Haar wavelet transforms and haar spectral transform decision diagrams for switching and multiple-valued functions

Abstract

In spectral interpretation, decision diagrams (DDs) are defined in terms of some spectral transforms. For a given DD, the related transform is determined by an analysis of expansion rules used in the nodes and the related labels of edges. The converse task, design of a DD in terms of a given spectral transform is relatively easy to solve for Kronecker product representable spectral transforms. However, in other cases, it appears a problem of decomposition of basic functions in spectral transform to determine the corresponding expansion rules and labels at the edges. We point out that this problem relates to the assignment of nodes in Pseudo-Kronecker DDs (PKDDs). Then, we show that in the case of Haar transform, the expansion rules and assignment of nodes in Haar Spectral transform DDs (HSTDDs) can be determined from a study and comparison of FFT and DD methods for calculation of the Haar spectrum. This consideration permits to generalize the definition of HSTDDs to multiple-valued (MV) functions considered as functions in finite fields or as subsets of complex-valued functions. Conversely, from such defined HSTDDs, we derive various Haar transforms for MV functions related to Fourier series and polynomial expressions for MV functions.