Three Mathematically-equivalent Fourier Series Forms


x (t ) 
n  
 C o m p le x -e x p o n e n t a l:
1
T0
cn 

T0
cn e
x (t ) e
x (t )  a 0 
a0 
1
T0

T0
an 
2
T0

T0
bn 
2
T0

 T rig o n o m e tric F o rm :
T0
An 
D C : a 0  A0  c 0
;
AC:
0

n 1
t
, w h e re 
 jn
a
x (t )  A 0 
 A m p li t u d e - p h a s e :
jn
n
0
t
0

2
T0
[S y n th e s is E q u a tio n ]
dt
[ A n a ly s i s E q u a t i o n ]
co s(n 
0
t )  bn s in ( n 
x (t ) d t


n 1
0
0
[ S y n . E q .]
t) dt
A n cos (n 
a n  j bn
;
2

t) dt
0
t n)
[S y n th e s is E q u a tio n ]
  bn 
a n 2  bn 2 ;  n  ta n 1 

 an 
cn 
t)
[ A n a ly s i s E q u a t i o n s ]
x (t ) c o s( n 
x (t ) s in ( n 
0
cn 
An e
2
j n
EEE EE2010/IM2004: Signals and Systems
[ A n a ly s i s E q u a t i o n s ]
(w h e re n    )
;
c  n  c *n
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