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An Algorithm for Construction of Error-Correcting Symmetrical Reversible Variable Length Codes

An Algorithm for Construction of Error-Correcting Symmetrical Reversible Variable Length Codes. Chia-Wei Lin, Ja-Ling Wu, Jun-Cheng Chen Presented by Jun-Cheng Chen 2004/09/24. Outline. Introduction Notations and preliminaries The proposed algorithm Experimental results Conclusion.

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An Algorithm for Construction of Error-Correcting Symmetrical Reversible Variable Length Codes

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  1. An Algorithm for Construction of Error-Correcting Symmetrical Reversible Variable Length Codes Chia-Wei Lin, Ja-Ling Wu, Jun-Cheng Chen Presented by Jun-Cheng Chen 2004/09/24

  2. Outline • Introduction • Notations and preliminaries • The proposed algorithm • Experimental results • Conclusion

  3. Introduction to RVLC (1/3) • Variable length codes (VLC) are of prime importance in the efficient transmission of digital signals. • High compression efficiency • There are other criteria that may be important in the application environment . • Channel bit-error resilience , maximum codeword length limitation • Reversibility of variable length codes makes instantaneous decoding possible both in the forward and backward directions .

  4. Symbol Probability C1 C2 A B C D E 0.33 0.30 0.18 0.10 0.09 00 01 11 100 101 00 11 010 101 0110 Average code length 2.19 2.46 Introduction to RVLC (2/3) • A reversible variable length code (RVLC) must satisfy the prefix-free and suffix-free condition for instantaneous forward and backward decoding . Table 0: Huffman code and reversible variable length codes(RVLCs) C1: Huffman code C2: Symmetrical RVLC

  5. Introduction to RVLC (3/3) • If a bit error occurs, VLC will propagate the bit error and the data after the bit error becomes useless. However, RVLC can recover data after the bit error. VLC RVLC Bit error

  6. Notations and preliminaries (1/3) • n source symbols: • probabilities of source symbols: • n codewords: • length of codeword : • :codeword sequences • : hamming distance

  7. Notations and preliminaries (2/3) • (minimum block distance) • In the case of RVLCs Note: if a VLC does not have equal-length codewords, then its minimum block distance is undefined. , if is undefined , if is defined

  8. codewords 00 11 010 101 0110 1001 Block distance:2 Block distance:3 (minimum block distance): 2 Block distance:4 Find minimum blcok distance Find minimum hamming distance for each level Minimum Block Distance

  9. Notations and preliminaries (3/3) • R(c, CLd, d) is the replacement of a codeword c in the codeword list CLd • children(c) are defined by all of the first symmetrical codewords on paths from the codeword c to leaf codewords. • CLdis a codeword list in which the constraint dbd holds for all its codewords. • children(c, d) is a subset of the symmetrical children of a symmetrical codeword c in which the constraint dbd holds for all the children in the subset. • children(c, CLd, d) is a subset of symmetrical children of c that the union of the subset and CLd is still a codeword list in which the constraint dbd holds for all its codewords.

  10. T-List: (1,00,010, 0110) 00 000 0000 codeword ’00’ children(‘00’,T-List, 2) = {} R(‘00’,T-List,2) ={0000} children(‘00’) = {000} 0 1 children(‘00’, 2) = {000}

  11. The Proposed Algorithm (1/4) • Goal: The free distance of the proposed RVLC is always greater than one, which can result in certain improvement in symbol error rate relative to VLCs with free distance one.

  12. The Proposed Algorithm (2/4) • The algorithm • Step1: Assign the initial codeword list (“1”, “00”, “010”, “0110”,…) to the target list, T-List. • Step2: If any codeword c in T-List satisfies the condition of replacement, replace the codeword c with R(c, T-List, 2) that results in the smallest average codeword length. If the number of codewords in T-List is more than n, the number of source symbols, keep the first n codewords in T-List and discard the others. (Notice that the block distance is still greater than one after the codeword replacement) • Step3: Repeat Step2 until there is no codeword in T-List satisfying the condition of replacement.

  13. U PU(U) After step 1 After step 2 a1 0.286 1 00 a2 0.214 00 11 a3 0.143 010 010 a4 0.143 0110 101 a5 0.143 01110 0110 a6 0.071 011110 1001 Average codeword length 2.856 2.714 The Proposed Algorithm (3/4) • Six symbols with probabilities (0.286, 0.214, 0.143, 0.143, 0.071) are given Table 1: An example of the proposed algorithm, where PU(U) denotes the probability of the source symbol U.

  14. Iteration #1 c R(c,T-List, 2) Merge result Average code length after replacement 1 11, 101, 1001, 10001, 100001 00,11, 010,101, 0110,1001, 01110,10001, 011110,100001 2.714 00 0000, 00100, 001100 1, 010 0000, 0110 00100,01110 001100,011110 3.142 The Proposed Algorithm (4/4) • T-list: (1, 00, 010, 0110, 01110, 011110) • Table 2: Some temporary results of the proposed algorithm. Others are 3.285, 2.856, 2.856 and 2.856

  15. Experimental Results (1/3) • Test corpuses are from Canterbury Corpus File Set. (available in http://corpus.canterbury.ac.nz/) • Table 3 respectively lists various symmetrical RVLCs constructed by other algorithms, and the proposed algorithm for the English alphabet set. • Table 4 lists the results of the Canterbury Corpus file set compressed by using other algorithms and the proposed algorithm

  16. Experiment Results (2/3) Table 3: Symmetrical RVLCs for the English alphabet set

  17. Experiment Results (3/3) Table 4: the average codeword lengths of various symmetrical RVLCs for the Canterbury Corpus file set

  18. Conclusion (1/2) • The optimal design of RVLC with good distance properties and compression efficiency is still an open problem. • The proposed algorithm is suitable for any VLC-involved applications to enhance the error-correcting capability of critical time-constrained applications. • It should be pointed out that symmetrical RVLCs with cannot be surely obtained by the proposed algorithm.

  19. Conclusion (2/2) • The major contribution of the proposed algorithm is that it yields more efficient symmetrical RVLCs with the same free distance than other known algorithms.

  20. Thank You

  21. Level 4 No. of Available Candidate Codewords at level 5 No. of Available Candidate Codewords at level 6 No. of Available Candidate Codewords at level 7 0000 0 00 000 Candidate codewords MSSL 0110 1 8 6 15 MSSL:3 1001 1 8 6 15 MSSL:1 not symmetrical 0000 3 7 6 14 1111 3 7 6 14 MSSL (Maximum Symmetrical Suffix Length) 0110 0 10 110 .

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