A practical guide to error-control coding using Matlab by Yuan Jing

By Yuan Jing

This functional source offers engineers with a complete figuring out of mistakes regulate coding, an important and extensively utilized sector in smooth electronic communications. The target of blunders regulate coding is to encode info in this kind of approach that no matter if the channel (or garage medium) introduces mistakes, the receiver can right the blunders and recuperate the unique transmitted info. This e-book comprises the main helpful glossy and vintage codes, together with block, Reed Solomon, convolutional, rapid, and LDPC codes. pros locate transparent suggestions on code development, deciphering algorithms, and mistake correcting performances. in addition, this targeted e-book introduces computing device simulations integrally to assist readers grasp key suggestions. together with a significant other DVD with MATLAB courses and supported with over 540 equations, this hands-on reference offers an in-depth therapy of a variety of useful implementation matters. DVD is incorporated! It comprises rigorously designed MATLAB courses that practitioners can observe to their tasks within the box.

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The maximum-likelihood decoding principle was explained in Chapter 1. Next we present a method for error correction. 3 The MATLAB function decode decodes linear block codes. Take the (7,4) code as an example. Suppose that we have received the word r = (1001110) containing one error in its fourth position. 2 Error Correction with Standard Array Standard array decoding is a table look-up (LUT) decoding technique for block codes. For an (n, k) binary block code C, there exist 2n possible ntuples that the decoder may receive, 2k of which are the codewords.

Interestingly, the roots of an mth degree primitive polynomial j(X ) over GF( p) are primitive elements of some extension field GF( pm) [5, p. 208]. We have already stated that an extension Galois field can be completely constructed on its primitive element. Consequently, we can also say that an extension field is built on a primitive polynomial. 1 We use MATLAB to verify that 1 + X + X 3 is a primitive polynomial; that is, it divides X 7 - 1 but not X n - 1 for 0 < n < 7. >> p1 = [1 1 0 1]; % polynomial 1 + x + x^3 >> p2 = [1 0 0 0 0 0 0 1]; % polynomial 1 + x^7 >> p3 = [1 0 0 0 0 0 1]; % polynomial 1 + x^6 >> % for polynomials over GF(2), x^7 – 1 = 1 + x^7, >> % x^6 – 1 = 1 + x^6, ...

The actual transmitted codeword can only be one of them. It is the job of the decoder to determine which one is the true codeword. Based on the maximum-likelihood decoding principle, the codeword that makes r most likely should be regarded as the true codeword. For a BSC channel, it is the codeword closest in the Hamming distance to r. The error pattern e in this case will have the smallest number of 1’s. The maximum-likelihood decoding principle was explained in Chapter 1. Next we present a method for error correction.

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