( ( T I ( I ! R " ( & ) 3

P H 3 U ! " ) T Q & ! W ! ( ! & 0 R T ( ! V & W " & ! W

2.2.15b E G

t0Q " H & ! W " ' "

x1(t), x2(t) % R V ( ! ( ! 0 R U Q R ! ( R ( ( T I ( I ! R " ( V & W " U I R R P

7

0 ! U T V

x1(t) = x2(t) t < t0, 1 1

2

x1(t) = x2(t) t > t0. 1 1 e

! 3 (

x1(t) x2(t) % R V & ! W V W ! R ( ! & 0 R T " xi(t) % R V T ( V & W " ( ! 3 ( ( 0 yi(t) d Q " % R

y1(t)

V W ! R ( ( T I ( I ! R " ( R ! V ( "

t < t0& ! W

x1(t)

x2(t) = 0

d U Q ! ! V

y1(t)

y2(t)

(

! T ( R 0

t0Q " H ! 3 ( d ! 0

t < t0& ! W

y1(t) = y2(t)d U V 0 ! R T '

t < t0& ! W

y2(t) = 0 ( ( % ( V & W " & " ! T V % R V T U ) ( W V & W

' "

x(t) % R V ( ! 0 W ' R ( ( T I ( I ! R " ( ! 0 & T ) V ( ( % ( V & W " U I R R R U Q ! ! V

Q " % R !

x1(t) 0 & ) 3 R t < t0 & ! W y(t) = 0 ( " ' " y(t) ! 3 ( 3 V 0 & R ! t < t0 & ! W x(t) = 0d T ! ! U

x(t) x1(t) % R V T y(t) y1(t) ! 3 ( V W ! R ( ! ( % " x1(t) d T ! 3 ( ( 0 y1(t) d ( & 0 R T ( V & W " ( ! 3 (

y1(t) 0 ( ! & 0 R T " R U ) " H ! I ( H ( ! ( ! 0 Q V U t < t0 & ! W y(t)Y @ Y G Y

( ( T I ( I ! R " ( V & W " !

t < 0& ! W

y(t) = 0Q V T % S 0 ) " ( 0 % S ( % R V T

W ( S " & ! 3 R ( T T ! V ( ( T I ( I ! R " R 0 U & h ) " V " 3 ! ) & ) ( W V 0 & R

% $ $ %

2.2.16` B F

y(t) + 2y(t) = x(t) . 1 1 # g

2.1.2 ' ' ' (

0

&

y(0) = 0 % # $ ! # ! $ &

y(0) = 0 # % & %

x1(t) =

1 1 t 00 otherwise.

1 1 # #

x2(t) 0 . 1 1 # 1

x1' ! $ &

y1(t) =

0 t 012

1 e2t 1 t 0

12

e2 1 e2t t 0 .

1 1 # 4

$

t% & !

t < 1 y2(t) 0 ' %



21/188

( ! V & W " & h ) "

! U I 0 ! ) ! ' S U " ( ! & 0 R T 0 R ( V 0 & R

" #

2.2.17` B F

y(t) = x(t)x(t + 1) . 1 1 # X

" #

y(t) = x(t) + 1 . 1 1 #

s % & & &

s' %

2.2.18` 8 F B `

sx(t).

= x(t + s).(2.2.16)

Time InvariantTI% &

(sx) = s(x)(2.2.17)

& & # ! & & !

H H & ! & S ! 0 W T I ( " T U ! S " 0 Q " H W ! ' 0 ( V & W " " ( " Q S ! 0

Linear Time Invariant( ! V & W " d d d h ' T Q " H ( ! W ! ' ( ! & 0 R T ( ! V & W " Q ' ! % W R Q ( ! V & W " ! &

(LTI) & ) 3 S T V Q " H W ! ' R 0 Q ! " V

(x)(t) = t x(t) ( V & W "

s ([x]) (t) = s (tx(t)) 1 1 #

2

= (t + s)x(t + s) 1 1 # e

= tx(t + s) 1 1 1 g

= [sx](t) . 1 1 1 #

R U ) "

% & $

2.2.19` D 9 b

( ! 0 & T T '

y = xd Q " % R R " ( ! 0 0 !

sx 3 H 0 R " ( ! 0 0 !

x 0 V T U R !

sW ' R P ` 7

syQ V T

sd H H ! " Q " H 0 H T I ( 0 R ( % ! R

sx % R V & ! W

( 2.1.1)( 0 & ( ! S

sy V

T U "

sx % R V T ! 3 ( 0 !

& H 3 & ! & S ! 0 ( 0 Q " % "

D V & V H R

Version 1.11 g

September 2006


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( " V % " " 3 ! ) & h ) "

# ! & ! & ! ! # $

2.2.20` D 9 b

d

dtx

=

d

dt(x) .

1 1 1 1

( V & W " & ! & S ! 0 W T I ( " & H 3 & ! & S ! 0 & " ! T V

y = (x)d Q " % R

dx/dt 3 V H 0

t0& " 0 R ! % " W 3 & ) W ( % S 0 ( "

x % R V 0 V T U R P

7

" ' "

y1 ! 3 ( !

x1 % R V U T ' R !

( 2.1.1)d

y1 = dy/dtd !

x1 = dx/dt R

x % R V T ! 3 ( ( 0

Y @ Y G Y 0 ! ! U " ( 0

%

2.2.21` 8 F B `

x " # & #

([x]) = x . 1 1 1 4

2.2.22` B F

(x) = x 1 1 1 X

$

y(t) = xn(t 1) 1 1 1

! &

n " # ! &

n

y(t) =dx(t)

dt

1 1 1 f

%

[x] = [x + ] 1 1 1

$

y $ "

x & % %

dy(t)

dt= x

1 1 1 2

( " R ) ( V & W " ( & 0 ( " ( ( T I ( I ! R " ( 0 " R & h ) " ) T W ( & 0 ! ( " ( V & W " V ! R 0 & ! V % T

R 0 & U 0 Q " H W ! ' ! ( & 0 R T ( ( % Q ! & V H ( T W

SISO 0 % R V ! S " 0 U & Q " H

V S I & V

Version 1.11 #

September 2006


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( " V % " " 3 ! ) & h ) "

