Here we have
(5.30)
Let us define the syndrome, S, as
(5.31) S r S
# HT
r S
C S
e S
e S
C S
r S
HT cInk
P d
H [InkPT ]
G # HT 0
The syndrome is a result of a parity check performed on the received code word to determine whether
is a valid member of the code word set. If the syndrome is zero, then is a member and/or may have
no errors. If the syndrome is not zero then is not a member of the vector space and it has errors. We can
construct the parity check matrix for the (7,4) example we provided above, it is given below
(5.32)
Suppose the transmitted vector is given as
(5.33)
And the received vector is given as
(5.34)
Based on our previous discussion we can rewrite this as follows:
(5.35)
The syndrome, S, to be calculated is given as
(5.36)
Searching the H matrix to find a column that matches S, we find the sixth column matches. So we
have determined the error vector is
(5.37)
Next we add the locally generated error vector to the received vector in order to correct the error present
and recover the transmitted code word.
(5.38)
Below we will show the syndrome is directly related to the error vector and not actually the
received vector.
S C S
# HT e S
# HT
S r S
# HT (C S
e S
) # HT
C S
[ 1 0 1 1 0 0 0 ] [ 0 0 0 0 0 1 0] [ 0 0 0 0 0 1 0]
r S
e ^
C S
e S
e ^
r S
C S
e S
e^
[ 0 0 0 0 0 1 0]
S [ 1 1 0 ]
S r S
# HT [ 1 0 1 1 0 1 0 ]# G1 0 0
0 1 0
0 0 1
1 0 1
1 1 1
1 1 0
0 1 1 W
r S
[1 0 1 1 0 0 0] [0 0 0 0 0 1 0]
r S
C S
e S
r S
[1 0 1 1 0 1 0]
C S
[1 0 1 1 0 0 0]
H [I3 PT ] C1 0 0 1 1 1 0
0 1 0 0 1 1 1
0 0 1 1 1 0 1 S r S
r S
r S
PERFORMANCE IMPROVEMENT TECHNIQUES 237
(5.
Pages:
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395