lings adder

Computer Arithmetic: Algorithms and Hardware Designs

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Huey Ling High-speed Binary Adder Based on the bit pair (ai, bi) truth table, carry propagate pi and generate gi have dominated carry-look- ahead formation process for more than two decades. This paper presents a new scheme in which new propa- gation is examined by including the neighboring pairs (ai, bi; ai+,, b,+l). This scheme not only reduces component count in design, but also requires fewer logic levels in adder implementation. In addition, this algorithm oflers an astonishingly uniform loading in fan-in and fan-out nesting. Introduction The traditional recursive formula for carry propagation has dominated the carry handling process in the computer industry for more two decades. Today, adder de- signs based on a similar technique include Amdahl V6, IBM 168, and IBM 3033. The recursive formulation of carry is based on the bit pair bi) truth table. By examining the local bit pair, carry propagate pi carry generate gi are formed. The high-order carries are generated by nesting the and g, together. By considering the adjacent bit pairs (a,, bi; ai+l, bi+l), a new recursive formula is obtained for new carry propagation. The comparison between this new scheme and the existing scheme will be discussed in the following sections. The detailed implementation, circuits, and logic level count are also included. Surprisingly, this method offers an astonishingly uniform loading in fan-in/ fan-out nesting. The formation of new carry and sum This paper introduces a new approach to represent the new carry formation and propagation based on concept of the complementing signal which was intro- duced in 1%5 [I]. To examine the impact of this com- plementing signal in performing binary addition and com- plementing signal look-ahead, one should evaluate the formation of Hi and Hi+, as a function of neighbor- ing bit pairs (i, i + 1). Let us consider adding two binary numbers A and B together, where A = ao2" + al2"-' + a2Y-' + . . + aiP + . . . + a,2' ; = b02" + b,2"" + b,2"-' + . . . + bi2n-i + * * . + bn2' . The relation among the new carry (Hi, Hi+,) the neighboring bit pairs (ai, b,; bi+J can be ex- pressed as in Table 1 [l]; all of these are generated by ai, b, or transmitted through the low-order bits, i + 1, i + 2, . . ., with the transmitting-enable switch ON. This sig- nal or new carry can only be terminated when the in- hibitor is ON (ai+l + bi+, = 0). H, plays both regular carry and complementing signal roles in performing binary addition. By grouping all the Hi, we obtain =f(l, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15) = aibi + Hi+l(Gi+lbi+l + ai+,Li+, + ai+,bi+,) = aibi + Hi+l(ai+l + bi+,) = ki + H,+,T,+, , (1) where ki is the new complementing signal, is the pre- vious complementary signal, and Ti+, is the previous carry enable switch or the previous stage propagate. Equation (1) shows that new carry can be formed locally by ki or produced remotely; can be produced with the remote stage carry inhibitor not ON (ai+l + Copyright 1981 by International Business Machines Corporation.
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lings adder - Huey Ling High-speed Binary Adder Based on...

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