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( ! 0 ( ! T ' " & " ! T V ( V & W " ( ! & 0 ( " V ( ! 0 ! ! U " T % I ( R H % & ! ' ) ! W ! 3 ! T R 0 T " U I T 3 W "
0 ! " ( ! 0 ( ! & " ! % R V
( ! R ! V ( ( " ' ! ) R ( ! V & W " T U ( ! R ! V ( T U ( ! T T V ( ! & ) 3 Q ( R Q ( ! R ! & ( S ! & ) " T W & H I & I 0 T
& ) " ) T W ( & 0 ! ( " ( V & W " ( ! " ' ( " ! T 0
& ) " ) ' % & ! ' % ! 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
N & ) % " ! 0 3 & ) " ! H V & ) " T U ( T T V & !
Nn=0
andny(t)
dtn=
Mm=0
bmdmx(t)
dtm(2.1.1)
aN = 0 V ) " ( I R R ! R 0 ! 3 ( 0 y(t) d ! d d d ( V & W " T % R V ! R ( R I " d d d Q ! ( R ( ! 0 0 ! x(t) & U 0 V ( ! I S ) I 0 & 0 W ! V % ( 0 & ) 3 T Q ( R ( & I 0 U Q ! ! V
bM= 0
Q V !
& ! ( W T ! H 0 ! ! U " W S ! " ( ! & S % P & W
Nn=0
aNndny(t)
dtn=
Mm=0
bMmdmx(t)
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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& H I & h ) "
%
t t0 & ! M x(t) % 2.1.1 b E G
{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
' R ! S ! H U W " ( U " ! d d d R ( U " & ! W ! ' " ( R " % " ) ! ' R & U 0 V
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)
},
x1 + x2
( 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)}.
" ' ( " Q V ! " V & H 3 ( T ! W S ( ! & 0 R T T T 3 ( " ' ( " 0 ! ! U " U T ' R !
( 2.1.1)d R P ` 7
T U " Q ! & ( S ! H Q V 0 V W ! R Q V T ! ) I Q ! & ( S U 0 ! ! U " T
2.1.1 S U " S T T I ( 0 R (
! 3 ( & ) 3 " T I ( 0 R ( W ) I & ) "
x
% R V ( ! 0 & ! W P 0 Q ! " ( V & W " & ) 3 " & ) "
0 & R U " ! 3 ( ! % R V Q & U ' T U & ! 0 ( ! H h P ( V & W " T U 3 U ! " % % Q ! W & ! H
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
x(t) = 0
0 x 0 % ( 2.1.1) ! % 2.1.3 ` 8 F B ` " #
N ! %
( 2.1.1) ! %
yh' & % % %
% #
x% %
( 2.1.1) %
yp' & %
"
( 2.1.1) $ % %
2.1.4` D 9 b
#
yp %
#
yh ! %
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( ! V & W " & h ) "
"
yh + yp
T U "
2.1.2 R W " W ! R P
7
% 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' & % %
#
x% %
( 2.1.1) %
yZSR' & % %
( 2.1.1) $ %
y.
= yZIR + yZSR 2.1.6` D 9 b
T U "
2.1.2 R W " W ! R P
7
y1 y2 H 0 R ! U T I ( 0 R ( 0 H % R V W & ) " ( 0 & ( ! S y2 d ! y1 0 V T U R T T V R 3 ! " ! Q ! & ( S 0 T Q ! " V 0 R 3 ! " ! Q ! & ( S ! ( & ) 3 S T 0 !
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) =
Mm=0
bmDmx(t)(2.1.3)
( $
0
! $ ( !
0
#
D
Nn=0
ann(2.1.4)
( 2.1.4) "
( ( U & ! 0 ! " T Q ( R R 3 ! " ! Q ! & ( S ( 0
! % & $
1, . . . , N' %
2.1.8b E G
yh(t) =Ni=1
Aifi(t)(2.1.5)
% # ( !
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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( ! 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
V 3 ( 0 H Q " % R ( W T ( ! ( ! 0 Q & U ' ( 0 U 3 ) " H Q ! " % & U 0 V
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 `
% & # % & $ $
$ $ % & % $ $ % &
& $
% & % % &
% & $ $ % & $ $
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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
$
( $
" # % %
2.2.7 ! $
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
( 2.2.6)( V & W " " ! ) & ! ( & 0 R T ( V & W " ( & 0 ( "
( 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 ! " !
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$
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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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( ! 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 ' %
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( ! 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
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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
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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 $
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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
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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 (
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& ! 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
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( 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 ( "
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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 !
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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
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2
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( ! 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 (
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( ! 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
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( ! 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.
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( ! 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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! 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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! 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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! 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