Coding of Real-Number Sequences for Error Correction: A Digital Signal Processing Problem

Abstract

Error-correcting codes defined over the real-number and complex-number fields are introduced. The possibility of utilizing realnumber arithmetic permits the codes to be implemented with operations normally available in standard programmable digital signal processors by methods which are discussed. Hadamard and discrete Fourier transform codes are presented for block coding, and the latter are seen to be cyclic and to include the class of BCH codes. It is shown that maximum distance separable real-number BCH ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N, K</tex> ) codes exist for all nontrivial values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> . A large class of block and convolutional real-number single-error-correcting codes, derived from similar codes over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF(p)</tex> , are presented. Both block and convolutional codes are seen to be describable by the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transform in a manner which emphasizes their similarities to conventional digital signal processing structures such as digital filters and digital filter banks. Methods for correcting weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t + 1</tex> errors in a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> error-correcting code are demonstrated and interpreted; in particular, the use of a VLSI digital signal processor for implementation of an algorithm for correcting almost all double adjacent error patterns in a single-error-correcting convolutional code is discussed.