Properties of Autocorrelation Coefficients for Single-Output Switching Functions

A variety of mathematical transforms have traditionally been used in various logic synthesis applications. This paper investigates the use of the autocorrelation transform: C(τ) = f (v) ∞ f (v ⊕ τ) v=0 2 n −1 ∑ Properties of the coefficient resulting from the application of this transform to switching functions are examined and detailed, including properties to identify symmetries and decompositions. The potential uses in logic synthesis of these properties and other observations based on the autocorrelation coefficients are explored, with emphasis on proofs as mathematical justification of the theorems relating the observed properties of the coefficients to properties of the underlying switching functions. We first present the definition and an explanation of the autocorrelation transform. We then introduce several theorems relating the values of the resulting autocorrelation coefficients to properties of the underlying switching function. A number of potential applications for these theorems are presented, and future directions for this work are discussed.