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Sec. 3–5 Line Codes and Spectra 175
where is a modulo 2 adder or an exclusive-OR gate (XOR) operation. The received encoded
data are decoded by
(3–48)
where the tilde denotes receiving-end data.
Each digit in the encoded sequence is obtained by comparing the present input bit
with the past encoded bit. A binary 1 is encoded if the present input bit and the past
encoded bit are of opposite state, and a binary 0 is encoded if the states are the same. This
is equivalent to the truth table of an XOR (exclusive-OR) gate or a modulo 2 adder. An
example of an encoded sequence is shown in Table 3–4, where the beginning reference digit
is a binary 1. At the receiver, the encoded signal is decoded by comparing the state of adjacent
bits. If the present received encoded bit has the same state as the past encoded bit, a
binary 0 is the decoded output. Similarly, a binary 1 is decoded for opposite states. As
shown in the table, the polarity of the differentially encoded waveform may be inverted
without affecting the decoded data. This is a great advantage when the waveform is passed
through thousands of circuits in a communication system and the positive sense of the
output is lost or changes occasionally as the network changes, such as sometimes occurs
during switching between several data paths.
Eye Patterns
'
dn = ' e n { ' e n-1
The effect of channel filtering and channel noise can be seen by observing the received line
code on an analog oscilloscope. The left side of Fig. 3–18 shows received corrupted polar
NRZ waveforms for the cases of (a) ideal channel filtering, (b) filtering that produces intersymbol
interference (ISI), and (c) noise plus ISI. (ISI is described in Sec. 3–6.) On the right
side of the figure, corresponding oscilloscope presentations of the corrupted signal are
shown with multiple sweeps, where each sweep is triggered by a clock signal and the sweep
width is slightly larger than T b . These displays on the right are called eye patterns, because
they resemble the picture of a human eye. Under normal operating conditions (i.e., for no
TABLE 3–4
EXAMPLE OF DIFFERENTIAL CODING
Encoding
Input sequence d n 1 1 0 1 0 0 1
Encoded sequence e n 1 0 1 1 0 0 0 1
Reference digit ——————————
c
Decoding (with correct channel polarity)
Received sequence
'
e n 1 0 1 1 0 0 0 1
(correct polarity)
Decoded sequence
'
d n
1 1 0 1 0 0 1
Decoding (with inverted channel polarity)
Received sequence
'
e n 0 1 0 0 1 1 1 0
(inverted polarity)
Decoded sequence
'
d n
1 1 0 1 0 0 1