Formulário M008 Capítulo 5

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FORMULÁRIO – M008 
RELAÇÕES TRIGONOMÉTRICAS: 
 ( )cos
2
j j
e e
θ θ
θ
−+= ( )sen
2
j j
e e
j
θ θ
θ
−−= 
 cos( ) cos( )cos( ) sen( )sen( )a b a b a b± = ∓ sen( ) sen( )cos( ) sen( )cos( )a b a b b a± = ± 
 ( ) ( )2 2sen cos 1x x+ = ( ) ( )2 1 cos 2sen
2
x
x
−
= ( ) ( )2 1 cos 2cos
2
x
x
+
= 
 ( ) ( ) ( )sen 2 2 sen cosx x x= ( ) ( ) ( ) ( )1 1sen cos sen sen
2 2
x y x y x y= − + + 
 ( ) ( ) ( ) ( )1 1sen sen cos cos
2 2
x y x y x y= − − + ( ) ( ) ( ) ( )1 1cos cos cos cos
2 2
x y x y x y= − + + 
PRINCIPAIS DIRETIVAS DE DERIVAÇÃO E INTEGRAÇÃO: 
 
[ ] 0=K
dx
d
 
[ ] )(')( xfKxfK
dx
d ⋅=⋅ 
[ ] )(')()( 1 xfxfmxf
dx
d mm ⋅⋅= − 
[ ] )(')ln()()( xfaaa
dx
d xfxf ⋅⋅= 
( )[ ]
)()ln(
)('
)(log
xfa
xf
xf
dx
d
a ⋅
= 
( )[ ] ( ) )(')(cos)( xfxfxfsen
dx
d ⋅= 
( )[ ] ( ) )(')()(cos xfxfsenxf
dx
d ⋅−= 
( )[ ] ( ) )(')(sec)( 2 xfxfxftg
dx
d ⋅= 
( )[ ] ( ) )(')(seccos)(cot 2 xfxfxfg
dx
d ⋅−= 
[ ] )(')()()(')()( xgxfxgxfxgxf
dx
d ⋅+⋅=⋅ 
 
CKxdxK +=⋅ 
C
m
xf
dxxfxf
m
m +
+
=⋅⋅
+
 1
)(
)(')(
1
 
C
a
a
dxxfa
xf
xf +=⋅⋅ )ln(
)('
)(
)( 
( ) +=⋅ Cxfdxxf
xf
)(ln
)(
)('
 
( ) ( ) +−=⋅⋅ Cxfdxxfxfsen )(cos)(')( 
( ) ( ) +=⋅⋅ Cxfsendxxfxf )()(')(cos 
( ) ( )[ ] +=⋅⋅ Cxfdxxfxftg )(secln)(')( 
( ) ( ) +=⋅⋅ Cxftgdxxfxf )()(')(sec2 
( ) ( ) +−=⋅⋅ Cxfgdxxfxf )(cot)(')(seccos 2 
 +




=⋅
+
C
a
xf
arctg
a
dx
xfa
xf )(1
)(
)('
22
 
  ⋅−⋅=⋅ duvvudvu 
FÓRMULAS DE PROCESSOS ESTOCÁSTICOS 
 
( ) ( ) ( ) ( ) ( )1 2 1 2 1 2 1 2,XR t t E X t X t X t X t X X= = =   
 
2 2( ) ( ) g(t) ( ) j ft j ftG f g t e dt G f e dfπ π
∞ ∞
−
−∞ −∞
= =  
 
( ) ( )X XR S fτ
ℑ
↔ ( ) ( )Y YR S fτ
ℑ
↔ 
( ) ( ) ( ) ( ) ( ),XR t t E X t X t X t X tτ τ τ+ = + = +   
 
