IJCA Template - International Journal of Computer Applications - IJCA

International Journal of Computer Applications (0975 – 8887)
Volume 7– No.7, October 2010
1. The j th component of each vector w i is a 1
2. For ij there is at most one vector whose i th component is a 1
These J vectors are said to be orthogonal on the j th digit position.
We call them orthogonal vectors. Now, let use form J parity-check
sums from these J orthogonal vectors,
For each i in {1,.., J}
A
i
p
j
e
p
e
we see that the error digit e j is checked by all the check sums
above. Because of the second property of the orthogonal vectors,
any error digit other than e j is checked by at most one check sum.
These J check sums are said to be orthogonal on the error digit e j .
If all the error digits in the sum A i are zero for ij, the value of A i
is equal to e j . Based on this fact, the parity-check sums orthogonal
on e j can be used to estimate e j , or to decode the received digit r j .
Table 1 shows some examples of OSMLD codes that can be
decoded with the proposed SISO decoder described above. In this
table we used the abbreviations DSC for Difference Set Cyclic
codes, EG for Euclidean Geometry codes and BCH for Bose
Chaudhuri and Hocquenghem codes. The EG codes used in this
study are 0-order and, for an extensive description of projective
geometry codes and Euclidean geometry can be found in [10].
Table 1. Set of OSMLD Codes
N k J d min Rate code
7 3 3 4 0.42 DSC
15 7 4 5 0.46 BCH
21 11 5 6 0.52 DSC
63 37 8 9 0.58 EG
73 45 9 10 0.61 DSC
255 175 16 17 0.68 EG
273 191 17 18 0.69 DSC
819 447 15 16 0.54 OSMLD
1057 813 33 34 0.76 DSC
4161 3431 65 66 0.82 DSC
16513 14325 129 130 0.86 DSC
2.2 Majority Logic Decoding Principle
The error digit ej is decoded as 1 if at least one-half of the check
sums orthogonal on ej, are equal to 1; otherwise, ej is decoded as
0 like majority rule. When C is a cyclic code, each ei can be
decoded simply by cyclically permuting the received word r into
the buffer store.
Example: let us consider the (7,3) code, which is the short code in
difference set codes class (see Table 1). This code is specified by
the perfect difference set P={0, 2, 3} of order 2. From this prefect
set, we can form the following three check sums orthogonal on e 7 :
A 1 = e 4 + e 5 + e 7
A 2 = e 2 + e 6 + e 7
A 3 = e 1 + e 3 + e 7
The If a simple error e=(000001) occur, than we have A 1 = A 2 =
A 3 = 1. If a double error occur, as an example e 7 =1 and one value
j
of e 1 , ..., e 6 is equal to 1, then two values of Ai are 1. So we can
say that :
- e 7 =1 if only and if at least 2 values of Ai are 1
- e 7 =0, otherwise.
2.3 Soft-input Soft-output Threshold
Decoding
Threshold decoding is simply the logical extension to soft
decisions of majority decoding. Soft-out decoding algorithm when
applied to OSMLD codes is stated as follows:
Let us consider a transmission of block coded binary symbols
{0,1} using a BPSK modulation over AWGN channel, the
decoder soft output for the j th bit position of a given soft input
R(r ,r ,..,r ) is defined as:
1 2 n
P(c 1/ R)
j
LLR
j
ln P(c
j
0 / R)
where C(c
1,c 2,...,c n) is the transmitted codeword. The hard
decision vector corresponding to the received vector r is denoted
by H(h 1 , h 2, ..., h n ) .
For a code with J orthogonal parity check equations, (1) can be
expressed as:
j
i
LLR
j
ln P(c
j
0 / B
i
P(c
1/ B
where B , i=1...J are obtained from the orthogonal parity check
i
equations on the j th bit as follows:
The term B 0 is defined to be B 0 = h j . Each of the B i=1,..,J is
computed by dropping the term h j from the i th orthogonal parity
equation. Thus, each of the B i can be written as:
n i
B c e
(3)
i j ik
k1
Where
eik
is the k th error term in the i th parity check equation
excluding e j ( n is the total number of terms in the i th orthogonal
parity equation without c ).
i
By applying BAYES rule, (2) becomes
i j j
LLR
j
ln P( B
i / c
j
0) P(c
j
0)
j
P( B / c 1) P(c 1)
Since the parity check equations are orthogonal on the j th symbol
the individual probabilities P(B / c 1 or 0) are all independent
and (4) can be rewritten as
J P(B
i
/ c
j
1) P(c
j
1)
LLR
j
ln ln
i0
P(B
i
/ c
j
0) P(c
j
0)
i
Assume that the transmitted symbols are equally likely to be 0 or
1, and thus the last term in (5) is null. As a result, we obtain
j
i
(1)
(2)
(4)
(5)
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