Outline Introduction System Description Bound Results Closure Codebook and Marker Sequence Design for Synchronization-Correcting Codes Victor Buttigieg 1 Johann A. Briffa 2 1 Department of Communications and Computer Engineering University of Malta Msida MSD2080, Malta 2 Department of Computing University of Surrey Guildford GU2 7XH, England 3 August 2011 Codebook and Marker Sequence Design for Synchronization-Correcting Codes 1 / 25
Outline Introduction System Description Bound Results Closure Overview Introduction and motivation 1 System Description 2 Bound on SEC Code Performance 3 Results 4 Conclusions 5 Codebook and Marker Sequence Design for Synchronization-Correcting Codes 2 / 25
Outline Introduction System Description Bound Results Closure Introduction System Description Concatenated code for insertion/deletion/substitution errors Inner code used to correct synchronization errors Outer code used to correct remaining substitution errors Similar scheme to Davey & MacKay (2001) and Ratzer (2005) Motivation Practical applications are emerging that require synchronization correcting codes Image watermarking Bit-patterned magnetic media Synchronization correcting codes still not very well understood Codebook and Marker Sequence Design for Synchronization-Correcting Codes 3 / 25
Outline Introduction System Description Bound Results Closure Synchronization-Error Channel Binary Substitution-Insertion-Deletion (BSID) Channel P i P d t i t i +1 1 P P 1 P i d s P s For every transmitted bit: We can have zero or more insertions Followed by one of the following Deletion Or a correct transmission Or a substitution Codebook and Marker Sequence Design for Synchronization-Correcting Codes 4 / 25
Outline Introduction System Description Bound Results Closure General System Overview Outer Codec Inner Codec d c f Information q -ary SEC + Source Encoder Encoder m Marker BSID Channel Sequence r Information Soft-input q -ary Inner Sink Decoder Decoder Codebook and Marker Sequence Design for Synchronization-Correcting Codes 5 / 25
Outline Introduction System Description Bound Results Closure Outer Code q -ary Outer Code Objective – to correct residual substitution errors Input and output symbols are in F q Takes K input symbols and emits N symbol blocks Pre-requisite – Soft input decoder available. Codebook and Marker Sequence Design for Synchronization-Correcting Codes 6 / 25
Outline Introduction System Description Bound Results Closure Inner Code Synchronization and Error Correcting (SEC) Code Maps each q -ary outer code symbol to a fixed length binary codeword Main objective – to correct for synchronization errors at bit level Marker Sequence Available at both encoder and decoder Added modulo-2 to the output from the SEC encoder Main objective – to maintain symbol level synchronization. Minimizes symbol insertion/deletion at the output of inner decoder Codebook and Marker Sequence Design for Synchronization-Correcting Codes 7 / 25
Outline Introduction System Description Bound Results Closure Inner Code SEC Codes SEC code has parameters ( n , q , d l min ) Fixed length n bits q codewords Minimum Levenshtein distance d l min (which is maximized) N input symbols result in a frame of Nn bits d l min determines number of correctable substitution, deletion and insertion errors � d l min − 1 � t = 2 Codebook and Marker Sequence Design for Synchronization-Correcting Codes 8 / 25
Outline Introduction System Description Bound Results Closure Inner Code Offset Codes Adding a modification vector to a code maintains its Hamming distance structure This is in general not true in the case of Levenshtein distance Most modification vectors will decrease d l min An Allowed Modification Vector (AMV) is a vector that retains the minimum Levenshtein distance of the code The result is an Offset Code Codebook and Marker Sequence Design for Synchronization-Correcting Codes 9 / 25
Outline Introduction System Description Bound Results Closure Marker Sequence Error effects Any errors in a codeword that are not corrected by the SEC code may results in the codeword decoded incorrectly but decoder remaining synchronized at codeword boundaries the decoder losing synchronization by a few bits at both boundaries the decoder deleting or inserting an entire codeword, maintaining synchronization at subsequent codeword boundaries Marker Sequence role Outer code may be able to correct first two types of errors Third type may result in an uncorrectable long burst of errors Marker sequence is introduced to address this problem Codebook and Marker Sequence Design for Synchronization-Correcting Codes 10 / 25