1
( ) ( ) g(t) ( ) 
2
j t j t
G g t e dt G e d
ω ωω ω ω
π
∞ ∞
−
−∞ −∞
= =  
 
( ) ( )X XR Sτ ω
ℑ
↔ ( ) ( )Y YR Sτ ω
ℑ
↔ 
2( ) ( ) | ( ) |
Y X
S f S f H f= ⋅ 
 
2 1(0) ( ) ( )
2
X X X X
P X R S f df S dω ω
π
+∞ +∞
−∞ −∞
= = = =  
 
( ) ( ) ( ),XYR t t X t Y tτ τ+ = + 
 
( ) ( )XY XYR S fτ ↔ 
 
( ) ( ) ( ) ( ) ( )XY X XR h u R u du h Rτ τ τ τ
∞
−∞
= − = ∗ 
 
( ) ( ) ( ) ( ) ( )Y XY XYR h w R w dw h Rτ τ τ τ
∞
−∞
= − − = − ∗ 
 
2( ) ( ) | ( ) |
Y X
S S Hω ω ω= ⋅ 
 
2 1(0) ( ) ( )
2
Y Y Y Y
P Y R S f df S dω ω
π
+∞ +∞
−∞ −∞
= = = =  
 
[ ] [ ] [ ]( ) ( ) ( ) ( ) (0)E Y t E X t h t dt E X t H
∞
−∞
= ⋅ = ⋅ 
 
 
 
( ) ( ) ( )XY XS f H f S f= ⋅ ( ) ( ) ( )XY XS H Sω ω ω= ⋅ 
 
 
( ) ( ) ( )Y XYS f H f S f∗= ⋅ ( ) ( ) ( )Y XYS H Sω ω ω∗= ⋅ 
 
PROPRIEDADES DA TRANSFORMADA DE FOURIER 
 
Propriedade )(tx )(wX )( fX 
Espelhamento 
 
)( tx − )( wX − )( fX − 
Simetria 
 
)(tX 2 ( )x wπ ⋅ − )( fx − 
Escalonamento )(atx 






a
w
X
a ||
1
 





a
f
X
a ||
1
 
Deslocamento 
no tempo 
)( dtx − jwdewX −)( jfdefX −)( 
Derivação no 
tempo )(txdt
d
n
n
 
( ) )(wXjw n ( ) )(2 fXfj nπ 
Integração no 
tempo 
∞−
t
dx λλ)( )()0()(
1
wXwX
jw
δπ+ 1 1( ) (0) ( )
2 2
X f X f
j f
δ
π
+ 
Deslocamento 
na frequência 
tjw
etx 0)( )( 0wwX − )( 0ffX − 
Derivação na 
freqüência 
)()( txjt n− 
)(wX
dw
d
n
n
 )(
2
1
fX
dw
d
n
nn






π
 
Convolução no 
tempo 
)()( thtx ∗ )()( wHwX )()( fHfX 
Convolução na 
frequência 
)()( thtx 
)()(
2
1
wHwX ∗
π
 
)()( fHfX ∗ 
Modulação )cos()( 0twtx )(
2
1
)(
2
1
00 wwXwwX −++ )(
2
1
)(
2
1
00 ffXffX −++ 
 
 
 
PRINCIPAIS PARES DE TRANSFORMADA DE FOURIER 
 
)(tx )(wX )( fX 
)(tδ 1 1 
A )(2 wAδπ )( fAδ 
)(tue at− 
0>a jwa +
1
 
1
2a j fπ+
 
||ta
e
− 
22
2
wa
a
+
 
2 2
2
(2 )
a
a fπ+
 
)(tu 
jw
w
1
)( +πδ 1 1( )
2 2
f
j f
δ
π
+ 
τ
||
)sgn(
t
t = 
jw
2
 
1
j fπ
 
tjw
e 0 )(2 0ww −πδ )( 0ff −δ 
)cos( 0tw )()( 00 wwww −++ πδπδ )(
2
1
)(
2
1
00 ffff −++ δδ 
0sen( )w t )()( 00 wwjwwj −−+ πδπδ )(
2
)(
2
00 ff
j
ff
j −−+ δδ 
1
rect
0
t
τ
  =  
  
 
2/||
2/||
τ
τ
>
<
t
t
 
 
Sa
2
wττ  ⋅  
 
 
 
( )Sa fτ π τ⋅ ⋅ ⋅