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Sec. 3–4 Digital Signaling 161
1.5
1
w 1 (t)
0.5
0
–0.5
0
1
2 3 4
t(ms)
5 6 7 8
(a) Rectangular Pulse Shape, T b =1 ms
1.5
1
w 2 (t)
0.5
0
–0.5
0
1
2 3 4
5 6 7 8
(b) sin(x)/x Pulse Shape, T b =1 ms
t(ms)
Mid-symbol
sampling
Figure 3–12
Binary signaling (computed). (See Example3_07.m.)
What is the bandwidth for the waveform in Fig. 3–12a? Using Eq. (3–32), we find that
1
the lower bound for the bandwidth is 2D = 500 Hz. In Sec. 3–5, it is shown that the actual null
bandwidth of this binary signal with a rectangular pulse shape is B = 1T s = D = 1,000 Hz. This
is larger than the lower bound for the bandwidth, so the question arises, What is the waveshape
that gives the lower bound bandwidth of 500 Hz? The answer is one with sin (x)x pulses, as
described in Case 2.
CASE 2. SIN (X)/X PULSE ORTHOGONAL FUNCTIONS From an intuitive
viewpoint, we know that the sharp corners of the rectangular pulse need to be rounded to reduce
the bandwidth of the waveform. Furthermore, recalling our study with the sampling theorem,
Eq. (2–158), we realize that the sin (x)x-type pulse shape has the minimum bandwidth.
Consequently, we choose
w k (t) =
sin e p T s
(t - kT s )f
p
T s
(t - kT s )
(3–33)