Spectral Logic and Its Applications for the Design of Digital Devices

CHAPTER 2

SPECTRAL TRANSFORMS FOR LOGIC FUNCTIONS

The subject of this chapter is the representation of systems of logic functions by orthogonal series. In the previous chapter, it has been shown that the classical Boolean representation as the sum-of-products, can be viewed as a particular Fourier series-like representation, and the same considerations can be extended to Boolean polynomial representations, that is, Reed-Muller expressions. Extensions of the same principle, achieved by changing the decomposition rules, equivalently, basis functions, lead to Fixed-polarity Reed-Muller expressions, and Kronecker expressions, for example, References 488,489,491, and 555. Coefficients c(w) in these expressions are logic values, thus, c(w) ∈ {0, 1} and, therefore, they are called bit-level expressions. In this chapter, we will discuss word-level expressions, which are defined in terms of basis functions borrowed from abstract harmonic analysis on finite groups, and having real or complex-valued (in the case of multiple-valued functions) coefficients, which are represented by computer words. These expressions will be very similar to the classical Fourier representations.

We describe the basic properties of various discrete functional transforms relating the original systems of functions to the spectral coefficients, introduce spectral and correlation characteristics of systems of logical functions, and demonstrate their use for the analysis and synthesis of digital devices. In short, ...