The Walsh Spectrum and the Real Transform of a Switching Function: A Review with a Karnaugh-Map Perspective

Abstract

Using a Karnaugh-map perspective, this paper investigates the definitions, exposes the properties, introduces new computational procedures, and discovers interrelationships between the Walsh spectrum and the real transform of a switching function. Appropriate Karnaugh maps explain the computation of Walsh spectrum as defined in cryptology. An alternative definition of this spectrum adopted in digital design and related areas is then presented together with procedures for its matrix computation. Then, the real transform of a switching function is defined as a real function of real arguments. This definition is clearly distinguished from similar ones such as the multi-linear form or the arithmetic transform. The real transform is visualized in terms of a particular version of the Karnaugh map called the probability map. Karnaugh maps are also used to demonstrate the computation of the spectral coefficients adopted in digital design as the weight of the switching function and weights of its subfunctions or restrictions. These maps match the earlier ones for the spectrum used in cryptology. Novel relations between the Walsh spectrum and the real transform are utilized in formulating two simplified methods for computing the spectrum via the real transform with some aid offered by Karnaugh maps. Finally, a solution is offered for the inverse problem of computing the real transform in terms of the Walsh spectrum.