Page 336 - Satellite Communications, Fourth Edition

316 Chapter Eleven

coding is employed, the distinction between P and BER becomes impor-
e
tant. P is still determined by conditions at the input, but the error con-
e
trol will, if properly implemented, make the probability of bit error at
the output (the BER) less than that at the input. Error control coding
applies only to digital signals, and in most cases the signal is in binary
form, where the message symbols are bits, or logic 1s and 0s.
Encoding refers to the process of adding coding bits to the uncoded
bit stream, and decoding refers to the process of recovering the original
(uncoded) bit stream from the coded bit stream. Both processes are usu-
ally combined in one unit termed a codec.

11.2 Linear Block Codes

A block code requires that the message stream be partitioned into blocks
of bits (considering only binary messages at this stage). Let each block
contain k bits, and let these k bits define a dataword. The number of
k
datawords is 2 . There is no redundancy in the system, meaning that
even a single bit error in transmission would convert one dataword into
another, which of course would constitute an error.
The datawords can be encoded into codewords which consist of n bits,
where the additional n k bits are derived from the message bits but
n
are not part of the message. The number of possible codewords is 2 , but
k
only 2 of these will contain datawords, and these are the ones that are
transmitted. It follows that the rest of the codewords are redundant, but
only in the sense that they do not contribute to the message. (The n k
additional bits are referred to as parity check bits). If errors occur in
transmission, there is high probability that they will convert the per-
missible codewords into one or another of the redundant words that the
decoder at the receiver is designed to recognize as an error. It will be
noted that the term high probability is used. There is always the pos-
sibility, however remote, that enough errors occur to transform a trans-
mitted codeword into another legitimate codeword in error.
The code rate r is defined as the ratio of dataword bits to codeword
c
bits (note that although it is called a rate, it is not a rate in bits per
second)

k
r n (11.1)
c
The code is denoted by (n, k) for example a code which converts a 4-bit
dataword into a 7 bit codeword would be a (7, 4) code.
A repetition code illustrates some of the general properties of block
codes. In a repetition code, each bit is considered to be a dataword, in
effect, k 1. For n-redundancy encoding, the output of the encoder is
n bits, identical to the input bit. As an example, consider the situation