18 Syndrome-based error correction block diagram for the (7,4) code.
When the code word is cyclic, the calculation of the syndrome can be performed by a shift register,
very similar to what was performed in the encoder discussion above. Recall the received code word,
y(p), is written as follows:
(5.41) y(p) C(p) e(p)
(5.42)
(5.43)
The division of y(p) by generator polynomial g(p) is carried out by a shift register structure, a general
division is shown in Fig. 5.19 [5].
y(p)
g(p) z(p)
r (p)
g(p)
y(p) x(p)g(p) e(p)
PERFORMANCE IMPROVEMENT TECHNIQUES 239
+
g0
S0 +
g1
S1 +
gn??“k??“1
Sn??“k??“1 . . . y(p)
FIGURE 5.19 General syndrome calculation for cyclic codes.
As we have done previously we will use the (7,4) code as an example to illustrate the syndrome
calculation. In Fig. 5.20, we show a method of calculating the syndrome.
The syndrome calculation procedure is relatively simple. After n shifts the contents of the registers
is equal to the syndrome. For the first n bits the switch is in position 1. Then the switch is placed into
position 2 for the next n k bits in order to extract the syndrome from the registers. It may be premature
to conclude the syndrome is a characteristic of the transmitted code word, but recall we have previously
shown this to not be the case.
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