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Sec. 1–11 Coding 21
Transmitter
Noise
Digital
source
m
Encoder
and other
signal processing
g(t)
Carrier
s(t)
Transmission medium
r(t)
circuits
(channel)
r(t)
Carrier
circuits
~
g(t)
Decoder
and other
signal processing
m
~
Digital
sink
Figure 1–4
Receiver
General digital communication system.
However, these extra bits have the disadvantage of increasing the data rate (bits/s) and,
consequently, increasing the bandwidth of the encoded signal.
Codes may be classified into two broad categories:
• Block codes. A block code is a mapping of k input binary symbols into n output binary
symbols. Consequently, the block coder is a memoryless device. Because n 7 k, the
code can be selected to provide redundancy, such as parity bits, which are used by the
decoder to provide some error detection and error correction. The codes are denoted by
(n, k), where the code rate R † is defined by R = k/n. Practical values of R range from
1 7
4 to 8, and k ranges from 3 to several hundred [Clark and Cain, 1981].
• Convolutional codes. A convolutional code is produced by a coder that has memory.
The convolutional coder accepts k binary symbols at its input and produces n binary
symbols at its output, where the n output symbols are affected by v + k input
symbols. Memory is incorporated because v 7 0. The code rate is defined by
R = k/n. Typical values for k and n range from 1 to 8, and the values for v range
1 7
from 2 to 60. The range of R is between
4
and
8
[Clark and Cain, 1981]. A small
value for the code rate R indicates a high degree of redundancy, which should
provide more effective error control at the expense of increasing the bandwidth of
the encoded signal.
Block Codes
Before discussing block codes, several definitions are needed. The Hamming weight of a code
word is the number of binary 1 bits. For example, the code word 110101 has a Hamming
weight of 4. The Hamming distance between two code words, denoted by d, is the number of
positions by which they differ. For example, the code words 110101 and 111001 have a
distance of d = 2. A received code word can be checked for errors. Some of the errors can be
† Do not confuse the code rate (with units of bits/bits) with the data rate or information rate (which has units
of bits/s).