082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

750 Chapter 23 Fourier series
which wecan write equivalently as
∞∑
f(t)= c n
e j2nπt/T
n=−∞
where
c n
= a n −jb n
c
2 −n
= a n +jb n
n=1,2,...
2
andc 0
=a 0
/2. Itcan be shown thatthe Fourier coefficients,c n
, arethen given by
c n
= 1 T
∫ T/2
−T/2
f (t)e −j2nπt/T dt
The integral can also be evaluated over any complete period as convenient. Further, if
we writeT = 2π
ω 1
then thiscomplex formcan be expressed as
where
f(t)=
c n
= ω 1
2π
∞∑
n=−∞
∫ π/ω1
c n
e jnω 1 t
−π/ω 1
f (t)e −jnω 1 t dt
Example23.20 FindthecomplexFourierseriesrepresentationofthefunctionwithperiodT definedby
{
1 −T/4 <t <T/4
f(t)=
0 otherwise
Solution We find
c n
= 1 T
∫ T/4
−T/4
1e −j2nπt/T dt
= 1 T
[ e
−j2nπt/T
−j2nπ/T
] T/4
−T/4
Therefore,
= −1
2nπ j (e−jnπ/2 −e jnπ/2 )
= 1 ( ) e jnπ/2 −e −jnπ/2
nπ 2j
= 1
nπ sinnπ 2
f(t)=
∞∑
n=−∞
1
nπ sinnπ 2 ej2nπt/T