Two Operand Binary Adders with Threshold Logic

• Authors: José Fernández Ramos;Alfonso Gago Bohórquez
• Venue: IEEE Transactions on Computers
• Year: 1999

The central topic of this paper is the implementation of binary adders with Threshold Logic using a new methodology that introduces two innovations: the use of the input and output carries of each bit for obtaining all the sum bits and a modification of the classic Carry Lookahead adder technique that allows us to obtain the expressions of the generation and propagation carries in a more appropriate way for Threshold Logic. In this way, it has been possible to systematize the process of design of a binary adder with Threshold Logic relating all its important parameters: number of bits of the operands, depth, size, maximum fan-in, and maximum weight. The results obtained are an improvement on those published to date and are summarized as follows: Depth 2 adder: $s = 2n$, $w_{max} = 2^n$, $f_{max} = 2n + 1$. Depth 3 adder: $s = 4n - 2\left\lceil {{n \over {\left\lceil {\sqrt n } \right\rceil }}} \right\rceil $, $w_{\max }= 2^{\left\lceil {{n} \over {\left\lceil {\sqrt n } \right\rceil }}} \right\rceil } $, $f_{\max }= 2\left\lceil {{n \over {\left\lceil {\sqrt n } \right\rceil }}} \right\rceil+ 1$. Depth d adder (asymptotic behavior): $s = O (n)$, $w_{\max }= O(2^{\root {d - 1} \of n } )$, $f_{\max }= O(\root {d - 1} \of n )$. If the weights are bounded by $w_{max}$: $n_{\max }= O\!\left( {\log ^{d - 1} w_{\max } } \right)$, $d_{\min }= O\!\left( {{{\log n} \over {\log \left( {\log w_{\max } } \right)}}} \right)$.