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Baseband Pulse and Digital Signaling Chap. 3
Example 3–6 VECTOR REPRESENTATION OF A BINARY SIGNAL
Examine the representation for the waveform of a 3-bit (binary) signal shown in Fig. 3–11a. This
signal could be directly represented by
where p(t) is shown in Fig. 3–11b and p j (t) ! pCt - (j - 1 2 ) TD.
The set {p j (t)} is a set of orthogonal functions that are not normalized. The vector
is the binary word with 1 representing a binary 1 and 0 representing a binary 0. The function p(t) is
the pulse shape for each bit.
Using the orthogonal function approach, we can represent the waveform by
Let { w j (t)} be the corresponding set of orthonormal functions. Then, using Eq. (2–78) yields
or
N=3
s(t) = a d j pCt - (j - 1 )TD N=3
2 = a d j p j (t)
j=1
w j (t) = p j(t)
=
3K j
1
,
w j (t) = c 2T
0,
d = (d 1 , d 2 , d 3 ) = (1, 0, 1)
N=3
s(t) = a s j w j (t)
j=1
(j - 1)T< t < jT
t otherwise
where j = 1, 2, or 3. Using Eq. (2–84), where a = 0 and b = 3T, we find that the orthonormal series
coefficients for the digital signal shown in Fig. 3–11a are
s = (s 1 , s 2 , s 3 ) = (52T , 0, 52T)
The vector representation for s(t) is shown in Fig. 3–11d. Note that for this N = 3-dimensional
case with binary signaling, only 2 3 = 8 different messages could be represented. Each message
corresponds to a vector that terminates on a vertex of a cube.
See Example3_06.m for the waveform generated by the orthonormal series. This is identical
to Fig. 3–11a.
p j (t)
T 0
p 2
C j (t) dt
L
0
j=1
= p j(t)
125T