Carry-lookahead Adder - Carry Lookahead Method

Carry Lookahead Method

Carry lookahead logic uses the concepts of generating and propagating carries. Although in the context of a carry lookahead adder, it is most natural to think of generating and propagating in the context of binary addition, the concepts can be used more generally than this. In the descriptions below, the word digit can be replaced by bit when referring to binary addition.

The addition of two 1-digit inputs A and B is said to generate if the addition will always carry, regardless of whether there is an input carry (equivalently, regardless of whether any less significant digits in the sum carry). For example, in the decimal addition 52 + 67, the addition of the tens digits 5 and 6 generates because the result carries to the hundreds digit regardless of whether the ones digit carries (in the example, the ones digit does not carry (2+7=9)).

In the case of binary addition, generates if and only if both A and B are 1. If we write to represent the binary predicate that is true if and only if generates, we have:

The addition of two 1-digit inputs A and B is said to propagate if the addition will carry whenever there is an input carry (equivalently, when the next less significant digit in the sum carries). For example, in the decimal addition 37 + 62, the addition of the tens digits 3 and 6 propagate because the result would carry to the hundreds digit if the ones were to carry (which in this example, it does not). Note that propagate and generate are defined with respect to a single digit of addition and do not depend on any other digits in the sum.

In the case of binary addition, propagates if and only if at least one of A or B is 1. If we write to represent the binary predicate that is true if and only if propagates, we have:

Sometimes a slightly different definition of propagate is used. By this definition A + B is said to propagate if the addition will carry whenever there is an input carry, but will not carry if there is no input carry. It turns out that the way in which generate and propagate bits are used by the carry lookahead logic, it doesn't matter which definition is used. In the case of binary addition, this definition is expressed by:

For binary arithmetic, or is faster than xor and takes fewer transistors to implement. However, for a multiple-level carry lookahead adder, it is simpler to use .

Given these concepts of generate and propagate, when will a digit of addition carry? It will carry precisely when either the addition generates or the next less significant bit carries and the addition propagates. Written in boolean algebra, with the carry bit of digit i, and and the propagate and generate bits of digit i respectively,