View File - University of Engineering and Technology, Taxila

When errors occur in bursts, the number of parity check bits needed tocorrect burst errors of length q has a lower bound given byn k 2q ð6:28ÞThe code rate is R c ¼ k n ¼ m0 k2 m0 k 1 : ð6:29ÞExample 6.4: Consider a (7, 4) BCH code. In this instance, n ¼ 7; k ¼ 4.Using the above properties,1. e 0 k ¼ 1:2. m 0 k ¼ 3 ðfrom ð6:25ÞÞ:3. d H;min ¼ 3 ðfrom ð6:26ÞÞ:These empirical results show that the Hamming code is a single-errorcorrecting BCH code.An important class of nonbinary BCH codes is the Reed–Solomoncodes in which the symbols are blocks of bits [13]. Their importance is theexistence of practical decoding techniques, as well as their ability to correctbursts of errors.One final note on error correction techniques. Where a series ofcodewords have good short-burst error correction properties, such as terrestrialmicrowave links with short but deep fades, interleaving of bits provides analternative technique for correcting the errors at the receiver. As an illustration,if we allow the individual bits of a given codeword to be spaced d c bits apart, aburst of length d c would corrupt only one bit in each of d c codewords. Thissingle error can easily be rectified at the receiver, making the transmissionimpervious to longer error bursts.6.2.2. Convolutional CodesA convolutional coder is a finite memory system. The name ‘‘convolutional’’refers to the fact that the added redundant bits are generated by mod-2convolutions. A generalized convolutional encoder is shown in Fig. 6.2. Itconsists of an L-stage shift register, n mod-2 adders, a commutator, andnetwork of feedback connections between the shift register and the adders. Thenumber of bits in the input data stream is k. The number of output bits for eachk-bit sequence is n bits. Since n bits are produced at the output for each input kbits, the code rate is still R c ¼ k=n, the same as (6.1).Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.