linear block code 指導教授：黃文傑 博士 學生：吳濟廷 2004.02.10. 2 outline...
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Linear Block Code
指導教授：黃文傑 博士學生：吳濟廷2004.02.10
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OUTLINE Introduction Encoding
Generator matrix Decoding
Parity check matrix Syndrome Error correction
Conclusion
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Source v.s. Channel Coding Why source coding ?
Eliminate redundancy in the data Send same information in fewer bits
Why channel coding ? Combat channel effects Coding gain
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Code taxonomy
Today
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Linear Block Code A (n,k) block code , where
Block : encoder accepts a block of message symbols and generates a block of codeword
Linear : addition of any two valid codeword results in another valid codeword
n k
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Code Rate Code rate
Code rate increases , error correcting capability decreases
more bandwidth efficiency Code rate decreases , error correcting capability increases
waste of bandwidth
k
n
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Vector Space In a binary field
011
100
10+
addition multiplication
101
000
10
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Vector Space and Subspace Vector space : set of all binary n-tuples , Vector subspace : subset S of vector space
All-zeros vector is in S Sum of any two vectors in S is also in S
nV
nV
0000 0001 0010 0011 0100 0101 0110 0111
1000 1001 1010 1011 1100 1101 1110 1111
= 16 4-tupels4
4 2V
4VSubset of
0000 0101 1010 1111
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Vector Space structure
Packing the space with as many codewords as possible
Codewords to be as far apart from one another as possible
Linear block-code structure
nV
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A (6,3) Linear Block Code
6-tuples 32 message vector 32
But ….. How to generate all these codewords ?
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Generator Matrix We could use look-up table for small k But for larger k, we use the generator matrix G For (6,3) code
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2
3
1 1 0 1 0 0
0 1 1 0 1 0
1 0 1 0 0 1
V
G V
V
1
4 2
3
1 2 3
1 1 0 1 1 0
1 1 0
1 1 0 1 0 0 0 1 1 0 1 0
1 0 1 1 1 0
V
U G V
V
V V V
The same method for the other messages….
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Generator Matrix
11 12 1,( )
21 22 2,( )
1 2 ,( )
1 0 0
0 1 0
0 0 1
k
n k
n k
k k k n k k n
G P I
P P P
P P P
P P P
where P is the parity array
and is the k*k identity matrix
(0,1)ijp
kI
(n-k)
k
k
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Codeword using Generator MatrixFor a (n,k) code
where
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Parity-Check Matrix Parity-check matrix H enable us to decode For (k*n) generator matrix G, there exist an (n-k)*n
matrix H
(n-k)
k
(n-k)
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Parity-Check Matrix It’s easy to verify that for each codeword U, ge
nerated by G and the
to check if each basis still has orthogonality
TH
1 2 1 211 21 1,( )
21 2,( )
1 2 ,( )
1 1 2 2
1 0 0
0 1 0
0
0 0 1, ,..., , , ,...,
, ,..., 0
Tn k k
n k
n k
k k k n k
n k n k
UH p p p m m mp p p
p p
p p p
p p p p p p
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Syndrome Testing Let r be a received vector,
where is an error pattern Syndrome of r is defined
r U e
1 2, ,..., ne e e e
( )
T
T
T T
T
S rH
U e H
UH eH
eH
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Syndrome Testing For example, if transmited
and received The syndrome
And the syndrome of error pattern
1 0 1 1 1 0U
0 0 1 1 1 0r error
1 3 1 6
6 3
1 0 0
0 1 0
0 0 10 0 1 1 1 0
1 1 0
0 1 1
1 0 1
1 1 1 1 1 1 0 0
TS rH
1 0 0 0 0 0 1 0 0T TS eH H
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Error Correction Stand array : represent possible received vectors
contains all the correctable error
=0
coset
Coset leader
2n k
2k
(n,k) standard array
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Standard Array Coset : a set of numbers having a common syndrome
Coset leader : correctable error patterns Received vector doesn’t mean the Tx mes
sage is for sure except the error pattern is
( ) T Ti j jS U e H e H
i jr U e
iU je
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Error Correction Decoding Procedure Calculate syndrome of r : Locate the coset leader : Corrected codeword :
TS rH
je
jU r e
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Example for (6,3) code
1 0 0
0 1 0
0 0 1
1 1 0
0 1 1
1 0 1
jS e
Standard array for (6,3) code
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Example for (6,3) code Assume transmitted
received Syndrome
Use look-up table
1 0 1 1 1 0U
0 0 1 1 1 0r
0 0 1 1 1 0 1 0 0TS H
ˆ 1 0 0 0 0 0eˆ ˆ
0 0 1 1 1 0 1 0 0 0 0 0
1 0 1 1 1 0
U r e
The same as transmitted codeword !!
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Conclusion Linear block code is easy to implement Will be extended to space-time block code Still some other kinds of code to introduce …
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