Tv(t)

Ei(t)

R(t)

( T " U I ( V & W " P " 3 ! ) P 1 1 & ! 0

v) 3 R I ( " 0 ! 3 (

R) 3 R & ) & W ! "

i & H 0 % R V ( V & W " Q R ! ( R

) T W Q ! ( R I ( " ! & H Q & U '

v(t) = R(t)i(t). 1 4 #

) 3 R T U R ! U 3 ! % & ! W ( ! V & W " ( ! R ! V ( ( 0 I ( R R

W ! '

R = R0

I ( & 0 R T ! Q " H W ! ' Q ! & V H ( & % I ! H ( V & W "

) 3 R ( ! R ' ) H ' W ( ! ) 3 R ( ! R U ( 0 ( & 0 ( "

f(t)& U 0 V

R(t) = R0 + f(t)

II R 0 ( V & W " Q ! & V H ( & % I ( V & W " Q V T ! I V ! R V & W & ) ' & % R V ! T ( ! 3 ( & W ! H ( V & W "

( 2.3.1) S T Q V U Q " H W ! '

s[i](t) = sv(t) = s[(R0 + f(t))i(t)] = (R0 + f(t + s))i(t + s)

= (R0 + f(t))i(t + s) = [

si](t).

1 4 1

P ( ! & 0 R T ' ! ) R

[x1 + x2](t) = (R0 + f(t))(x1(t) + x2(t)) 1 4 4

= (R0 + f(t))x1(t) + (R0 + f(t))x2(t) = [x1](t) + [x2](t) 1 4 X

( ! & 0 R T ( & ) 3 S T ' ! )

T 3 & ) 3 R ! & ' 0 ! ) 3 R H 0 ' S % " T ! ) 3

R0 0 & H ! T ( ! V & W U ) 3 R ! H

R(t) = R0 + i2(t)

III T ! ) 3 V & W & ! W 0 T 0 R ' & H V & W & ! W

Q V U Q " H W ! ' 0 Q ! & V H ( & % I 0 ! H ( V & W "

s[i](t) = sv(t) = s[(R0 + i2(t))i(t)] = (R0 + i

2(t + s))i(t + s) = [si](t). 1 4

P ( ! & 0 R T ' ! ) R

[x1 + x2](t) = (R0 + (x1(t) + x2(t))2)(x1(t) + x2(t))

1 4 f

= (R0 + x21(t))x1(t) + (R0 + x22(t))x2(t) = [x1](t) + [x2](t) 1 4 $

Version 1.11 1

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& h ) " T & W " P ( ! V & W " & h ) "

( & 0 R T R 0 ( V & W " Q V T !

R(t) = R0 + t i

2() d

IV R 0 0 Q " H W ! ' ( V & W " V ( ! 0 & T Q ( R " ) ! ' ! U I ! " V Q ! & V H W ( V & W " ( 0 H V & ! &

( & 0 R T

( ! ( % ( ! V & W " T V V W ! R % R V T U " ) ! ' V & W ' & ( ! ! T ( ( ! ! 3 ( T V U Q ! ! V

! R 0 ! R R ! % R " ( ! T 0 R S ) ( ! 0 ! ! U " ) T W ( ! V & W " & ! 0 ( ' S ( % T 0 T W ! ) " T 0 U ( U ' ( "

& 0 ( T Q ( R V ! R ) " T # ' H S % & ! ' ! T 0 V ( ! 0 ! ! U " & ) & 0 R T T " U I T 3 W " T V & 0 ( T Q ( R V

W ) !

( T 0 R & S ) 0 ! ! U " ) T W

S ! 3 T U W ! R (

S % ! R T V U S I T ! U I % ) R T ! ' U 0 R H W % & ) ( ! I ( & ! 0 ( T U ( ! U I T & W "

) V ! 3 ( ( ! 0 T ' " ! % R V ( ! 0 % R V " T 0 h & ! I U % S ! ' V ( V & W " & 0 ( T ! U I % ) R " T

h ( ! & ! I U ( ! 0 % S ! ' T U & ! I & U S 0 " T V & ! U " ' R ! R V ( U " ( U T ! T W Q ! R V ( T W 3 T

T W ( & 0 ! ( " ( I 0 T V U I R R

H1, H2d ! R " ! % & U 0 ( ! V & W " ( U ( ! R ! ( R U I R R ( ( " V % & ! ! T 0 V

( V & W " T % R V ( ! 0 0 & ! & ! ( ! V & W " & ! I & ) 3 R

( 2.1.1) & ! ( T 0 R & S ) 0 ! ! U " )

0 ! " !

H2 R U ( V & W " T % R V ! ! " ! H ! 3 (

y0 ( V & W " ( ! 3 (

x0 !

H1 R ! U 0 &

P 0 & ! H & ! I Q " % R

z0 ! R U ( V & W " "

EyH1 H2Ex Ez

& ! ( ! V & W " & ! I P 1 4 & ! 0

0 ! ! U " ( & H W ( ! 0 & 0 ( T Q ( R 0

z0 ! T U 0 ! " !

x ( % R V U ! H ( T T ! V ( V & W " T W ! U I R

( 0 H ( ! U W T ! U S U ! R T Q 0 " ) ' " ( 0 U I T ) V ! 0 ! ! U " & ) % ! " Q V 0 ( T 0 R & S )

0 ( " ! % " % R V T ! ! H U ( T ! U " ! 3 ( ( 0 U I T H 0 ! ) & S R ( V & W " T V & ! ( S T & R

( T T ! V ( V & W " T U & ! 0 ( ( R ( ! R R 0

0 ! & U ( ( " V % & ! & 0 ! ( " T ' " & ! I

Q ( R 0

y1 + y20 ! T U 0 ! " & U 0 ( " V % " ( V & W " & ) 3 R ( S ( ! U " % R V ( ! V & W " ( U T Q 0 V

" ) ' " ( 0 U I T ) V ! 0 ! ! U " & ) % ! "

