the weight of each impulse function equal to the area under a pulse. As a sec
ample of calculating Fourier coefficients, we consider a train of impulse tunr
ni
,a"i,'
Fourier series
for an impulse
train
The Fourier series
for the impulse
train shown
in Figure 4.10 will be calculated.
Fron
co=+
[x1r1eik.,ra1
l o J T "
t r T d 2 , l l
+ 
E1t1pi*,,'
47
= !ri*.",I
:
+
I
oJrlz
To
I
f,o
t
o
This result is based on the property
of the impulse function
l*
rur^,

todt
=
f(to),
166
Fourier
Series
Ieq(2.a1)]
provided
that
f(t)
is continuous
at t
:
t0.
The exponential
form of the Fourier series
is s
x(t)
=
3
!,'o'"'.
*1*Tu
A line spectrum
for this functionis given
in Figure 4.11. Because
the Fourier coeffici
real, no phase
plot is given. From (4.13),
the c;bined
trigonometric
form for the trair
pulse
functions
is given by
Figure 4.10
Impulse
train
Figure 4.11
Frequency
spectrum
t,
impulse
train.
+.fi+coska;6r.
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Sec. 4.3
Fourier
Series
and Frequency
Spectra
Note that this
is also
the trigonometric
form.
A
comparison of the frequency spectrum of the square wave (Figure 4.6)
with that of the train of impulse
functions (Figure 4.11)
illustrates an important
property of impulse functions.For the squarewave, the amplitudes of the har
monics decrease
by the factor l"lk,where
ft is the harmonic number. Hence, we ex
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 Spring '08
 CARR
 Fourier Series, Fourier Series Frequency, trair pulsefunctionsis givenby

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