A family of all essential Radix-2 addition/subtraction multi-polarity transforms: algorithms and interpretations in Boolean domain

Abstract

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.

Classification and properties of fast linearly independent logic transformations

“…A unified approach to the generation of butterfly structures for family of LI matrices can be also of interest for researchers developing efficient multiresolution digital signal processing systems using unconventional applications of butterfly LI decomposition techniques [1], [4], [5], [10], [18], [19], [25].…”

Fast transforms for multiple-valued input binary output PLI logic

“…If the sets of columns are linearly independent with respect to XOR operations (i.e. columns are bit-by-bit XORed), then M n has only one inverse in GF (2) and is said to be linearly independent. Lemma 1: Let M n be defined as in Definition 1.…”

“…It can be shown that among the 20 MVB PLI Transformation matrices listed in Table 1, only 6 MVB PLI transforms from Class MA satisfy Equations 10 and 11. The existence of Property 4 for PLI representation of MVB functions is important as it implies identical forward and inverse transform matrices over GF (2). Moreover, Property 5 is also satisfied which has eventually improved from the conventional means of calculating the spectral coefficient by matrix inversion and multiplication to calculation by fast transform.…”

“…Generally, if log 2 q ∉ Z (Z is a set of integers), there exist no fast transforms in Classes MB and MD. However, when q is a power of 2, GF(q) is simply an extension of GF (2). This provides applications to MVB PLI Transformation matrices of Classes MB, MC and MD, since for some MVB functions, they may result in more optimal representations of MVB functions.…”

“…The Reed-Muller Transform utilizes the algebra of GF (2) and any switching function may be completely realized by the modulo-2 sum-of-products expression which is known as the complement-free ring-sum [1]. For many important functions that are non-unate (e.g., parity checkers, adders and multipliers), Reed-Muller realizations are advantageous when area, speed, and testability are of main concern [14].…”