Find the Fourier series of the periodic rectangular pulse train shown in Fig. 13.6-6.
SOLUTION
The DC content of the waveform is . The Fourier series contains only cosine
terms since exponential Fourier series coefficients are real.
τ
T
v
T
vtt
T
t
jnT
n
jnt
T
T
jnt
j
n
j
n
===
−
−
−
−
−
−
∫
111
00
0
2
2
0
2
()ededee
ωω
ωτω
ω
00
00
2
2
2
22
0
2
2
τ
τ
τ
ωτωτ
ωτ
⎡
⎣
⎢
⎤
⎦
⎥
−=−
−
−
∫
ee by Euler's Fo
jnjn
j
n
sinrrmula.
∴===
⎛
⎝
⎜
⎞
⎠
⎟
⎛
v
T
n
nT
n
n
T
x
x
n
τ
ωτ
ωτ
τ
ωτ
ωτ
τ
2
22
2
0
0
0
0
sinsin
sin
⎝⎝
⎜
⎞
⎠
⎟
=,. where x
n
ωτ
0
2
Let v(t) be a periodic waveform with period T and let v
C
(t) be defined as v
C
(t) v(
α
t),
where
α
> 0. Show that v(t) and v
C
(t) have same coefficients in their Fourier series.
SOLUTION
v
C
(t) will be a time-compressed ...
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