Generalized Haar Spectral Representations and Their Applications

Haar transform is known to have the smallest computational requirement and has been used mainly for pattern recognition and image processing. Although the properties of Haar spectra of Boolean functions have considerable interest and attraction, the majority of publications to date have employed the Walsh rather than Haar transform in their considerations. It is mainly due to the fact that up to recently there was no efficient method of calculating Haar spectra directly from reduced representations of Boolean functions such as decision diagrams and cubes. Recently, efficient methods based on Decision Diagrams and cubical representation for the computation of Haar spectra have been developed. Two methods based on decision diagrams and a new data structure called the “Haar Spectral Diagram” is discussed. The method to calculate Haar spectra from disjoint cubes of Boolean functions is also presented. A concept of paired Haar transform for representation and efficient optimization of systems of incompletely specified Boolean functions will be discussed. Finally another form of Haar transform, so called “Sign Haar Transform” is discussed and basic properties of Boolean functions in its spectral domain are shown. Various applications of Haar transform in logic design are also mentioned.