Q V

0

( T 0 R & S ) 0 ! ! U " ( & H W ( ! 0 & 0 ( T

T V & R U Q ! " V

! T 0 V ( ! V & W " & ! I " ( T ' ( " ( V & W " ( 0 & 0 ( T Q ( R ) V

T T V Q S ! 0

& ( ! V & ! " ! T T V & ! I & U S 0 U

T ' " & ! I ! & ! & ! I & 0 ( T T ' H 3 ! V 0 & R ! T ! ! R ! ' ( ! V & W " T W

3' & S ) " T R & U 0 V

( ! V & W " V Q T U 3 ! ( 0 U I T & h ) " ) T W ( ! & 0 ! ( " ( ! V & W " T ' T Q V T T V ! R ( ! V & W " T U ! V % W

Version 1.11 4

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& h ) " T & W " P ( ! V & W " & h ) "

Ex

E

E

H1

H2 Ey1

Ey2

Ey

+

T ' " ( ! V & W " & ! I P 1 X & ! 0

U " 0 & R U S V ( ! T 3 " U ! H U T 3 0 ( ! V & W " & ! I T U 3 ! ( W ( 0 & ! ( S T V ! ! T ! ! R ! '

H 3 ! % " ( ! V & W " & ! I T ! S T I ! R ! T T V T V Q ( ! R

6d d

5 ' & S & ) ( ! I ( ( V & W " & ! 0 (

Version 1.11 X

September 2006


26/188

& ! S % & ! ' S T ! T ! ! R ! ' ' T I & 3 R 0 ( ! & 0 R T d T & 3 R 0 ( ! R ! V ( P H & ! W U T & H I

0 h ! ) I % & ! ' S T ( & H 3 R ( & ) 3 I ( " ! & H Q W " d d d T ' T " U I T 3 W " ! & ! S S T ( T ) ( ' R ! S

# ) " " R ( U " ( S T I ! & 3 0 ( & U & U T T V ! ( ! & H 3 ( ! ' R ! S

P T & 3 R 0 & ) ( ! & ) 3 ! " 0 % R V ( ! V & W " ' ! % W R H ' & S

y(t) =

K(t, s)x(s) ds . 4 g #

d " Q ! ) R & S & ' " V

kernel h Q W & 3 ( 0 & ' R 0 ! ( V & W " ( ! 3 R ( ( 0 Q ! " V ( W ! '

K ' R ! S

0 ( & ) 3 U ! T ! ! R ! ' ( V & W

y(t) =

h(t s)x(s) ds . 4 g 1

& ' I ( V & W " ( 0 ( 3 "

h ' R ! S Q T % R V ( ! 0 Q ! T ! ! R ! ' ) T W ( & ) 3 ! " ( 0 H ( V & W "

% ) R W ' & T W Q V 0 & ! H I R ' & ) T U ( T ) ( ' R ! S & ' R " ( " T V T W ( R W U R H 3 ! % " ( ! V & W " T U

! H ' R ! S T

P T 0 ) 0 I ( " & ! ' " ! T 0 ) 0 T ' W ( V & W " Q R ! ( R

Q ! T W " T 3 ( " & W ! "

t = 0W 3 & % S 0 I ( " & " ! T V Q W " 0 T T d d d 0 ! T ' T U ( T I ( "

! R 0 ! ( V & W " ( T " U I ( ! ) 3 R ( T V Q 0 T 0 ) 0 & ! ' " ! T ' U Q ! V Q W R T ' ! & ! ' " ( 0 & I "

T ' I ( "

0W 3 & & I 0 % S 0 0 ! & H ! % S 0 0 ! T ' T W I ( "

0W 3 & R S T 0 ( ! 3 R ( T S "

' " T ' T W Q W " R U ) " & H U ! Q " H ' & S ! U Q 0 V 0

0 ! U & H !

V0 !

Q =

i(t) dt = C V. 4 # #

W 3 & T & S % S 0 ( " ( ! 0 P ( ! R ! V ( ( U ( 0 ' U T ' H S 0 T T 0 ) 0 & H ( ! 0 T ' ! ' H ! R 0 V 0

' & ) T U ( T ) ( ' R ! S 0 & ' R ! 0 ' % S ) T W 0 " ! H V ( ! 0 % S 0 " R ! U ( ! 0 T W T & 3 R 0 !

0

(t)0 ! Q ! " %

1


27/188

( T ) ( ' R ! S ( ! T & 3 R 0 ( ! V & W "

s

V

i(t)

C

( T ) ( ' R ! S P U I " P 4 # & ! 0

10 ! ! I U U V ) 0 " ! 3 0

% S 0 % ) 0 " & ' % T ! S T U T 0 ) 0 & ! 0 ( 0 ( T ) ( ' R ! S

% S 0 T U R ' % T & S % S 0 ( " 0 !

! I ( ! 0 P Q H ! & ' T W

U ! I & U 0 V ( ! ! U I ( ! R ! V (

& " ! T V

10 ! T T ! V ! I U !

(t) dt = 1.

4 # 1

P V ! H ' R ! S Q " % R ! R V & T ! U I R 0 ( ' ! ) " ! ( & !

Et

T(t)

1 6

( ' R ! S P 4 1 & ! 0

) T W T U " T ( T ) ( ' R ! S ( 0 & ' T T ! ' "

fs(t) =

12 |t| <

0

0( & I 0

4 # 4

ft(t) =

12 (t + ) t 01

2 ( t) 0 t 0

( & I 0

4 # X

fb(t) =12

e12(

t)2

. 4 #

! & ' U T U ! U T ! U " ) T W ! & ' R U W ! & " T 3 ) T W ( T ) ( ' R ! S T ! & ' 0 ! Q ! U 0 & ( ! 0

R ! U 0 & ( ! ( ! 0 R U ( ! R ! V ( ( U T ! & ' Q V 0 ! H & ' " " ) I 0 T V V ( ! 0 & T T ' % ! 0 3 Q ! " W S

( 0 Q ' T Q ( R ! % S 0 T U R ' % T & S ) 0 " Q ' U T U !

