Complex composite spectra of Unified Complex Hadamard transform for logic functions

A method to evaluate the Unified Complex Hadamard spectra of AND, OR, and XOR for Boolean functions, directly from the spectra of the functions, is presented. The results are given using a general coding scheme, and different possible codings of Boolean functions are also discussed. A new definition of the convolution operation called complex convolution is derived. Different properties of such a convolution are presented. A theorem giving final formulas for the composite Unified Complex Hadamard spectra of Boolean functions is stated in terms of the complex convolution. Efficient representations of the spectra in the form of decision diagrams are presented. An application of Unified Complex Hadamard Transform in image watermarking is also discussed.