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Sec. 2–5 Fourier Series 77
But the impulse train may itself be represented by its Fourier series † ; that is,
q
a
n=-q
d(t - nT 0 ) =
q
a c n e jnv 0t
n=-q
(2–115)
where all the Fourier coefficients are just
Eq. (2–114), we obtain
c n = f 0 . Substituting Eq. (2–115) into
q
w(t) = h(t) * a f 0 e jnv 0t
n=-q
(2–116)
Taking the Fourier transform of both sides of Eq. (2–116), we have
W(f) = H(f) a
q
= a
q
n=-q
n=-q
f 0 d(f - nf 0 )
[f 0 H(nf 0 )] d(f - nf 0 )
(2–117)
Comparing Eq. (2–117) with Eq. (2–109), we see that Eq. (2–112) follows.
This theorem is useful for evaluating the Fourier coefficients c n when the Fourier transform
of the fundamental pulse shape h(t) for the periodic waveform is known or can be
obtained easily from a Fourier transform table (e.g., Table 2–2).
Example 2–13 FOURIER COEFFICIENTS FOR A PERIODIC RECTANGULAR WAVE
Find the Fourier coefficients for the periodic rectangular wave shown in Fig. 2–12a, where T is the
pulse width and T 0 = 2T is the period. The complex Fourier coefficients, from Eq. (2–89), are
c n = 1 T 0 L0
T 0 /2
Ae -jnv 0t dt = j
A
2pn (e-jnp - 1)
(2–118)
which reduce, (using l’Hospital’s rule for evaluating the indeterminant form for n = 0), to
A
2 , n = 0
c n = e A
-j
np , n = odd
0, n otherwise
(2–119)
See Example2_13.m for the evaluation of these FS coefficients and for a plot of the resulting FS.
† This is called the Poisson sum formula.