Orthogonal Functions for the Logical Design of Switching Circuits

A new approach to the mathematical representation of switching functions is presented. It was developed in connection with a theoretical study of magnetic-core logic, but the results are considered to be more basic and general than the core-logic problem. The ampere-turns (MMF) expression for core switching is shown to be part of a special type of Fourier series expansion of a switching function, in which the turns are directly related to the spectrum of the function. Fouriers transform methods, used for analysis of X-ray diffraction, have been adapted to the representation of switching functions. The method leads not to Boolean algebra, but to ordinary algebra in terms of the orthogonal functions (-1)k
₁
x
₁
+k
₂
x
₂
+...+k
_n
x
_n
, where X
₁
, X
₂
,...,X
_n
= 0,1, and k
₁
,k
₂
,...,k
_n
= 0,1. Methods of application are described for magnetic-core logic and for character recognition.