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Universidade Tecnológica Federal do Paraná 
Campus Toledo 
Curso de Engenharia Eletrônica 
ET45A – Sinais e Sistemas 
Prof. Eduardo Vinicius Kuhn 
 
 
 1 de 3 
 
FORMULÁRIO 
Matemática básica 
j je e
cos( )
2
 − +
 = 
2 1
cos ( ) [1 cos(2 )]
2
 = +  
j je e
sen( )
2j
 − −
 = 2 1
sen ( ) [1 cos(2 )]
2
 = −  
jj ea b r + = 2 2r a b= + sen( )
sinc( )=



 
j( /2)j e  = 
je cos( ) jsen( ) =  +  
1tan ( / )b a− = 
j | |e ( 1)k k  = − 
cos( ) cos( )cos( ) sen( )sen( ) =     sen( ) sen( )cos( ) cos( )sen( ) =      
1
sen( )d cos( )ax x ax
a
= − 
1
cos( )d sen( )ax x ax
a
= 
1
e eax axdx
a
= 
2
e
e ( 1)
ax
axx dx ax
a
= − 
2 2
e
e sen( ) [ sen( ) cos( )]
ax
ax bx dx a bx b bx
a b
= −
+
 
2 2
e
e cos( ) [ cos( ) sen( )]
ax
ax bx dx a bx b bx
a b
= +
+
 
cos( ) sen( )
d
ax a ax
dx
= − sen( ) cos( )
d
ax a ax
dx
= 
2
u vdu udv
d
v v
− 
= 
 
 ( )d uv udv vdu= + 
1
, 1
1
m nn
k
k m
r r
r r
r
+
=
−
= 
−
 
1
2
0
[ ( 1) 1]
, 1
( 1)
nn
k
k
r n r r
kr r
r
+
=
+ − −
= 
−
 
0
1
, | | 1
1
k
k
r r
r

=
= 
−
 
Operações elementares 
( ) ( ) ( )y t x h t d

−
=  −   
2| ( ) |xE x t dt

−
=  
/2 2
/2
1
lim | ( ) |
T
x TT
P x t dt
T −→
=  
par ímpar( ) ( ) ( )x t x t x t= + 
par
( ) ( )
( )
2
x t x t
x t
+ −
= ímpar
( ) ( )
( )
2
x t x t
x t
− −
= 
Série de Fourier 
0j
( ) e
k t
k
k
x t c


=−
=  0
0
j
0
1
( )e
k t
k T
c x t dt
T
− 
=  
0( ) 2 ( )k
k
X c k
+
=−
 =   −  2| |x k
k
P c

=−
=  
Transformada de Fourier 
j( ) ( )e tX x t dt
 − 
−
 =  j1
( ) ( )e
2
tx t X d
 
−
=  
 
 
j ( )( ) | ( ) | eX X   =  
| ( ) | | ( ) |X X− =  e ( ) ( ) − = −  
( ) ( ) ( ) ( )x t y t X Y    
1
( ) ( ) ( ) ( )
2
x t y t X Y   

 ( ) 2 ( )X t x  − 
1
( )
| |
x at X
a a
 
  
 
 
0j
0( ) ( )
t
x t t X e
− 
−   
( ) ( j ) ( )
n
n
n
d
x t X
dt
   
( )
( ) (0) ( )
j
t X
x d X
−

  +   
 
0j
0e ( ) ( )
t
x t X

 − 
(0) ( )X x t dt

−
=  
1
(0) ( )
2
x X d

−
=  
 
 2 21
| ( ) | | ( ) |
2
x t dt X d
 
− −
=  
  
rect sinc
2
t T
T
T
   
   
   
 sinc( ) rect
2
W
Wt
W
 
  
  
 
2
sgn( )
j
t 

 
1
( ) ( )
j
u t   +

 
1
e ( ), Re( ) 0
j
atu t a
a
−  
+ 
 
2
1
e ( ), Re( ) 0
( j )
att u t a
a
−  
+ 
 
( ) 1t  
1 2 ( )   
0 0 0cos( ) [ ( ) ( )]t    − +  + 0 0 0sin( ) j [ ( ) ( )]t    + −  − 
 
 
Universidade Tecnológica Federal do Paraná 
Campus Toledo 
Curso de Engenharia Eletrônica 
ET45A – Sinais e Sistemas 
Prof. Eduardo Vinicius Kuhn 
 
 
 2 de 3 
 
Transformada de Laplace 
( ) ( )e stX s x t dt
 −
−
=  
j
j
1
( ) ( )e
2 j
c st
c
x t X s ds
+ 
− 
=
 
 
0
0( ) e ( )
st
x t t X s
−
−  (0) lim ( )
s
x sX s
→
= 
0
0( ) ( )
s t
x t e X s s − 
0
( ) lim ( )
s
x sX s
→
 = 
1
( )
| |
s
x at X
a a
 
  
 
