The generator polynomial of (n,k) cyclic code
is a factor of pn 1 and is given as
(5.21)
The message vector can be expressed as
(5.22)
The code word to be transmitted is then given as
(5.23)
For the (7,4) code we have been discussing, the generator polynomial is given as g(p) p3
p 1. This polynomial can be translated to a generator matrix given below. Note that this matrix form
is for a nonsystematic code, but as mentioned earlier this matrix can be transformed to become
systematic.
(5.24)
A systematic code word can be generated in the following way: multiply the message polynomial,
x(p), by pn??“k, then divide the result by the generator polynomial, g(p), to obtain the remainder, r(p).
Lastly, add the remainder to the shifted data sequence. This is shown mathematically below
(5.25)
Equivalently stated as
(5.26)
The division is accomplished using a feedback shift register structure shown in Fig. 5.16. For the
first k bit times the switch is in the #1 position so the message bits are sent to the output, since we are
considering a systematic coder. Then the switch is moved to the number 2 position where the feedback
path is disconnected and the parity bits are read out.
pnkx(p) z(p)g(p) r(p)
pnkx( p)
g( p) z( p)
r( p)
g( p)
G D1 0 1 1 0 0 0
0 1 0 1 1 0 0
0 0 1 0 1 1 0
0 0 0 1 0 1 1 T
C(p) x(p) # g(p)
x( p) xk1pk1 xk2 pk2 c x1p x0
g( p) pnk gnk1pnk1 c g1p 1
C[Cn2Cn3cC0Cn1 ]
[Cn1Cn2cC1C0 ]
PERFORMANCE IMPROVEMENT TECHNIQUES 235
TABLE 5.
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