Fast Algorithms for Signal Processing and Error Control

Abstract

Algorithms for computation are a central part of digital signal processing and of decoders for error-control codes. When restricted to the study of their computational algorithms, there is not much to distinguish those two subjects. Only the arithmetic field is different; in one case the real or complex field, and a Galois field in the other. Even this distinction is hard to defend; signal processing problems may use Galois fields, and error-control codes in the real or complex field are now under study.

We will survey the state of modern fast algorithms for certain types of computations used in digital processing and error-control including tasks such as linear convolution, cyclic convolution, discrete Fourier transforms, and spectral estimation. In particular we will discuss the decoding of error-control codes as a problem in spectral estimation, and see how this viewpoint leads to the development of efficient decoder algorithms. The Winograd FFT, a fast Berlekamp-Massey algorithm, and a time-domain Reed-Solomon decoder will be described.