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A theory has been developed to calculate the Rademacher-Walsh transform from a cube a r r a y specification of incompletely specified Boolean functions. T h e importance of representing Boolean functions a s a r r a y s of disjoint ON-a n d Ix-cubes has been pointed out, a n d a n efficient new algorithm to generate disjoint cubes f r o m nondisjoint ones has been designed. The t r a n s f o r m algorithm makes use of the properties of a n a r r a y of disjoint cubes a n d allows the determination of the spectral coefficients in a n independent way. The programs for both algorithms use advantages of C language to speed u p the execution. The comparison of different versions of the algorithm has been carried out. T h e presented algorithm a n d its implementation is the fastest a n d most comprehensive program (having many options) known t o us for the calculation of R ademacher-Walsh transform. It successfully overcomes all drawbacks in the calculation of the t r a n s f o r m f r o m the design automation system based on spectral methods-the SPECSYS system from Drexel University that uses Fast Walsh T r a n sform.

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Novel discrete orthogonal transforms are introduced in this paper, namely the unified complex Hadamard transforms. These transforms have elements confined to four elementary complex integer numbers which are generated based on the Walsh-Hadamard transform, using a single unifying mathematical formula. The generation of higher dimension transformation matrices are discussed in detail.Index Terms-Digital signal processing, discrete transforms, fast algorithms, orthogonal transforms, unified complex Hadamard transforms.

New classes of recursive transforms over GF (3) have been introduced here. They are based on simple recursive equations what allows to obtain corresponding fast forward and inverse transforms and very regular butterfly diagrams. The classification is further extended into various transforms with horizontal and vertical permutations. The relations between various classes of introduced ternary transforms are also discussed.

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By investigating some family of elementary order-2 mavices, new transforms of real vectors are introduced. When used for Boolean function transformations. these transforms are one-to-one mappings in a binary I tcmary vecwr space. The concept of different polarities of considered Arithmetic and Adding transforms has been introduced.

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