 ( ) ( ) ( ) ( )x t h t X s H s  1 2 1 2
1
( ) ( ) ( )* ( )
j2
x t x t X s X s

 ( ) ( )
d
tx t X s
ds
−  
1
( ) ( )
t
x d X s
s−
  ( ) ( ) (0 )
d
x t sX s x
dt
− − 
2
2
2
( ) ( ) (0 ) (0 )
d
x t s X s sx x
dt
− − − − 
1
( ) , Re( )ate u t s a
s a
−  −
+
 
1
( ) , Re( )ate u t s a
s a
−− −   −
+
 
1
( )u t
s
 
2 2
cos( ) ( ) , Re( )
( )
at s a
e bt u t s a
s a b
− +
  −
+ +
 
2 2
sen( ) ( ) , Re( )
( )
at b
e bt u t s a
s a b
−   −
+ +
 
Operações elementares (tempo discreto) 
( ) ( ) ( )
k
y n x k h n k

=−
= − 
2| ( ) |x
n
E x n

=−
=  
0
0
0
2
0
1
lim | ( ) |
2 1
N
x
N
n N
P x n
N→
=−
=
+
 
par ímpar( ) ( ) ( )x n x n x n= + 
par
( ) ( )
( )
2
x n x n
x n
+ −
= ímpar
( ) ( )
( )
2
x n x n
x n
− −
= 
Série de Fourier de tempo discreto 
0
0
j
( ) e
k n
k
k N
x n c

= 
=  0
0
j
0
1
( )e
k n
k
n N
c x n
N
− 
= 
=  j
0(e ) 2 ( )k
k
X c k
+

=−
=   −  
0 0j
0( )
k n
kx n n c e
− 
−  0j
( )
M n
k Me x n c

− ( ) kx n c−−  
0
2| |x k
k N
P c
= 
=  
Transformada de Fourier de tempo discreto 
j j(e ) ( )e n
n
X x n

 − 
=−
=  j j
2
1
( ) (e )e
2
nx n X d 

= 
 
 
jj j j (e )(e ) | (e ) | e XX X
  = 
j j| (e ) | | (e ) |X X−  = 
j j(e ) (e )X X−   = − 
j j( ) ( ) (e ) (e )x n y n X Y   
j j1
( ) ( ) (e ) (e )
2
x n y n X Y  

 
0j j
0( ) ( )
n
x n n e X e
−  −  
0 0j j( )
e ( ) (e )
n
x n X
 −
 
j( ) j (e )
d
nx n X
d


 
2 j 2
2
1
| ( ) | | (e ) |
2n
x n X d



=−
= 

  
j( ) ( )kx kn X e  
* * j( ) ( )x n X e−  
j
1
( ) , | | 1
1 e
nu n
− 
   
− 
 
j 2
1
( 1) ( ) , | | 1
(1 e )
nn u n
− 
+    
− 
 1 2 ( 2 )
k
k

=−
   −  
1, 0 | |
sinc( ) rect
0, | |2
WW
Wn
WW
   
 =  
      
 
j
1
( ) ( 2 )
1 e k
u n k

− 
=−
 +  − 
−
 
0 0 0cos( ) [ ( 2 ) ( 2 )]
k
n k k

=−
   − −  +  + −  ( ) 1n  
0 0 0sen( ) [ ( 2 ) ( 2 )]
j k
n k k

=−

   − −  − + −  0j
0e 2 ( 2 )
n
k
k


=−
   − −  
 
Universidade Tecnológica Federal do Paraná 
Campus Toledo 
Curso de Engenharia Eletrônica 
ET45A – Sinais e Sistemas 
Prof. Eduardo Vinicius Kuhn 
 
 
 3 de 3 
 
Transformada z 
( ) ( ) n
n
X z x n z

−
=−
=  
11
( ) ( )
2 j
nx n X z z dz−=
 
 
1
( )x n X
z
 
−   
 
 
(0) lim ( )
z
x X z
→
= 
1
( ) lim( 1) ( )
z
x z X z
→
 = − 
( )* ( ) ( ) ( )x n y n X z Y z 
( )n z
x n X
 
   
 
 ( ) ( )
d
nx n z X z
dz
− 
( ) ( )kx n k z X z−−  
1
( ) ( ) ( ) ( )
m
m n m
n
x n m u n z X z x n z− −
=
−  + − 
1
0
( ) ( ) ( ) ( )
m
m n m
n
x n m u n z X z x n z
−
− +
=
+  − 
 
( ) , | | | |n z
u n z
z
   
− 
 ( 1) , | | | |n z
u n z
z
− − −   
− 
 ( ) kn k z− −  
2
( ) , | | | |
( )
n z
n u n z
z

   
− 
 
2
( 1) , | | | |
( )
n z
n u n z
z

−  − −   
− 
 
0
0 2
0
[ cos( )]
cos( ) ( ) , | | 1
2cos( ) 1
z z
n u n z
z z
− 
  
−  +
 0
0 2
0
sen( )
sen( ) ( ) , | | 1
2cos( ) 1
z
n u n z
z z

  
−  +