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Baseband Pulse and Digital Signaling Chap. 3
where T s = T b for the case of binary signaling. The resulting waveform that is transmitted is
shown in Fig. 3–12b.
Once again, the data can be detected at the receiver by evaluating the orthogonal-series
coefficients. Because sin (x)x orthogonal functions are used, Eq. (2–160) shows that the data can
be recovered simply by sampling † the received waveform at the midpoint of each symbol interval.
Referring to Fig. 3–12 and sampling at the midpoint of each T s = 1-ms interval, the correct 8-bit
word 01001110 is detected.
For Case 2, the bit rate and the baud are still R = 1 kbits and D = 1 kbaud. The absolute
bandwidth of Eq. (3–33) can be evaluated with the help of Fig. 2–6b, where 2W = 1T s . That is,
B = 1(2T s ) = 500 Hz. Thus, the lower-bound bandwidth (as predicted by the dimensionality
theorem) is achieved.
Note that when the rectangular pulse shape is used, as shown in Fig. 3–12a, the digital
source information is transmitted via a binary digital waveform. That is, the digital signal is a
digital waveform. However, when the sin (x)x pulse shape is used, as shown in Fig. 3–12b, the
digital source information is transmitted via an analog waveform (i.e., an infinite number of
voltage values ranging between -0.4 and 1.2 V are used).
See Example3_07.m for the plots shown in Fig. 3–12.
Multilevel Signaling
In the case of binary signaling discussed in Example 3–7, the lower-bound bandwidth of
B = N(2T 0 ) was achieved. That is, for Case 2, N = 8 pulses were required and gave a bandwidth
of 500 Hz, for a message duration of T 0 = 8 ms. However, this bandwidth could be
made smaller if N could be reduced. Indeed, N (and, consequently, the bandwidth) can
be reduced by letting the w k ’s of Eq. (3–27) take on L 7 2 possible values (instead of just the
two possible values that were used for binary signaling). When the w k ’s have L 7 2 possible
values, the resulting waveform obtained from Eq. (3–27) is called a multilevel signal.
Example 3–8 L = 4 MULTILEVEL SIGNAL
Here, the M = 256-message source of Example 3–7 will be encoded into an L = 4 multilevel signal,
and the message will be sent, once again, in T 0 = 8 ms. Multilevel data can be obtained by encoding
the -bit binary data of the message source into L-level data by using a digital-to-analog
converter (DAC), † as shown in Fig. 3–13. For example, one possible encoding scheme for an
= 2-bit DAC is shown in Table 3–3. = 2, bits are read in at a time to produce an output that is
one of L = 4 possible levels, where L = 2 .
† The term analog in digital-to-analog converter is a misnomer because the output is an L-level digital signal.
However, these devices are called digital-to-analog converters in data manuals.