0T U R ' % T & S % S 0 ( "

Version 1.11 f

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( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

I R R V " & (

10 ! ( ! ' R ! S " ( I 0 T V T W T & 3 R 0 % ! R Q '

( & I ) T W ! R R ! & V %

! & ' " ) I 0 T V T H 0

0 % S & ' R ! S 0

gd U

f(t)g(t) dt =

f(t)g(t) dt

f(t)g(0) dt = g(0) 4 # f

f( & ) 3 " ! U U T U !

gT U ( ! S & T T 3 R U

f( & ) 3 " 0 ! Q ! U 0 & Q ! ! ! U & U 0 V

gd 3 0

T U T ) ! 3 ! T (

( 3.1.6)d ! & ' ( ! V 0 U Q ! " V

T V T V T U R ( U 0 & U ! " U ( 0 0 & R ( T ) ( ' R ! S T U & ( ! ' " W " R Q " H W ' U R U R S T

g S & ' R ! S

g(t s)(s) ds = g(t) . 4 1 #

T & 3 R 0

s R ( U " T U ' R ! S V

g(t

s)

T W R R ! ( " ! R 0 & U 0 V ( T ) ( ' R ! S ( & ) 3 " W ! R Q ! ! ! U

( V & W " ( & ) 3 T W H Q ! ! ! U T W S R & S

g(t)& " ! T V

s = 0& U 0 V

g(t s) ' R ! S & W ( 0 Q V T Q ( T ' R !

( 3.0.2) ! T ! ! R ! '

h(t) =

h(t s)(s) ds . 4 1 1

( V & W " ( ! 3 ( 0

h ' R ! S I & V H 0

( 3.0.2) ) T W ( & 0 ! ( " ! T ! ! R ! ' ( V & W " 0 & " ! T V

( T ) ( % R V T

R 0 ( ! T T V ! " ( ! ' R ! S T U ( T T V & ! 0 ( ' & ) T U ( T ) 0 ( I 0 ( T T V ! " h ' R ! S ! R 0 &

( ! T T V ! " ( ! ' R ! S W ( ! U W T & ! % 0 " ! & ( ! " " h P ' I U " ' ! I Q V ! T T V & ! 0 ( H V & ( R Q V T ! U S

& S I & T ! & V " ! R 0 U T V U " ( U T T V ! R U V ' R ! S 3 U ! " T U T T V 0 ! " T 0 U R

& ) 3 T & R & ) " ) T W ( & 0 ! ( " ( V & W " T % R V V T 0 % I ( T T V ! R U ) V ( T ) T S T & R

" ! ) 3 ! % " & ! ) ! W & ! S ( ! & " ( T U & U ' U " 0 & R % ! R ( T ) T U ( ! & H 3 R

( ! ' R ! S Q ( ! & 3 T 0 ( ! T ! W S P ( ! T 3 & ( ! R ! V ( ( 0 ! " ' ! T 0 ( ! T T V ! " ( ! ' R ! S U & R U Q ! " V

! V ! T & 3 R 0

T & 3 R 0

f ' R ! S Q ( R T & 3 R 0 & ) ( ! ' R ! S Q S 0 T Q ( R V R I 0 T I ( R

f(t)(t) dt.(3.3.1)

Q " H R ( U " T U {} ( ! ' R ! S % ! 0 0 ! X T U % R V ( ! ( ! 0 I & " & U 0 ( & 0 R T ( V & W " & ) 3 " Q ! "

f ' R ! S T U Q ! S 0 V U ! T T & S % " T T V ( 0 W ) ! 0 & S % " 0 ( ! ( ! 0 !

Version 1.11

$

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( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

& I R & ( ! T ! ) 3

f T U " T !

[T, S]d Q " % R U T ! ! & R 0 0 " R H 0 ( ! ! U Q R 0

f, g( ! ' R ! S ( U 0 0

P V

( 0

(t) =

1 T t S 0

0 ( & I 0

4 4 1

T ' R ! H ' R ! S & ! W

(t)f(t) dt >

(t)g(t) dt . 4 4 4

! T 0 V ( ! & 0 R T ( ! V & W " T W V ( ! T T V ! " ( ! ' R ! S T W ! U I T 0 V 0 ( 0 H T T V T ( W & )

! U I " 3 ! ) h & 0 R T T R ! ' R ! S 0 & ! ' 0 ' " ( " U " T U & S & ' " ! T 0 V ( ! V & W "

T R ! ' R ! S T U T ! W S & " ! T V T & 3 R 0 ! ( ( T W ! S " ! % I ' R ! S 0 ! H V T R ! ' R ! S ( V & W " T

T T V 3 U ! " 0 ! " T ) V

( 3.3.1)T & 3 R 0 & ) ( & ) 3 ! "

' R ! S T W T W ! S & U 0

f ' R ! S T & ! U '

& ! ! U T U Q I ! ( ! ' R ! S 0 & ' R % ! 0 T % R V ( ! ( ! 0 I & " ( 0 T 3 R Q ! U 0 & T U ' R ! S " & ( !

% ! 0 ! H

Sd H % ! 0 Q " % T 3 ! R

Laurent Schwartz 0 ' " ( " U T W

Schwartz Test functions ! R T ! S T V " & ( ! & " T ! ) 3 |t| & U 0 V ( ! V W ! ) Q ( ! & H 3 R ! Q % ! R ! & ) % T V " ( ! & H 3 R Q T & U 0 ( ! ' R ! S

0 ! ( & ) 3

# $ & !

'

S

3.3.1` 8 F B `

n, m ! &

lim|t|

tnDm(t) = 0.

& ) 3 ! " V T & 3 R 0 V & ( % " U " "

d ! T T U 0 T

kT V & ! W

Sd Q

tket2 , e(t2jt)( ! ' R ! S " 3 ! ) T

0 Q ! "

f( 0 ( W " U " ) I & ) 3 "

( 3.3.1)d

% %

f, g

3.3.2` D 9 b

f(t)(t) dt =

g(t)(t) dt

! ! # % $

f(t) = g(t) & # %

% %

0 ( T T V ! " ' R ! S T U ( ' ! ) " & ) 3

%

f (Tempered distribution)

3.3.3` 8 F B `

n, m % % $

k % # )

limk

supt

|tnDmk(t)| = 0, 4 4 X

limk k = 0 &

k

f

k

Version 1.11

2

September 2006


30/188

( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

( ! 3 ( ) T W ( & ) 3 ! " ' R ! S & " ! T V Q I ! ( ! ' R ! S T W ( T ! W S ) T W ( & ) 3 ! "

f( T T V ! " ' R ! S

( ' R ! S Q R 0 U ( ! ' R ! S T 3 ( ! T T V ! " ( ! ' R ! S ( T ! W S ( 0 I & R U " Q I ! ( ! ' R ! S Q U ( ! % R V T

T U % R V T Q ( ! 3 ( " ' ! ) T ' S % " ( ! H Q ( ! T T V ! " ( ! ' R ! S ( U 0 ' ! ) T ) V U V " W ! R Q I !

