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USING THE MATLAB COMMUNICATIONS TOOLBOX TO LOOK AT CYCLIC CODING

USING THE MATLAB COMMUNICATIONS TOOLBOX TO LOOK AT CYCLIC CODING. Wm. Hugh Blanton East Tennessee State University http://faculty.etsu.edu/blanton. The Theory. General Concepts. Channel Coding.

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USING THE MATLAB COMMUNICATIONS TOOLBOX TO LOOK AT CYCLIC CODING

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  1. USING THE MATLAB COMMUNICATIONS TOOLBOXTO LOOK AT CYCLIC CODING Wm. Hugh Blanton East Tennessee State University http://faculty.etsu.edu/blanton

  2. The Theory General Concepts

  3. Channel Coding • Designed to improve communications performance by enabling the transmitted signals to better withstand the effects of various channel impairments, such as noise, interference, and fading.

  4. 110 010 011 111 000 100 001 101 Hamming Distance The number of bit positions in which two binary words differ. This figure has a minimum distance between code words of 1.

  5. Hamming Distance of 2 A code word that differs by a Hamming distance of 2 will detect an erroneous code word. 0011 0001 0111 r = 1 0000 0110

  6. 1 1 1 If the minimum distance is 3, the code words are separated by at least three edges. Note that if a single error occurs, the erroneous code word will be closer to one of the correct code words. The error is corrected by assigning the received code word to the nearest (in Hamming distance) valid code word.

  7. The important concepts associated with Hamming distance are: • When the Hamming distance is 1, it is impossible to detect an error, much less correct the error. • When the Hamming distance is 2, the error can be detected, but not corrected. • When the Hamming distance is 3, a single error can be detected and corrected.

  8. More Theory Mathematical Concepts

  9. LINEAR BLOCK CODES • Definition • A systematic (n,k) linear block code is a mapping from a k-dimensional message vector to an n-dimensional codeword in such a way that part of the sequence generated coincides with the k message digits.

  10. n-dimensional information vector, where n = 7. The linear block code that everyone uses is the (7,4) code. k-dimensional information vector, where k = 4.

  11. Generator Matrix • How do you generate a (7,4) code? • Of course with a Generator Matrix! Note that m = n - k, the difference

  12. A (7,4) code is generated by the matrix

  13. The (7,4) block code is generated using the matrix equation • where • u is a [1 × k] vector representing the information word • v is a [1 × n] vector representing the code word • [G] is a [k × n] generating matrix

  14. For the information word (0 0 0 1), the code word is computed as follows: Transformation from 4-bit information word to 7-bit code word Note that the additions are modulo-2 additions

  15. The Hamming distance for the (7,4) code is 3. • Thus, this code can detect double bit errors and correct single bit errors

  16. Parity Check Matrix • Another important matrix associated with block codes is the parity check matrix, [H].

  17. The parity check matrix is formed by starting with the identity matrix and appending the transpose of the nonidentity portion of [G]:

  18. For the (7,4) block code,

  19. The Syndrome • The parity check matrix has the property v[H] T = 0 • That is, any errorless, received code word multiplied by the transpose of the parity check matrix, [H], yields a zero vector, or syndrome. • If the received code word contains an error, the resulting vector will match the corresponding bit that caused the error.

  20. Consider one of the valid code words of the (7,4) block code, v = 1100101 and the transpose of the (7,4) parity check matrix.

  21. Modulo-2 Addition

  22. Syndrome The null vector indicates no error.

  23. Suppose the valid code words, v = 1100101 gets corrupted to 1000101.

  24. Modulo-2 Addition

  25. The second row corresponds to the second bit from left. Syndrome Match syndrome with corresponding code in parity check matrix

  26. Each row in the transposed parity check matrix corresponds to a bit in the code word. • By matching the non-zero syndrome with the contents contained in the rows of the transposed parity matrix, the corresponding corrupted bit can be detected and and subsequently corrected.

  27. The process will convert the erroneous code 1000101 to the correct code 1100101.

  28. Cyclic Codes

  29. CYCLIC CODES • The implementation of linear block codes requires circuits capable of performing matrix multiplication and of comparing the result of various binary numbers. • Although integrated circuits have been developed to implement the most common codes, the circuitry can become quite complex for very long blocks of code.

  30. A special case of the block code, the cyclic code can be electronically implemented relatively easily.

  31. A (7,4) cyclic code generating matrix

  32. A (7,4) cyclic code generating matrix Non-systematic—No Identity matrix.

  33. A nonsystematic (7,4) cyclic code generating matrix is made systematic by performing row, column operations until you obtain the Ik partition.

  34. MATLAB COMMUNICATIONS TOOLBOX DERIVATION

  35. MATLAB COMMUNICATIONS TOOLBOX DERIVATION • Although row,column matrix manipulation is a straightforward process for students that have taken a linear algebra course, many engineering technology students are not familiar with the process.

  36. Although, the discussion of linear block codes is not mathematically complicated, it is mathematically tortuous with many mathematical turns.

  37. Finally, the discussion for the (7,4) cyclic code is relatively benign with a manageable number of calculations.

  38. For larger codes, the calculations become overwhelming, especially for the engineering technology student.

  39. In order to • better manage larger codes, • relieve the mathematical stress on engineering technology students, and • focus on the application of block codes rather than the mathematics; • the MATLAB Communications Toolbox provides a set of cyclic code functions.

  40. Consider the problem with a (15,7) cyclic code with the following generator polynomial. that corresponds to the code word 1 x1 x2 x3 x4 x5 x6 x7 x8

  41. The nonsystematic cyclic code generating matrix, G, is:

  42. And the systematic cyclic code generating matrix is

  43. By using the function cyclpoly, MATLAB yields a set of all generating polynomials >> p=cyclpoly(15,7,'all') p = 1 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1

  44. By using the function cyclgen, the nonsystematic generating matrix, genmat, is derived as follows: >> genpoly=[1 1 0 1 1 1 0 1 1]; >>[parmat,genmat]=cyclgen(15,genpoly,'nonsys')

  45. genmat = Columns 1 through 13 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 Columns 14 through 15 0 0 0 0 0 0 0 0 0 0 1 0

  46. which is in matrix form.

  47. Likewise, the systematic generating matrix, genmatsys, can be computed using cyclgen without the conditional parameter (nonsys): >> genpoly=[1 1 0 1 1 1 0 1 1]; >>[parmat,genmat]=cyclgen(15,genpoly)

  48. genmatsys = Columns 1 through 12 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 Columns 13 through 15 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

  49. which is in matrix form.

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