FEC

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FEC
The Reed-Solomon Theory
Project Members Alon Keren Sagi Linsker
Instructor Omer Gurewitz
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FEC The Reed-Solomon Method
How does FEC work? (Cont.)
k symbols
Each symbol contains a predefined No. of bits.
Data block
FEC encoding
n
Data
Redundant
k
l
k
FEC decoding
Noisy channel
Data block
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FEC The Reed-Solomon Method
What is RS method?
Reed Solomon codes are a subset of BCH codes and
are linear block codes. A Reed-Solomon code is
specified as RS(n,k). RS will correct code as
long as 2s r lt 2t. gt Up to t errors or up to
2t erasures. t half the number of redundancy
symbols s errors in block r erasures in block
(occur when the position of an erred symbol is
known.)
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FEC The Reed-Solomon Method
Why Reed-Solomon?
1. Good for handling burst-errors (using symbols
and not bits).
2. Does not require a back-channel (as opposed to
ARQ). Therefore, an excellent solution for
multicast.
3. Can be good for streaming (unlike ARQ, in
which a message can arrive later than expected).
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FEC The Reed-Solomon Method
Galois Fields
???? ?? Reed-Solomon ????? ?? ???????? ?? Finite
fields ?? ??? ??? Galois Fields.
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FEC The Reed-Solomon Method
Galois Fields
?? ??? ????? ???????? ????? ?? m ?????, ???? ????
?????? ????? ?????? ???????.

• ??????
• ?? ????? ???? ?????? ????? ????? ?? ????
• ????? "???? ????". gt ????? ???? ???? ??????.
• ????? ??? ?????? 2. gt ????? ???? ?????? ??????.

???? ???...
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FEC The Reed-Solomon Method
Galois Fields
??????? p(x) ????? m, ??? ??????? ???? ?? 1)
???????? ??? ???? ???? ??? . 2) ?????
?????? ????? n ??? p(x) ???? ?? ???
.
?? ???? ???? ??? ??? ???? ??????? ????
????? ?? ?? ????? ????.
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FEC The Reed-Solomon Method
Galois Fields
????? ????
??? ???? ?? ???????? ?????
gt ???? ???? ???? ?? GF(8). ?????, ????
????? ?????? ?? ?? ????? ????.
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FEC The Reed-Solomon Method
Galois Fields
???? ???
??? ???? ?? ????????.
(????? ?????? 2)
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FEC The Reed-Solomon Method
Encoding
???? ??? ??? ???? ??? ????? k ????????, ?? ??????
????? m ?????. ???? ????? ????? ???????? ?????
k-1 ???? ??????? ?????? ???? . ?? ????
????? ?????? ????? m.
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FEC The Reed-Solomon Method
Encoding
????? ??? ????? ?? k ????????? ????? ????? n
???????? ( ), ?? ????? ???? ?? t(n-k)/2 ??????
????? ?????. ???? ?????? ????? ???? ??
2t symbols (redundant )
k symbols (original)
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FEC The Reed-Solomon Method
Encoding
??? ????? ?? v(x) ????? ????? ?? "???????? ?????"
?? ??????
???, ????? ?? ???? ???? ?- . ????? ?????? ???
????? n-1 ???? ???? ????? ??
????, ?????
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FEC The Reed-Solomon Method
Encoding
K ????????? "??????" ?-2t
2t ????????? ???????
????, ???? ?????
????? ?????? ?????
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FEC The Reed-Solomon Method
Encoding
2t ??????? ?? ???????? ????? ???? ?? ?????? ??
v(x).
?????? ?? v(x).
????? ?? ?????? ?????? ??????!
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FEC The Reed-Solomon Method
Decoding
???? ????? ?? ???????? ?????? ?????? ??????
?????? ?????? ?? ???? ?????? v(x) ???????? ?????
e(x)
????? ?-i ?? ???????? e(x) ???? 0 ???? ??? ?????
??????? ?-i. ????? ????, ???? ? v ??????,
,??????? ??????? ???? ??????. ???, ??????? ??????
???? ?????? ??
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FEC The Reed-Solomon Method
Decoding
??? ??????, ????? g(x), , ?????? ?? v(x). ???
???, ????? ????? Sj ????????
??? ???
?? ?? ?????????? ???????, ??? ??? ??????, ??????
?????? ??????.
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FEC The Reed-Solomon Method
Decoding
????? ?? ????? ????? ???????
??? ???? ???????
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FEC The Reed-Solomon Method

Decoding
???, ????? ????? ?? ?????? ?? 2t ??????? ??
????????, ?? 2v ??????, ??? ????? ?? ?????
??????? ?????. ?????, ?? ?????? ????? ?? ?????
????? ??? ??????? ?????? ???
1. ??????? ??????? ?????? ??????
2. ????? ??????????? Berlekamp-Massey ??
Euclidean Algorithm, ???? ????? ???????? ??"?.
3. ?????? ?? ?? ?????? ???????? ???????? ?? ???
????? ?? ??????.
4. ????? ????????? ?? Forney, ?????? ?? ????
???????
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FEC The Reed-Solomon Method

Decoding
????? ????? ??? ?? ??????? ?????? ?? e(x),
?????? ????? ??? ?? ???? ??????. ?????,
????? ?? ??????? ?????? e(x). ???-??, ????? ?????
???????, ??? ??? p(x).
??, ????? v(x). ?????
????? ??????? ???? ?????? ???? ????? k ??????
???????
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FEC The Reed-Solomon Method

Decoding Results
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FEC The Reed-Solomon Method

Decoding Results