Digital Signal Processing, 10th Edition

16Error-Correcting Codes

Systems for information processing and transmission constitute a major field of application for signal-processing techniques. Error-detection and correction techniques are also widely used in these systems and, therefore, they coexist and even interact with signal processing in communication equipment.

Generally, coding is presented and taught with a mathematical approach [1]. However, some of the most commonly used coding techniques exploit signal-processing concepts, results, and algorithms [2]. For example, Reed–Solomon coding uses the discrete Fourier transform and linear prediction, convolutional coding is FIR filtering, and turbo codes are related to IIR filtering.

This chapter provides an introduction to some important error-correcting codes from a signal-processing perspective, allowing readers to gain an understanding of these codes and assess their strengths and weaknesses. Moreover, a unified view of communication techniques may result.

16.1 Reed–Solomon Codes

Reed–Solomon codes are extensions of the BCH (Bose–Chaudhury–Hocquenghem) codes to multi-bit symbols [3, 4]. They exploit predictable signals generated by line errors to identify these errors and subtract them from the received signal so as to recover the transmitted signal.

Before the codes are described, some addition information about linear prediction is provided.

16.1.1 Predictable Signals

A signal is said to be predictable if it satisfies a linear recurrence equation. For example, ...