Outline Introduction System Description Bound Results Closure Marker Sequence Objective of Marker Sequence Required to combat insertion/deletion of complete codewords If decoder inserts/deletes complete codeword, marker sequence would be out of sync This results in a larger number of perceived errors by the decoder Marker sequence improves synchronization at the codeword level Previous Work Gallager (1961) used a pseudo-random sequence together with convolutional codes Davey & MacKay (2001) also used a pseudo-random sequence together with a sparse code Codebook and Marker Sequence Design for Synchronization-Correcting Codes 11 / 25
Outline Introduction System Description Bound Results Closure Marker Sequence Choice of Marker Sequence Pseudo-random sequence would destroy SEC code properties Instead, a sequence of Allowed Modification Vectors are used Simulation results have shown that a sequential use of AMVs leads to adequate performance Performance at high error rates is improved if a larger number of AMVs is used Codebook and Marker Sequence Design for Synchronization-Correcting Codes 12 / 25
Outline Introduction System Description Bound Results Closure Inner Code SEC Code Construction Two different construction techniques were used SEC codes with parameters ( n , M , 3 ) using the Varshamov-Tenegolts construction (Varshamov, 1965; Levenshtein, 1966) SEC codes with larger d l min using Simulated Annealing adopting similar techniques as given by Gamal et al. (1987) for error-correcting codes Construction of good ( n , q , d l min ) SEC codes with a maximum number of AMVs is still an open problem Codebook and Marker Sequence Design for Synchronization-Correcting Codes 13 / 25
Outline Introduction System Description Bound Results Closure Inner Code Inner Decoder Decoder uses the symbol-level algorithm given by [Briffa et al. 2010] APPs for symbol d ∈ F q at block index 0 ≤ ι < N obtained using � L ι ( d ) = α ι ( x 1 ) β ι + 1 ( x 2 ) γ ι ( d , x 1 , x 2 − x 1 ) . x 1 , x 2 Codebook and Marker Sequence Design for Synchronization-Correcting Codes 14 / 25
Outline Introduction System Description Bound Results Closure Inner Code Inner Decoder The forward and backward metrics are respectively defined by r | n ι + x � � α ι ( x ) = Pr , ς n ι = x 0 r | ρ � � β ι ( x ) = Pr n ι + x | ς n ι = x where ς n ι represents the assumed channel drift at the beginning of the codeword at block index ι . The joint probability of the received and transmitted sequence segments corresponding to block index ι is defined as � � r | n ( ι + 1 )+ x + δ x , f | n ( ι + 1 ) γ ι ( d , x , δ x ) = Pr . n ι + x n ι Codebook and Marker Sequence Design for Synchronization-Correcting Codes 15 / 25
Outline Introduction System Description Bound Results Closure Comparison with Previous Work Davey-MacKay and derivatives System similar to the Davey-MacKay (2001) construction DM construction uses a sparse code with minimum Hamming weight codewords instead of SEC code Synchronization is entirely achieved through pseudo-random marker sequence Sparse code ensures that marker sequence is not changed much Principal disadvantages Sparse code is a liability in the synchronization process Difficult to distinguish between sparse codewords Briffa et al. (2010) improved the inner decoder by taking into consideration the sparse codebook used (symbol-level decoder) This symbol-level decoder is used in the current work Codebook and Marker Sequence Design for Synchronization-Correcting Codes 16 / 25
Outline Introduction System Description Bound Results Closure Bound on SEC Code Performance Assumption Decoder is synchronized on the codeword boundaries Symbol-error probability bounded by considering the effect of errors on individual codewords Probability of having n i , n d and n s Errors The probability of having an error pattern with, respectively, n i , n d and n s insertion, deletion and substitution errors is · ( P t P s ) n s · ( P t ( 1 − P s )) n − n d − n s Pr { n i , n d , n s } = P n i · P n d i d where P t = 1 − P i − P d is the probability of transmitting a bit Codebook and Marker Sequence Design for Synchronization-Correcting Codes 17 / 25
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