3.3.6 コンボリューション（畳み込み） 2 つの関数 g(t)，w(t)について，その

3.3.6
ࢤ࣭ࣤ࣍ࣛࣖࢨࣘࣤ㸝␒ࡲ㎲ࡲ㸞
2 ࡗࡡ㛭ᩐ g(t)㸡w(t)࡞ࡗ࠷࡙㸡ࡐࡡ␒ࡲ㎲ࡲࢅ௧ୖ࡞ᏽ⩇ࡊ㸡 g(t)*w(t)࡛⾪ࡌ㸣
g(t) * w(t) =
$
%#$ g(" )w(t # " )d"
᫤㛣࠽ࡻࡦ࿔ἴᩐ㡷ᇡ࡚ࡡ␒ࡲ㎲ࡲ࡞㛭ࡊ࡙㸡௧ୖࡡᛮ㈻࠿࠵ࡾ㸣
!
ࢤ࣭ࣤ࣍ࣛࣖࢨࣘࣤᏽ⌦
(3.43)
g1 (t) * g2 (t) " G1 (# )G2 (# )
1
g1 (t)g2 (t) "
G1 (# ) * G2 (# )
2$
(3.44)
ッ᪺㸯
!
&"% e" j#t '() &"% g1($ )g2 (t " $ )d$ *+,dt
%
%
= & & e" j#t g1 ($ )g2 (t " $ )d$ dt
"% "%
F {g1 (t) * g2 (t)} =
%
%
=
&"% g1($ )')( &"% e" j#t g2 (t " $ )dt*,+d$
=
&"% g1($ )G2 (# )e" j#$ d$
%
%
%
= G2 (# ) &
%
g ($ )e" j#$ d$
"% 1
g(t) " G(# )
g(t $ t0 ) " G(# )e $ j#t 0
= G1 (# )G2 (# )
!
!
!
!
% 1
( 1 , j$t . 1 ,
1
F "1 & G1 ($ ) * G2 ($ )) =
e 0
G1 (+ )G2 ($ " + )d+ 3d$
",
",
' 2#
* 2#
/2#
2
,
, j$t
1
=
- - e G1(+ )G2 ($ " + )d+ d$
4 # 2 ", ",
. 1 , j$t
1
1 ,
=
G1 (+ )0
e G2 ($ " + )d$3d+
",
",
/2#
2
2#
g(t) " G(# )
j# 0 t
1 ,
g(t)e
" G(# $ # 0 )
=
- g1(+ )g2 (t)e j+t d+
2# ",
1 ,
= g2 (t)
- g1(+ )e j+t d+
2# ",
= g1 (t)g2 (t)
3.3.7
᫤㛣᚜ฦ࡛᫤㛣✒ฦ
ࡵࡊ㸡 g(t) " G(# ) ࡝ࡼࡣ㸡
᫤㛣᚜ฦ
dg
" j#G(# )
dt
!
(3.46)
᫤㛣✒ฦ
t
%#$ g(" )d" &
!
G(' )
+ (G(0)) (' )
j'
(3.47)
ッ᪺㸯
᫤㛣᚜ฦ
!
㏣ࣆ࣭࢙ࣛንᥦࡡᏽ⩇ᘟࡻࡽ
g(t) =
1
2"
%
&$% G(# )e j#t d#
୦㎰ࢅ᫤㛣᚜ฦࡌࡿࡣ
dg 1 %
de j#t
=
G(
#
)
d#
&
dt 2" $%
dt
1 %
=
& j#G(# )e j#t d#
2" $%
dg
"
# j$G($ )
dt
!
!
ྜྷ᧧షࢅ⧖ࡽ㏁ࡌࡆ࡛࡞ࡻࡖ࡙
dng
n
" ( j# ) G(# )
n
dt
!
(3.48)
᫤㛣✒ฦ
%1 # $ t
u(t " # ) = &
!
'0 # > t
g(t) * u(t) =
$
ࢤ࣭ࣤ࣍ࣛࣖࢨࣘࣤᏽ⌦ࡻࡽ !
!
g(t) * u(t) " G(# )U(# )
&1
)
= G(# )( + $% (# )+
' j#
*
"
!
!
t
t
!
G(# )
+ $G(0)% (# )
j#
!
G(( )
&$% g(# )d# ' j( + )G(0)*(( )
t
"
%#$ g(" )u(t # " )d" = %#$ g(" )d"
!
=
0
!
u(t) "
1
+ $% (# )
j#