Martin Tomlinson - Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

The Author(s) 2017

Martin Tomlinson 1 , Cen Jung Tjhai 2, Marcel A. Ambroze 1, Mohammed Ahmed 1 and Mubarak Jibril 3

The sphere packing bound by Shannon [] presented his coding theorem for the average FER for the ensemble of all random binary ( n , k , d ) codes. There are possible binary combinations for each codeword which in terms of the n -dimensional signal space hypercube corresponds to one vertex taken from possible vertices. There are codewords, and therefore different possible random codes. The receiver is considered to be composed of matched filters, one for each codeword and a decoder error occurs if any of the matched filter receivers has a larger output than the matched filter receiver corresponding to the transmitted codeword. Consider this matched filter receiver and another different matched filter receiver, and assume that the two codewords differ in d bit positions. The Hamming distance between the two codewords is d . The energy per transmitted bit is , where is the energy per information bit. The noise variance per matched filtered received bit, , where is the single sided noise spectral density. In the absence of noise, the output of the matched filter receiver for the transmitted codeword is and the output of the other codeword matched filter receiver is . The noise voltage at the output of the matched filter receiver for the transmitted codeword is denoted as , and the noise voltage at the output of the other matched filter receiver will be . The common noise voltage arises from correlation of the bits common to both codewords with the received noise and the noise voltages and arise, respectively, from correlation of the other d bits with the received noise. A decoder error occurs if that is, a decoder error occurs when .

The average noise power associated with is and as the noise is Gaussian distributed, the probability of decoder error, , is given by This may be expressed in terms of the complementary error function (erfc) and Each of the other codewords may also cause a decoder error but the weight distribution of the code is usually unknown. However by averaging over all possible random codes, knowledge of the weight distribution of a particular code is not required. The probability of two codewords of a randomly chosen code , differing in d bit positions, is given by the binomial distribution