Computing linear transforms of symbolic signals

Signals that represent information may be classified into two forms: numeric and symbolic. Symbolic signals are discrete-time sequences that at, any particular index, have a value that is a member of a finite set of symbols. Set membership defines the only mathematical structure that symbolic sequences satisfy. Consequently, symbolic signals cannot be directly processed with existing signal processing algorithms designed for signals having values that are elements of a field (numeric signals) or a group. Generalizing an approach due to Stoffer (see Biometrika, vol.85, p.201-213, 1998), we extend time-frequency and time-scale analysis techniques to symbolic signals and describe a general linear approach to developing processing algorithms for symbolic signals. We illustrate our techniques by considering spectral and wavelet analyses of DNA sequences.

I. Introduction

Much of signal processing has focused on analyzing numeric information. Real and complex-valued data satisfy the mathematical properties of a field. Arithmetic operations between numbers can subsequently be derived from the definition of a field [8]. Specifically, the values of most data form an ordered field. Signal processing techniques all rely on this mathematical structure.