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Coding for the Correction of Synchronization Errors. ASJ Helberg CUHK Oct 2010. Content. Background Synchronization errors and their effects Previous approaches Resynchronization Concatenation Error correction Algebraic insertion/deletion correction Single error correcting

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Presentation Transcript
content
Content
  • Background
    • Synchronization errors and their effects
  • Previous approaches
    • Resynchronization
    • Concatenation
    • Error correction
  • Algebraic insertion/deletion correction
    • Single error correcting
    • Multiple error correcting
  • Problems and applications
synchronization errors
Synchronization errors
  • Due to timing or other noise and inaccuracies,
  • Manifests as the insertion or deletion of symbols
  • Examples:
    • PPM (Pulse position modulation) in optic fibres,
    • Terabit per square inch magnetic recording
    • Optic disc recording due to ISI and ITI
    • Multipath effects in radio
types of synchronization errors
Types of Synchronization errors
  • Insertion or deletion, excluding additive errors
    • Additive errors are special case of deletion and insertion in same position of bits of opposite value
    • Repetition/duplication error: copies bit
    • Bit/peak shift: 01 becomes 10
    • Bit/peak shift of size a: 0a1 becomes 10a
effects
Effects
  • A single synchronization error causes a catastrophic burst of additive errors

Tx: 0001010011011110

Rx: 0001100001110

  • Boundaries of data blocks are unknown to receiver

e.g. 001100 becomes 0000

synchronizable codes
Synchronizable codes
  • Comma-free codes
  • Prefix synchronised codes
  • Bounded synchronisation delay
  • Synchronisation with timing
  • Marker codes
  • Corrupted blocks are discarded!!
sync error correcting codes
Sync error correcting codes
  • Binary Algebraic block codes
  • Nonbinary and perfect codes
  • Bursts of sync errors
  • Weak synchronization errors
  • Convolutional codes
  • Expurgated codes (Reed Muller/ LDPC)
binary algebraic block codes
Binary Algebraic block codes
  • VarshamovTenengoltz construction:
  • One asymmetric error
levenshtein codes
Levenshtein codes

With 2n > m >= n + 1, s=1 correcting code

With m >= 2n, s=1 and t=1 correcting code

Partition a=0 was proven to have the maximum cardinality

Largest common subword obtained from two valid codewords is

hamming distance properties of levenshtein codes
Hamming distance properties of Levenshtein codes
  • Proposition 1 : A Levenshtein code C has only one code word of either weight w = 0 or weight w  = 1.
  • Proposition 2 : In a Levenshtein code there is a minimum Hamming distance, dmin 2 between any two code words.
  • Proposition 3 : Code words in a Levenshtein code have a dmin 4 if they have the same weight.
  • Proposition 4 : Levenshtein code words that differ in one unit of weight have dmin 3.
weight distance diagram
Weight distance diagram

0

dmin = 2

2

dmin = 4

dmin = 3

3

dmin = 4

n-3

dmin = 4

dmin = 3

n-2

dmin = 4

dmin = 2

n

generalised structure
Generalised structure

Proposition 5

  • Code words of weights w = 0, 1, 2, ..., s do not occur together in an s ‑ correcting code.

Proposition 6

  • The minimum Hamming distance of an s ‑ insertion/deletion correcting code is dmins + 1.
  • Again, the proof of propositions 5 and 6, is straight forward when considering the resulting subwords after s deletions.
slide14

Proposition 7

  • Any two number theoretic s ‑ insertion/deletion correcting code words which differ in weight by i, 0 is, have a Hamming distance of d 2(s + 1) ‑ i.

dmin = (w2‑ x) + (w1‑ x)

= w2+ w1‑ 2x

= w2+ w2‑ w ‑ 2x

= 2(w2‑ x) ‑ w

From Proposition 6, dmins + 1 corresponding to number of “1’s” by which w2 differ from w1 i.e. (w2‑ x) thus d 2(s + 1) ‑ i

weight distance diagram1
Weight-distance diagram

0

dmin = s+1

s+1

dmin = 2(s+1)

dmin = 2s+1

s+2

dmin = 2(s+1)

dmin = 2(s+1) - i

n-s+2

dmin = 2(s+1)

dmin = 2s+1

n-s+1

dmin = 2(s+1)

dmin = s+1

n

bounds for the general algebraic construction
Bounds for the general algebraic construction
  • Lower bound on s correction capability
upper bound on cardinality
Upper bound on Cardinality
  • Hamming type upper bound
modified fibonacci
Modified Fibonacci
  • S=1:
  • 1, 2, 3, 4, 5, 6, 7, …
  • S=2
  • 1, 2, 4, 7, 12, 20, 33, …
  • S=3:
  • 1, 2, 4, 8, 14, 23, 38, …
  • S=4:
  • 1, 2, 4, 8, 16, 31, 60, …
  • Partitioning 2n into , thus in limit, cardinality bounded by
  • 2n / m with

(non-empty partitions)

problems
Problems
  • Very low cardinalities
  • Does not scale well
  • No decoding algorithm
  • Codeword boundaries assumed
  • Validity not proven in general
lower bounds on the capacity of the binary deletion channel
Lower bounds on the capacity of the binary deletion channel

A Kirsch and E Drinea, “Directly lower bounding the information capacity for channels with i.i.d. deletions and duplications, IEEE Transactions on Information Theory, vol. 56, no. 1, January 2010, pp 86-102

connection with network coding
Connection with network coding?
  • Synchronization in NC environments is assumed
  • Especially on physical layer NC
  • “Pruned/punctured” codes may be useful ?
  • Superimposed codes that are also sync error correcting?