By H. B. Bakoglu

ISBN-10: 0201060086

ISBN-13: 9780201060089

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**Extra resources for Circuits, Interconnections, and Packaging for Vlsi**

**Example text**

1 2 1 24 2. FUNCTIONAL EXPRESSIONS FOR MULTIPLE-VALUED FUNCTIONS In matrix notation, for a function f specified by the function vector F = [f (0), . . , f (3n − 1)]T , the coefficients ai in the Galois field expression are calculated as Sf,3GF = G3GF (n)F, where ⎡ n G3GF (n) = G3GF (1), G3GF (1) = (X3GF (1))−1 i=0 ⎤ 1 0 0 =⎣ 0 2 1 ⎦ 2 2 2 in GF (3). 11) may be written as f (x1 , . . , xn ) = X3GF (n)G(n)F. 2 1 0 2 1 0 2 0 0 0 2 0 1 0 2 2 0 2 2 0 0 0 0 1 1 0 1 2 0 1 2 0 0 0 0 2 1 ⎤ ⎡ ⎦⊗⎣ 0 0 0 2 0 1 2 0 1 0 0 0 0 1 1 0 1 1 1 0 2 0 0 0 0 2 1 0 2 1 0 2 2 0 0 0 1 0 2 2 0 1 ⎤ 0 1 ⎦ 2 0 0 0 0 0 0 0 0 2 1 2 2 0 0 1 2 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥.

6 EFFICIENCY OF REPRESENTATIONS Efficiency of an analytical representation of switching functions can be considered from two different aspects: 1. Regarding the complexity of calculation of the considered representation. In the case of polynomial representations that means the calculation of the coefficients of the representation. 2. Relative to the complexity of realization of the given function starting from this particular representation. In the case of polynomial representations, that means the number of non-zero coefficients, since each product term in the representation requires a circuit in the network.

This matrix we denote by A −1 (n). We take the inverse of it over the field of rational numbers Q, as the arithmetic transform matrix A(n) which is used to define the coefficients in the arithmetic expressions. Since coefficients in arithmetic expressions are integers, which means computer words are required to represent them, these expressions belong to the broad class of various word-level functional expressions for binary-valued functions. Since the arithmetic expressions are defined with respect to the same set of basis functions as the Reed-Muller expressions, the optimization of arithmetic expressions is performed by selecting polarities of variables in the same way as in the Reed-Muller expressions.

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