# Computer and Adders

Topics: Binary numeral system, Computer arithmetic, Computer Pages: 20 (3829 words) Published: July 13, 2013
FAST VLSI BINARY ADDITION
Dept. of Electrical and Computer Engineering University of Minnesota, Minneapolis, MN 55455, USA E-mail: parhi@ee.umn.edu Keshab K. Parhi

INTRODUCTION

leads to much faster and smaller binary adders. The use of redundant arithmetic in fast binary arithmetic was rst published in 5]. Many approaches to RB conversion have been published in 6]- 9]. The equivalence between binary addition and RB conversion is now clearly understood. Thus, algorithms known for one problem can be directly applied for the other. This paper considers two types of RB converters referred to as tree-based and carry-select which are obtained exactly in the same way as binary look-ahead and carry-select adders are designed. In this paper, it is shown that fastest binary adders can be designed using tree-based RB converters obtained using look-ahead approaches 3] 10]. It is shown that fastest binary addition can be performed using (Wlog2 W +1) multiplexers in time (log2W +1)tmux. If fastest design is not needed, then a family of adders can be designed using tree-based and carry-select converters. These adders can be analyzed for power consumption and the design with least power can be selected.

Consider the operation X = A + B where X = xW ?1 :::x0 , A = aW ?1 :::a0 , and B = bW ?1 :::b0 , represent W bit binary numbers. This addition can be expressed as X = A + B = A ? (?B ) = A ? (B + 1) = (A ? B ) + (?1). The negative of B is rst expressed in terms of sum of one's complement which corresponds to bit inversion and 1 in the least signi cant bit (lsb) position. The composite number (A ? B ) can be interpreted as a redundant number where each...

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