Minimization of Circuit Design Using Permutation of Haar Wavelet Series

There is an apparent renewed interest in application of the Haar wavelet transform in switching theory and logic design.1–6 In particular, the discrete Haar transform appears very interesting for its wavelets-like properties and fast calculation algorithms. The applications have been found in circuit synthesis, verification and testing.1–9 For applications of the Haar transform in logic design, efficient ways of calculating the Haar spectrum from reduced forms of Boolean functions are needed. Recently, such methods were introduced for calculation of the Haar spectrum from disjoint cubes, and different types of decision diagrams.10,12–16 In this paper, we extend our results from Ref. 7 and discuss the efficiency of circuit synthesis through the minimized Haar series in the number of coefficients count. We point out that the minimization of Haar series by permutation of switching variables may result in further savings in circuit synthesis. The presentation in this paper is organized as follows. In Sec. 2, we briefly review basic definitions of Haar functions and Haar wavelet series. Section 3 discusses applications of Haar wavelet series in VLSI design. Efficient ways of calculation of