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Sec. 2–5 Fourier Series 79
The spectrum of the square wave is easily obtained from Eq. (2–109). The magnitude spectrum
is illustrated by the solid lines in Fig. 2–12b. Since delta functions have infinite values, they
cannot be plotted; but the weights of the delta functions can be plotted as shown by the dashed line
in Fig. 2–12b. Also see Example2_14.m
Now compare the spectrum for this periodic rectangular wave (solid lines in Fig. 2–12b)
with the spectrum for the rectangular pulse shown in Fig. 2–6a. Note that the spectrum for the periodic
wave contains spectral lines, whereas the spectrum for the nonperiodic pulse is continuous.
Note that the envelope of the spectrum for both cases is the same ƒ(sin x)/xƒ shape, where
x = pTf. Consequently, the null bandwidth (for the envelope) is 1/T for both cases, where T is
the pulse width. This is a basic property of digital signaling with rectangular pulse shapes. The
null bandwidth is the reciprocal of the pulse width.
The other types of Fourier coefficients may also be obtained. Using Eqs. (2–101) and
(2–102), we obtain the quadrature Fourier coefficients:
A
a n = c 2 , n = 0
0, n 7 0
(2–121a)
2A
b n = c pn , n = odd
0, n = even
(2–121b)
Here all the a n = 0, except for n = 0, because if the DC value were suppressed, the waveform
would be an odd function about the origin. Using Eqs. (2–106) and (2–107), we find that the polar
Fourier coefficients are
D n = e
A
2 , n = 0
2A
np ,
n = 1, 3, 5, Á
0, n otherwise
(2–122)
and
w n = -90° for n Ú 1
(2–123)
In communication problems the normalized average power is often needed, and for the
case of periodic waveforms, it can be evaluated using Fourier series coefficients.
THEOREM.
For a periodic waveform w(t), the normalized power is given by
P w = 8w 2 (t)9 =
n=q
where the {c n } are the complex Fourier coefficients for the waveform.
a
n=-q
ƒc n ƒ 2
(2–124)