Download Fourier Series: Deriving and Finding Coefficients - Prof. Alexander Parlos and more Study notes Mechanical Engineering in PDF only on Docsity!
364 Recitation Notes
9/5/
EXAMPLE 1.
Derive the Fourier Series of the following function:
First Identify the function:
( ) 1 , xT t T+xT, x= ...,-2,-1,0,1,2,...
t F t for T
Then plug this equation into the Fourier transform equation:
0
o o
T jw kt jw kt k T
t c F t e dt e T T T
Integrating over one period. Solving for co:
0
0 0
2
o
T (^) jkw T
o
o
t t c e dt dt T T T T
t c T T T
Continue solving for the series coefficients:
0 0 0
o o o
T T T jw kt jw kt jw kt k
t t
c e e dt e dt
T T T T T
Solving the first integral:
1
T
t
F(t)
0 0 0 0
o o o
T jw kt jw kt T jw kT
e dt e e
T T jw k T jw k
^ ^ ^
To solve the second integral us integrate by parts:
0
2 0 0
2
o
o
o
o o
o o
T jw kt
jw kt
jw kt
o
jw kt T^ T jw kt
o o
jw kT jw kt
o o
t e dt T T
u t dv e
e du v jw k
e e u dv uv v du t dt T jw k jw k
Te e jw kT jw k
^
Adding this result to the result of the first integral:
2 0
2 0
2 2 2 0 0
o o o
o
o
jw kT jw kT jw kt k o o
jw kt k o o
jw kt k
c e Te e T jw k jw kT jw k
c e T jw k jw kT jw k
c e Tjw k T w k
^
^
since (^0)
2 w T
ck becomes:
2
jw kt o
ck e
j k k
This result is then plugged into the Fourier series:
1 2 2
1
j kt j kt T T k k k k
c e c e
^
