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Error Control Decoding for Symmetric Erasure Channels

2. Introduction to Digital Communication. Technology of exchanging information from the source to the destination by the use of finite set of signals.Characteristics of the physical channels through which information is transmitted are of prime importance in the analysis and design of communication

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Error Control Decoding for Symmetric Erasure Channels

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    1. 1 Error Control Decoding for Symmetric Erasure Channels

    2. 2 Introduction to Digital Communication Technology of exchanging information from the source to the destination by the use of finite set of signals. Characteristics of the physical channels through which information is transmitted are of prime importance in the analysis and design of communication systems.

    3. 3 Reasons for increasing popularity of digital communications Digital messages are more easily encrypted and hence unreadable by unintended user and allows electronic authentification of sender. Storage and retrieval with measurable accuracy. Different forms of information such as video, audio, image , numeric are easily accommodated in digital transmission. Digital messages can be reconstituted at each stage of a transmission system with several stages. Supports electronic addressing and routing of messages in multi-user systems.

    4. 4 Need for Error Control and Correction Transmission medium introduces noise and interference into the transmitted signal. Hence, the transmitted data is incorrectly received at the receiver end. Error control coding emphasizes on methods of delivering information from the transmitter end to the receiver end with a minimum of errors.

    5. 5 Digital Communication System

    6. 6 Definitions and Notations in error correcting codes Codes constituting of only two symbols 1s and 0s : Binary Codes. Codes constituting greater than 2 symbols: Non-binary codes. Length of the codeword is denoted by n and information bits are denoted by k. k information bits are mapped into codeword of length n to construct (n,k) code.

    7. 7 Definitions and Notations in error correcting codes Hamming Distance If u and v are two codewords, then Error correction capability of a code

    8. 8 Binary Symmetric Channel

    9. 9 Binary symmetric erasure channel model

    10. 10 Constructing the probability of error expression t is the number of errors. u is the number of erasures. Decoder error likely if Probability of error expression is written as,

    11. 11 Erasure decoding for Hamming (7,4) code Steps involved : Developing the probability of error expression. Analyzing performance of erasure decoding as measured by probability of error criterion. Optimization of erasure parameter d

    12. 12 In BPSK, demodulators output for each bit in signal space is, , Erasure region is parameterized by Probability of bit error Probability of erasure

    13. 13 Probability of error Versus Erasure parameter d for A=3

    14. 14 Probability of error Versus Erasure parameter d for different A

    15. 15 d/A Versus SNR

    16. 16 Probability of error using optimum d Versus SNR

    17. 17 Erasure decoding for BCH (15,7) code

    18. 18 Probability of error Versus Erasure parameter d for different A

    19. 19 d/A Versus SNR

    20. 20 Probability of error using optimum d Versus SNR

    21. 21 Erasure decoding for BCH (15,11) code

    22. 22 d/A Versus SNR

    23. 23 Probability of error using optimum d Versus SNR

    24. 24 Erasure decoding for BCH (15,5) code

    25. 25 d/A Versus SNR

    26. 26 Probability of error using optimum d Versus SNR

    27. 27 Erasure decoding for Golay (23,12) code by simulation

    28. 28 Results of simulation for Golay (23,12) code

    29. 29 Erasure decoding for a non-binary code: Ternary Symmetric Erasure channel model

    30. 30 Erasure decoding for ternary (11,6) Golay code

    31. 31 Computation of error and erasure probabilities Probability density function assuming A is transmitted, As a function of ?,

    32. 32 Probability of error Versus erasure parameter d in degrees for A=4

    33. 33 Geometry 2 : Parameterization of erasure region by distance

    34. 34 Computation of error and erasure probabilities

    35. 35 Probability of error Versus d for A=4

    36. 36 Optimum d from p=q^2 approximation

    37. 37 Probability of error Versus d for different values of A

    38. 38 Probability of error Versus SNR

    39. 39 Results For Codes with larger error correcting capability, Probability of error can be improved significantly by erasure decoding with low complexity of the decoder. p=q^2 provides a good approximation to the optimum d. For larger codes, the probability of error for erasure decoding can be studied by simulation. Different geometries of erasure region yield different probability of error performance.

    40. 40 References

    41. 41 Thanks to Dr. Swaszek for his guidance and support.

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