563489578934

160
Baseband Pulse and Digital Signaling Chap. 3
Bandwidth Estimation
The lower bound for the bandwidth of the waveform representing the digital signal, Eq. (3–27),
can be obtained from the dimensionality theorem. Thus, from Eqs. (2–174) and (3–28), the
bandwidth of the waveform w(t) is
B Ú N = 1 (3–32)
2T 0 2 D Hz
If the w k (t) are of the sin (x)x type, the lower bound absolute bandwidth of N(2T 0 ) = D2
will be achieved; otherwise (i.e., for other pulse shapes), the bandwidth will be larger than this
lower bound. Equation (3–32) is useful for predicting the bandwidth of digital signals,
especially when the exact bandwidth of the signal is difficult (or impossible) to calculate.
This point is illustrated in Examples 3–7 and 3–8.
Binary Signaling
A waveform representing a binary signal can be described by the N-dimensional orthogonal
series of Eq. (3–27), where the orthogonal series coefficients, w k , take on binary values. More
details about binary signaling, including their waveforms, data rate, and waveform bandwidth,
are illustrated by the following example.
Example 3–7 BINARY SIGNALING
Let us examine some properties of binary signaling from a digital source that can produce M = 256
distinct messages. Each message could be represented by n = 8-bit binary words because M = 2 n =
2 8 = 256. Assume that it takes T 0 = 8 ms to transmit one message and that a particular message
corresponding to the code word 01001110 is to be transmitted. Then,
w 1 = 0, w 2 = 1, w 3 = 0, w 4 = 0, w 5 = 1, w 6 = 1, w 7 = 1, and w 8 = 0
CASE 1. RECTANGULAR PULSE ORTHOGONAL FUNCTIONS Assume that
the orthogonal functions w k (t) are given by unity-amplitude rectangular pulses that are T b = T 0 n =
88 = 1 ms wide, where T b is the time that it takes to send 1 bit of data. Then, with the use of
Eq. (3–27) and MATLAB, the resulting waveform transmitted is given by Fig. 3–12a.
The data can be detected at the receiver by evaluating the orthogonal-series coefficients as
given by Eq. (3–30). For the case of rectangular pulses, this is equivalent to sampling the waveform
anywhere within each bit interval. † Referring to Fig. 3–12a, we see that sampling within
each T b = 1-ms interval results in detecting the correct 8-bit data word 01001110.
The bit rate and the baud (symbol rate) of the binary signal are R = nT 0 = 1 kbits, and
D = NT 0 = 1 kbaud, because N = n = 8 and T 0 = 8 ms. That is, the bit rate and the baud are equal
for binary signaling.
† Sampling detection is optimal only if the received waveform is noise free. See the discussion following
Eq. (3–30).