Family of complex Hadamard transforms: relationship with other transforms and complex composite spectra

Abstract

Relationship of the recently introduced family of unified complex Hadamard transforms with other transforms used in binary and multiple-valued logic design are investigated in this paper. All the complex Hadamard matrices can be generated by a common unifying formula presented here. Only half of all possible 64 unified complex Hadamard transforms have half-spectrum property. The existence of such a property for 32 transformation matrices is proven for the first time in this paper. Half-spectrum property is important as h reduces the required computer storage by half when compared to other transforms operating on complex numbers. The method is also presented to evaluate complex Hadamard spectra of AND, OR, and XOR of Boolean functions directly from the spectra of the separate Boolean functions. The results are given using a general coding scheme, and different possible codings of Boolean functions are discussed. Moreover, new definition of the convolution operation called complex convolution is derived. Different properties of such a convolution are presented. Theorem giving final formulae for composite complex Hadamard spectra of Boolean functions is stated in terms of complex convolution. By using presented methods, many of the Boolean operations in original domain can be represented much simpler in terms of their composite complex spectra.