Q I ! ( ! ' R ! S

( ! T T V ! " ( ! ' R ! S T W ( ! " ! ) ( ! T ! W S Q " % T ) V & V ! " R ! " % R U ( W V U " ( U R

!

f % # %

'

f

3.3.4` 8 F B `

f(t)(t) dt.

f " $ $ !

f = f1 + f2% "

f1, f2 % ! $ !

f(t)(t) dt.

=

f1(t)(t) dt +

f2(t)(t) dt.

$ $ !

1%

(1(t)f(t)) (t) dt.

=

f(t) (1(t)(t)) dt.

#

, ' # $ ! % & % % % $ ! % $

f(t + )(t) dt.

=1

||

f(s)

s

ds.

0 ! T & 3 R 0 & U 0 V " ' ( " T W T R ! ! ! U T V H 0 ( ! " ! % I ( ! ' R ! S Q

f, f1, f2 0 V ( ! 0 & T T '

R ! U 0 ! ( " ( " ! Q I ! ' R ! S T U T ! W S T Q ! " % ' & 0 ! H T & 3 R 0 V U 3 ) T ! U I T 3 & Q ! "

( ! R ! V ( ( " ' " Q I ! ( ' R ! S ( T ! W S ( ! & ) 3 " & ! & U S V 0 & V " ! R 0 U & I 0 T & 3 R 0 "

W ! R ( & 0 R T 0 ( T T V ! " ' R ! S & ) 3 S T U Q ! ! V ! % T I ! R 0 ! H Q ! " % Q V T ! T & 3 R 0 T ( ! " ! )

V

f(t)[1(t) + 2(t)] dt =

f(t)1(t) dt +

f(t)2(t) dt.

Q " % ( I ( ( ! ' R ! S T U T 3 & T T V T " 0 ( " ( T T V ! " ' R ! S T U ( ! 3 R ( V 0 ! & ! R 0 ! V % T

& ) 3 ! " 0 T ( ! T T ! T W ( ! T T V ! " ( ! ' R ! S Q T S V V T U T U T & 3 R 0

& ! T T 3 & & 3 R 0 ! & H 3 T T V T V ( 0 ( ! T T V ! " ( ! ' R ! S T W T I T Q ( R 0 Q ! " V I ! R

R ( R U & ) 3 T 0 ( " ! R 0 U Q ! " & S ! T 3 & ' R ! S T U ( & H 3 R " R ! U Q ! " ( & H 3 R & ) 3 T U V

! H & ) 3 ( I ( V & R & S T 3 & ( & H 3 R T U T 0 T ( ! " ! ) ! H ( & H 3 R T U ( ! R ! V ( T ! 0 0 h ! ) I % & ! '

H 0 ( ! S & & H 3 ( T T V ! " 0 T T 3 & ' R ! S 0

f 0 S ' ( ( ' T I & 3 R 0 ( I % ! R

d

dtf(t)(t) dt = f(t)(t)|

f(t)d

dt(t) dt

4 4

=

f(t)d

dt(t) dt

4 4 f

Q V T T ! ) 3 |t| d % S 0 T ( S 0 ! U (t) d U U & ) T T 3 T ! ) 3 |t| & ! W ) " & " T ) 3 R 0 fd U 0 R (



31/188

( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

$ $ ! & !

3.3.5` 8 F B `

d

dtf(t)(t) dt

.=

f(t)d

dt(t) dt.(3.3.7)

#

1

%

(

0

!

( 3.3.7) & !

3.3.6` D 9 b

d

dt(f1 + f2) =

d

dtf1 +

d

dtf2,

4 4 2

d

dt(f) =

d

dtf

4 4 e

d

dt(f 1) = f d

dt1 + 1

d

dtf.

4 4 # g

Q V U ( ! & ) 3 " W ! R Q ! U 0 & Q ! ! ! U P 7

d

dt(f1(t) + f2(t)) (t) dt =

(f1(t) + f2(t))d

dt(t) dt

4 4 # #

=

f1(t)d

dt(t) dt

f2(t)d

dt(t) dt

4 4 # 1

=

d

dtf1(t)(t) dt +

d

dtf2(t)(t) dt.

4 4 # 4

! % T " ! ) R U Q ! ! ! U ( I V !

d

dt(1(t)f(t)) (t) dt =

(1(t)f(t))

d

dt(t) dt

4 4 # X

=

f(t)

1(t)

d

dt(t)

dt

4 4 #

=

f(t)

d

dt[1(t)(t)] (t) d

dt1(t)

dt

4 4 # f

=

f(t)d

dt[1(t)(t)] dt +

f(t)

(t)

d

dt1(t)

dt

4 4 # $

=

d

dtf(t)

1(t)(t) dt +

f(t)

d

dt1(t)

(t) dt

4 4 # 2

=

1(t)d

dtf(t) + f(t)

d

dt1(t) (t) dt.

