Analogue BCH Codes and Direct Reduced Echelon Parity Check Matrix Construction

Abstract

Many information sources are naturally analogue and must be digitised if they are to be transmitted digitally. The process of digitisation introduces quantisation errors and increases the bandwidth required. The use of analogue error-correcting codes eliminates the need for digitisation. It is shown that analogue BCH codes may be constructed in the same way as finite-field BCH codes, including Reed–Solomon codes, except that the field of complex numbers is used. It is shown how the Mattson–Solomon polynomial or equivalently the Discrete Fourier transform may be used as the basis for the construction of analogue BCH codes. It is also shown that a permuted parity check matrix produces an equivalent code, using a primitive root of unity to construct the code, as in discrete BCH codes. A new algorithm is presented which uses symbolwise multiplication of rows of a parity check matrix to produce directly the parity check matrix in reduced echelon form. The algorithm may be used for constructing reduced echelon parity check matrices for standard BCH and RS codes as well as analogue BCH codes. Gaussian elimination or other means of solving parallel, simultaneous equations are completely avoided by the method. It is also proven that analogue BCH codes are Maximum Distance Separable (MDS) codes. Examples are presented of using the analogue BCH codes in providing error-correction for analogue, band-limited data including the correction of impulse noise errors in BCH encoded, analogue stereo music waveforms. Since the data is bandlimited it is already redundant and the parity check symbols replace existing values so that there is no need for bandwidth expansion as in traditional error-correcting codes. Future research areas are outlined including the possibility of an analogue, maximum likelihood, error-correcting decoder based on the extended Dorsch decoder of Chap. 15. Steganography is another future application area for analogue BCH codes.