Lookahead carry unit explained

A lookahead carry unit (LCU) is a logical unit in digital circuit design used to decrease calculation time in adder units and used in conjunction with carry look-ahead adders (CLAs).

4-bit adder

A single 4-bit CLA is shown below:

16-bit adder

By combining four 4-bit CLAs, a 16-bit adder can be created but additional logic is needed in the form of an LCU.

The LCU accepts the group propagate (

P _{G} ) and group generate (G _{G} ) from each of the four CLAs. P _{G} and G _{G} have the following expressions for each CLA adder:^[1] P _{G} = P _{0} ⋅ P _{1} ⋅ P _{2} ⋅ P _{3} G _{G} = G _{3} + G _{2} ⋅ P _{3} + G _{1} ⋅ P _{2} ⋅ P _{3} + G _{0} ⋅ P _{1} ⋅ P _{2} ⋅ P _{3}

The LCU then generates the carry input for each CLA.

Assume that

P _{i} is P _{G} and G _{i} is G _{G} from the i^th CLA then the output carry bits areC _{4} = G _{0} + P _{0} ⋅ C _{0} C _{8} = G _{4} + P _{4} ⋅ C _{4} C _{12} = G _{8} + P _{8} ⋅ C _{8} C _{16} = G _{12} + P _{12} ⋅ C _{12}

Substituting

C _{4} into C _{8} , then C _{8} into C _{12} , then C _{12} into C _{16} yields the expanded equations:C _{4} = G _{0} + P _{0} ⋅ C _{0} C _{8} = G _{4} + G _{0} ⋅ P _{4} + C _{0} ⋅ P _{0} ⋅ P _{4} C _{12} = G _{8} + G _{4} ⋅ P _{8} + G _{0} ⋅ P _{4} ⋅ P _{8} + C _{0} ⋅ P _{0} ⋅ P _{4} ⋅ P _{8} C _{16} = G _{12} + G _{8} ⋅ P _{12} + G _{4} ⋅ P _{8} ⋅ P _{12} + G _{0} ⋅ P _{4} ⋅ P _{8} ⋅ P _{12} + C _{0} ⋅ P _{0} ