Page 339 - Satellite Communications, Fourth Edition

Error Control Coding 319

It will be noted that the matrix has 7 columns and 4 rows correspon-
ding to the (7, 4) code, and furthermore, the first four columns form an
identity submatrix. The identity submatrix results in the dataword
appearing as the first four bits of the codeword, in this example. In gen-
eral, a systematic code contains a sequence that is the dataword, and the
most common arrangement is to have the dataword at the start of the
codeword as shown in the example. It can be shown that any linear block
code can be put into systematic form. The remaining bits in any row of
G are responsible for generating the parity bits from the data bits. As
an example, suppose it is required to generate a codeword for a dataword
[1010]. This is done by multiplying d by G

1 0 0 0 1 1 1
0 1 0 0 1 1 0
C [1 0 1 0] ≥ ¥
0 0 1 0 1 0 1
0 0 0 1 0 1 1

[1010010]
The dataword is seen to appear as the first four bits in the codeword,
and the end three bits are the parity bits. The parity bits are generated
from the data bits by means of the last three columns in the generator
matrix. This submatrix is denoted by P:
1 1 1

1 1 0
P 5 ≥ ¥ (11.4)
1 0 1
0 1 1
The transpose of P, which enters into the decoding process is formed
by interchanging rows with columns, that is, row 1 becomes column 1,
and column 1 becomes row 1, row 2 becomes column 2 and column 2
T
becomes row 2, and so on. In full, the transpose of P, written as P is
1 1 1 0
T
P 5 C1 1 0 1S
1 0 1 1
What is termed the parity check matrix (denoted by H) is now formed
T
by appending an identity matrix to P :

1 1 1 0 1 0 0
H 5 C1 1 0 1 0 1 0S (11.5)
1 0 1 1 0 0 1