4 4 # e

Y @ Y G Y

T U ( & H 3 R T 3 & & H 3 ! " V ' ! I ( ! 0 ( 0 ( ! " ' " ( ! T T V ! " ( ! ' R ! S T U & H 3 ( ! T ! W S ! V % T

T W ( T T V ! " ' R ! S T U ( ! & ) 3 ( 0 T W S R 0 R U ) " ( T T V ! " ' R ! S ) " ( 0 ( T T V ! " ' R ! S

( ! R ! V ( " 0 ( T U R ! V R ( 0 ! ( T 3 & ' R ! S

Version 1.14 g

September 2006


32/188

( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

( ! T T V ! " ( ! ' R ! S T U T ! W S ( 0 I & T Q ( R ! R T H " T ( ! T T V ! " ( ! ' R ! S Q S 0 T ) V ' & ( U " U "

S

S T ( T T V ! " ' R ! S T V T ( ! & H ( ! U W T U ( 0 H Q I ! ( ! ' R ! S Q R 0 U ( ! ' R ! S W T & 3 R 0 T 3

( & ) 3 ! " ( T ! W S U ) V

0d & W ' & ! T ( ( T ) ( ' R ! S ( T ! W S V T U R T U " T V ( ! R ! V (

I % ! R Q V T

0d ( ! S & & ! U

g(t)(t) dt = g(0)

( ! T T V ! " ( ! ' R ! S T U & ) 3 ( 0 I & T Q ( R " ! ) & !

0d S & & U 0

g ' R ! S T V T S ' (

" V ) W & ! " 0 V T ! 0 ( ! ' R ! S T U & ( ! I & I S U " & ! W ( ! & " U R T & 3 R 0 ( ! R ! V ( T V 3 ( ! & I 0

( T T V ! " ' R ! S ( ! R ! V ( ! T ( d d d I & T Q ( R

P ) " T ! U S & ! ( ! T T V ! " ( ! ' R ! S & 0 ( T & U S 0 U W " T P ( " ( " & W

' "

N

g

f !

3.3.7b E G

f(t)(t) dt =

g(t)dN(t)

dtN dt.

T U ( ! & " ( W R ! & " ( ( & ) 3 ( 0 I & R U & ! S ( ! & " ( T W ' & S ( ! T T V ! " ( ! ' R ! S T W ) ! W

( ! T T V ! " ( ' R ! S

( ! & H 3 R ( ! 3 R ( ( 0 Q S 0 R ! ' & ) T U ( T ) ( ' R ! S T U ( ' ! ) " & ) 3 T ( W V & ! H I R

$

3.3.8` 8 F B `

(t)(t) dt = (0).

( T T V ! " ' R ! S ! H Q V T !

( ! S & ( R ! V ( ( 0 ' ! ) T T ' Q V ! ( & 0 R T T ! W S ! H Q V 0 V ' ! ) T T '

& " ! T V 3 & ) " ( ' R ! S ( 0

ud Q " % R

u(t) = 1 t > 0,

0 t < 0.

& ) 3 ! " ! R 0

t = 0d & W

3.3.9` D 9 b

(t) =du(t)

dt.

Version 1.14 #

September 2006


33/188

( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

( T ) ! " V ' ! ) ( T W ! S 3 & ) " ( ' R ! S T U ( & H 3 R 0 U ( T T V ! " ' R ! S V I V ! T ! R T W P 7

( & H 3 R ( & ) 3 S T T ! 0 Q I ! ( ' R ! S T V T W

(t)du(t)

dtdt =

d

dt(t)

u(t) dt

4 4 1 g

=

0

d

dt(t) dt

4 4 1 #

= () + (0) = (0). 4 4 1 1

Y @ Y G Y ( T ) ( ' R ! S ( T ! W S ' ! ) ! H

& " ! T V ( T ) T U ! % " 0 T T & 3 R 0 0 3 & ) " ( ' R ! S V ( ! 0 & T 3 T '

t

(s) ds =

0 t < 0

1 t > 0 .

4 4 1 4

3 & ) " ( 0 ! R ( & ) 3 T 0 (

0 ) ! ' R & ) 3 ! " ! R 0 H T & 3 R 0 Q V 0 !

P Q I ( ! ' R ! S T W ( T ! W S & ) ( & ) 3 ! " ( & H 3 R

3.3.5

& ) 3 S T ( T ) ( ' R ! S T U

( & H 3 R "

(t)(t) dt =

(t)d

dt(t) dt

4 4 1 X

= (0) 4 4 1

( T ) ( ' R ! S T U ( & H 3 R ( T ) ( ' R ! S ( & ) 3 " ! Q I ( ' R ! S 0 Q I ( ' R ! S T U ( & H 3 R U Q ! ! V

P V & U T 3 ! R !

doublet h T ! ) ( 0 & ' R

Et

T(t)

1

1

6

?

doubletP

T U ( & H 3 R P 4 4 & ! 0

( 0 H V ' R ! S & ! W P

0

d S & ( & H 3 R ( T W 0 U

f

' R ! S T V T I & T Q ( R ( 0 H R ! V (

(t)f(t) dt = f(0) . 4 4 1 f

H 0

0d ( ! S & ( ! & H 3 R

n( T W ' R ! S 0

f 0 " ! ) & !

f(t)(n)(t) dt =

f(1)(t)(n1)(t) dt 4 4 1

$

= (1)nf(n)(0) . 4 4 1 2

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( ! T T V ! " ( ! ' R ! S ( ! T & 3 R 0 ( ! V & W "

H 0 R ( ( I (

0 ) ! ' R S &

fd U 0 R ( ( & ) 3 ! "

f T 3 & ' R ! S ( T ) ( ' R ! S T U T S V "

((t)f(t))(t) dt =

(t)(f(t)(t)) dt = f(0)(0) =

(f(0)(t))(t) dt 4 4 1 e

( ! ! U Q !

( ! T T V ! " ( ! ' R ! S

Q ! ! ! U ) R U P

f(t)(t) = f(0)(t)Q V T !

f(t)(t)! " U I R " ! ) & !

(f(t)(t))(t) dt =

(t)(f(t)(t)) dt 4 4 4 g

=

(t)d

dt(f(t)(t)) dt

4 4 4 #

=

(t)f(t)d

dt(t) dt

(t)(t)d

dtf(t) dt

4 4 4 1

= f(0)

(t)d

dt(t) dt f(0)

(t)(t) dt 4 4 4 4

= f(0)

d

dt (t)(t) dt f

(0) (t)(t) dt

4 4 4 X

f(t)(t) = f(0)(t) f(0)(t) Q 0 V " ! V ( ! 0 & T Q ( R " ! ) & !

f(t)(n)(t) =n

k=0

(1)k

n

k

f(k)(0)(nk)(t) .

