Modern Digital Radio Communication Signals and Systems by Sung-Moon Michael Yang

Author:Sung-Moon Michael Yang
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(6.32)

where pS is RS code symbol (m bits) error rate. If BPSK is used, pS may be obtained from bit error rate p, assuming random errors, p S = 1 − (1 − p) m . When other modulations are used, its expression may be found accordingly.

6.2.6 Algebraic Decoding of BCH

In Fig. 6.13, we showed the decoding of any cyclic codes. The complexity of decoding is reduced substantially, as shown in the exercises. However, by utilizing the cyclic nature of code, algebraic decoding is possible, and thus it extends the usefulness of such codes. The algebraic decoding of BCH and RS was intensely studied during the 1960s to the 1970s. Even though there are a variety of techniques available, here we outline the decoding method by showing a double error (t = 2)-correcting BCH as an example. The same method is applicable to RS codes. Extending a single error-correcting code to multiple error-correcting codes is a nontrivial jump, considerable sophistication of code construction using finite field arithmetic, as shown briefly in the previous sections, and furthermore in decoding as well. However, this example contains all the necessary steps for algebraic decoding.

Example 6.12

We use (15, 7) double error-correcting BCH as an example to describe the algebraic decoding process. We need to go back to the construction of generator polynomial, g(X). The elements in GF (24) with p(X) = 1 + X + X4 and its corresponding minimal polynomial are listed in the table below.

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