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Sec. 2–2 Fourier Transform and Spectra 59
In terms of quadrature notation, Eq. (2–57) becomes
V(f) = [T Sa(pfT) cos(pfT)] + j[-T Sa(pfT) sin(pfT)
(2–58)
w
w
X(f)
Examining Eq. (2–57), we find that the magnitude spectrum is
and the phase spectrum is
sin pfT
|V(f)| = T `
pfT
0,
u(f) = le -jp/T + lSa(pfT) = -pfT+ d
p,
Y(f)
n
T 6 ƒfƒ 6 n + 1
(2–59)
T , n even
(2–60)
n
T 6 ƒ f ƒ 6 n + 1
T , n odd
The rectangular pulse is one of the most important and popular pulse shapes, because it
is convenient to represent binary one and binary zero data by using rectangular pulses. For
example, TTL logic circuits use 5-volt rectangular pulses to represent binary ones and 0 volts
to represent binary zeros. This is shown in Fig. 3–15a, where A = 5. Other examples of
rectangular pulse encoding are also shown in Fig. 3–15.
The null bandwidth for rectangular pulse digital signaling is obtained from Eq. (2–55)
or Eq. (2–59). That is, if the pulse width of a digital signal is T seconds, then the bandwidth
(i.e., the width of the frequency band where the spectral magnitude is not small) is approximately
1/T Hz. A complete discussion of signal bandwidth is involved, and this is postponed
until Sec. 2–9.
Example 2–7 SPECTRUM OF A TRIANGULAR PULSE
The spectrum of a triangular pulse can be obtained by direct evaluation of the FT integral using
equations for the piecewise linear lines of the triangle shown in Fig. 2–6c. Another approach,
which allows us to obtain the FT quickly, is to first compute the FT for the second derivative of
the triangle. We will take the latter approach to demonstrate this technique. Let
Then
w(t) =(t/T)
dw(t)
dt
= 1 T u(t + T) - 2 T u(t) + 1 u(t - T)
T
and
d 2 w(t)
dt 2 = 1 T d(t + T) - 2 T d(t) + 1 d(t - T)
T