4 4 4

Q ( Q ! ! ! U T U T 0 " U ) Q V 0 !

f(t)(n)(t)(t) dt = (

1)ndn

dtn(f(t)(t))

t=0

4 4 4 f

=(1)nn

k=0

n

k

f(k)(0)(nk)(0)

4 4 4 $

P I V ! T ! R T W U Q ! ! ! U T U Q " ) R U ) " !

n

k=0

(1)k

n

k

f(k)(0)(nk)(t)

(t) dt

4 4 4 2

=

nk=0

(1)knkf(k)(0)

(nk)(t)(t) dt

4 4 4 e

=(1)nn

k=0

n

k

f(k)(0)(nk)(0)

4 4 X g

! ! U R " 0 ! !

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Q W & 3 ( ! V & W " ( ! T & 3 R 0 ( ! V & W "

n 3 sin(t)(n)(t) 3.3.10 ` B F

sin(t)(t) = sin(0)(t) = 0 4 4 X #

sin(t)(t) = sin(0)(t) cos(0)(t) 4 4 X 1

= (t) 4 4 X 4

sin(t)

(2)

(t) = sin(0)

(2)

(t) sin(1)

(0)

(1)

(t) + sin

(2)

(0)(t)

4 4 X X

= 0 cos(0)(1)(t) sin(0)(t) 4 4 X

= (t) 4 4 X f

sin(t)(3)(t) = sin(0)(3)(t) 3sin(1)(0)(2)(t) + 3 sin(2)(0)(1)(t) sin(3)(0)(t) 4 4 X $

= 0 3 cos(0)(2)(t) 3 sin(0)(1)(t) + cos(0)(t) 4 4 X 2

= (2)(t) + (t) . 4 4 X e

( ! R V ( ! T 3 " & ! 0 ( T ( R ( R ( & 0 R T 0 U & Q " H 0 % R V ( V & W " T V U " 0 & R U S V

0 & ! " ( T & 3 R 0 ( V & W " V Q 0 V ) ! " W R 0 T Q T W ( ! T '

y(t) =

K(t, s)x(s) ds . 4 X #

0 ( V & W " Q W & 3 ( 0 & ' R U " " R ( U " R U T U ( V ! & " ! 0 ( U " " ' R ! S 0 U

K ' R ! S

( ! 0 ! ( ( 0 ( ! T ' I & T Q ( R U ( ! & " T ) T

SISO( V & W " ' ! % W R ( T T V ! " ' R ! S ( ! T T ! V

MIMO

( V & W " T

'

( 3.4.1) $

3.4.1b E G

d U " & ) & ) 3 ! " 0 ( ! 0 0 ( " % R V ( ! 0 T V T U Q ! ! V 0 % R V ! S " 0 ( V & W " P 7

H 0

, ,x1, x2( ! ( ! 0 ! W ! ' ( W V & I R

( 3.4.1) 0 ! !

y(t) = [x1 + x2](t) 4 X 1

=

K(t, s)(x1(t) + x2(t)) dt 4 X 4

=

K(t, s)x1(t) dt +

K(t, s)x2(t) dt 4 X X

= [x1](t) + [x2](t) 4 X

Y @ Y G Y T & 3 R 0 ( ! R ! V ( " W ! R U T U Q ! ! ! U & U 0 V

U T V ( & 0 R T 0 % R V ( V & W " & I R ( V & W " T V W " V & 0 ( " T T V 3 ! ! H U ( ! 0 & T ) V

) T W

K ' R ! S & ) 3 R

! S " ) T W ( & ) 3 ! "

K(t, ).

= [( )](t) = [](t) . 4 X f

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Q W & 3 ( ! V & W " ( ! T & 3 R 0 ( ! V & W "

( ' R ! S W T & 3 R 0 ) T W % R V ( 0 3 R

Q " H T W S & U 0 T ( % R V T

tW 3 & ( V & W " ( ! 3 ( ! H

3.6' & S 0 & ( T )

x(t) =

x()(t ) d . 4 X $

' ( " {fi()} ( ! ' R ! S ! {xi} W ! ' % ! 0 T V T ( & 0 R T 0 ( V & W " U Q ! ! V

i

xifi

(t) =

i

xi[fi](t) . 4 X

2

" ! ) & ! T ' R Q " & T 3 & R 0 & V H R ! V % T U 3 ! % 0 ! T & 3 R 0 U Q ! ! V

y(t) = [x](t) 4 X e

=

x()( ) d

(t) 4 X # g

=

x()[( )](t) d 4 X # #

=

x()K(t, ) d .

4 X # 1

Q W & 3 ( V & W " V ( V & W " T U 3 ! ! R T '

( ! & 0 R T ( R ! V ( ( U 0 & T ! S V 0 ( ! 3 ( % 0 ! " ! h S U " ! R 0 H ! U I U % P ( " ( " & W

( T T V ! " ' R ! S ( 0 H " ! R & ) 3 0 T ( R U T & 3 R 0 T 0 T ! S ! % & ! I " & S % " W " ! V % T ( & ) 3 ! "

( ! V & W " Q T ! 0 Q W & 3 ( ! V & W " V Q 3 T Q ( R 0 T & U 0 ( ! & 0 R T ( ! V & W " Q R U Q V 0 ! R ( U " R U T U

h ( ! 3 ! T ! ( S

Q ! & V H W ( & 0 R T ( " R ) ( V & W " 0 Q W & 3 ( V & W " ' & ' " T & S

% &

K(t, s) = k(t)(t s) % ! 3.4.2 @ A B 8 6

t

K% !

3.4.3b E G

t+

|K(t, s)| ds = 0 . 4 X # 4

& & #

s > t

K(t, s) = 0

H 0 ' ( " 0 R ( 0 P 7

y(t) =

K(t, s)x(s) ds =

t+

K(t, s)x(s) ds 4 X # X

t < T " ' H 0 ' ( " ! R 0 0 R ( 0 R U ) " ) T

s t & ! W 0 x(s) V & W ( ! T ( Q V T ! ' ( " ! U

[T, S]d Q " % R & U 0 T ! ! & R 0 !

