參數(shù)資料
型號： AN2407
廠商： 飛思卡爾半導(dǎo)體(中國)有限公司
英文描述： Reed Solomon Encoder/Decoder on the StarCore SC140/SC1400 Cores, With Extended Examples
中文描述： 里德所羅門編碼器/的StarCore SC140/SC1400核心解碼器，以擴展實例
文件頁數(shù)： 7/48頁
文件大?。?/td> 306K
代理商： AN2407
Theory
Reed Solomon Encoder/Decoder on the StarCore SC140/SC1400 Cores, With Extended Examples, Rev. 1
Freescale Semiconductor
7
The redundant symbols are obtained from the redundant polynomial
p(x),
which is the remainder
obtained by dividing
x
2T
m(x)
by the generator polynomial
g(x):
where
is the generator polynomial. We choose the most frequently used generating polynomial:
g x
( )
x
+
(
=
The code-word polynomial c
(x)
is defined as follows:
Since in
GF(2
m
)
algebra, plus (+), and minus (–) are the same, the code word actually equals the polynomial
x
2T
m(x)
minus its remainder under division by
g(x)
. It follows that c
(x)
is a multiple of g
(x).
Since there is a total of
2
mK
different possible messages, there are
2
mK
different valid code words at the encoder output. This set of
2
mK
code words of length
N
is called an
(N,K) block code
.
2.2.2 Decoding
When a received block is input to the decoder for processing, the decoder first verifies whether this block appears
in the dictionary of valid code words. If it does not, errors must have occurred during transmission. This part of the
decoder processing is called
error detection.
The parameters necessary to reconstruct the original encoded block
are available to the decoder. If errors are detected, the decoder attempts a reconstruction. This is called
error
correction
.
Conventionally, decoding is performed by the Petersen-Gorenstein-Zierler (PGZ) algorithm, which
consists of four parts:
1.
Syndromes calculation.
2.
Derivation of the error-location polynomial.
3.
Roots search.
4.
Derivation of error values.
The error-location polynomial in this implementation is found using the Berlekamp-Massey algorithm, and the
error values are obtained by the Forney algorithm. The four decoding parts are briefly outlined as follows:
1.
Syndromes calculation
: From the received block, the received polynomial is reconstructed, denoted as
c(x).
The received polynomial is the superposition of the correct code word
c(x)
and an error polyno-
mial
e(x):
The error polynomial is given in its most general form by:
p x
( )
x
2
T
m x
( )
(
)
modg x
( )
=
α
p
1
)
x
(
α
p
2
+
)
x
(
α
p
3
+
)…
x
α
p
2
T
+
(
)
g x
( )
x
α
+
(
)
x
(
α
2
+
)
x
(
α
3
+
)…
x
α
2
T
+
(
)
=
c x
( )
x
2
T
m x
( )
p x
( )
+
=
r x
( )
c x
( )
e x
( )
+
=
e x
( )
e
i
0
x
i
0
e
i
1
x
i
1
e
i
2
x
i
2
…
e
i
ν
1
–
(
)
x
i
ν
1
–
(
)
,
+
+
=
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