ST

K(t, s) ds = = 0 . 4 X #

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September 2006


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! T ! ! R ! ' ( ! V & W " ( ! T & 3 R 0 ( ! V & W "

% R V ( ! 0 & I R

x(s) =

T s S0

( & I 0

4 X # f

0 !

tW 3 & ! 3 ( & W

y(t) =

K(t, s)x(s) ds =ST

K(t, s) ds = 2 . 4 X #

$

Y @ Y G Y ( ( % R 0 ( V & W " Q V T ! ) ( W % R V V & W ! T ( ! 3 (

( ! ( ! 0 T V T ! 3 ( ( ! 0 Q ( ! R

K(t, s) = 1Q W & 3 T U " T Q T ! V 0 T 0 ( ! V S Q U Q W & 3 ( ! V & W " U

V S R 0 Q V T ! T & 3 R 0 ! ( ! 0 T U & U 0

0 W %

tQ " H ( U & ! S " ( ! T ( T U Q " H W ! ' R 0 Q W & 3 ( V & W " V ( 0 R 0 & ! &

Q " H W ! ' 0 Q W & 3 ( V & W " ( " 0 & R

) T W ( & ) 3 ! " ! T ! ! R ! ' ( V & W "

y(t) =

h(t s)x(s) ds . 4 #

V W ! R

3.4.3 S U " " ( & 0 R T 0 % R V ( V & W " ! H Q V T ! Q W & 3 ( V & W " T U & S & ' " Q ! " V ! H

0 ( V T U R S U " & 0 &

t < 0T V T

h(t) = 0d U 0 ! ( ! ( % T I & V ! ' S % " 0 R (

Q V 0 ( V & W " !

y = h x = x H & ' " Q V 0 ! P ( ( % 0 h = & ! W ( V & W " V T ' ( " 0 R ( T ( ( %

% & '

( 3.5.1) $

3.5.1b E G

& ) 3 S T ' ! ) R Q " H ( ! W ' I V ! T ' & & ! U P 7

y(t) =

h(t s)x(s) ds

4 1

=

h(s)x(t s) ds

4 4

=

h(s)x(t s) ds 4 X

= [x](t) 4

Y @ Y G Y

T U & ) 3 " U T U !

3.6' & S 0 & ! T ! ! R ! ' ( ! R ! V ( " W ! R R U Q ! ! ! U & U 0 V

P ! T ! ! R ! ' ( V & W " I & V 0 Q " H W ! ' Q W & 3 ( V & W " R U ) "

#

x &

h(t).

= K(t, 0) $ ! % & % !

K

3.5.2b E G

K(t, s)x(s) ds =

h(t s)x(s) ds . 4 f

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September 2006


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! T ! ! R ! ' ( ! T & 3 R 0 ( ! V & W "

& ) 3 " P 7

(t) =

K(t, s)(s ) ds = K(t, ) . 4 $

V W ! R Q " H ( ! W ' "

(t) = [] (t) 4 2 =

K(t, s)(s) ds 4 e

= K(t, 0) 4 # g

= K(t , 0) . 4 # #

Y @ Y G Y I V ! S U " !

K(t, ) = K(t , 0) V W ! R ( ! 0 ! ! U " ( U " Q ! ) ( 0 I ) R Q V T ! ( ! & " ( ( & H W H 0 U ! R I ( R T I ! R P V S 0 ! T ! ! R ! ' ( V & W " T V 0 T

( ! R ! V ( ! ! T ! ! R ! ' ( T ! W S T U ' " W ! & H I T Q " H U ) ' R ! R T ( ! ! U I Q ! T ! ! R ! ' ( ! V & W " U Q ! ! V

" $ ! %

3.6.1` 8 F B `

(f g)(t) .=

f()g(t ) d.

( ! 0 ( ! R ! V ( ! T ! ! R ! ' ( T ! W S T

& " ! T V ! U I ! R 0 R ( U " & ) % ( ! ! " ! '

f g = g f . 4 f #

= t R ( U " ( S T I ( & H W & ) 3 ( 0 ! U & R P I V !

(f g)(t) .=

f()g(t ) d 4 f 1

=

f(t )g() d() 4 f 4

=

f(t )g() d 4 f X

.= (g f)(t) . 4 f

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$

September 2006


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! T ! ! R ! ' ( ! T & 3 R 0 ( ! V & W "

P ( ! ! T ! ! R ! ' ( & U & U & 3 ! % ! ' " & " ! T V ( ! 0 ! % 0

(f g) h = f (g h) . 4 f f

P R ( U " ( S T I T 3 & ( 0 I V !

(f

g)(t) =

f(t

)g() d

4 f

$

((f g) h)(t) =

f(t s)g()h(s) d ds 4 f 2

(g h)() =

g( s)h(s) ds 4 f e

(f (g h))(t) =

g( s)h(s)f(t ) d ds . 4 f # g

d = d! H S T I V T U R ! ( 0 ! ! U !

= +s R ( U " ( S T I " ( W V ( T ' ( " 0 ! (

V ) " W ! R T & 3 R 0 ( ! R ! V ( " P ( & 0 R T T ! W S 0 ! T ! ! R ! '

f (g + h) = f g + f h . 4 f # #

T W V

tT V & ! W

g(t ) T W ! U I T Q ( R Q V U ( ! T T V ! " ( ! ' R ! S T 3 ! H & ) 3 T W S T Q ( R V & ! & T ' T ! & ) 3 T Q ( R H 0

t ) ! ' R S &

g(t) 0 ) T

R ( U " T U ' R ! S

( g)(t) =

()g(t ) d = g(t).0

g& U 0 V 3 R ! V R ! H ( ! H ( ! S & ( ) ! ' R T V

g ' R ! S T V T

( g)(t) = g(t) V V 0 ! R 0 & 0 ( ! ) ! ' R U I ( R 0 T Q V T ( ! T T V ! " ( ! ' R ! S Q Q ! ! ! U T U Q ! " 0 ! Q ! ! ! U & U 0 V ( T T V ! " ' R ! S

T W

Q I ! ( ' R ! S T ! " T & 3 R 0 & ! ' " ' ! ) R ( 0 H ' ) T ) V

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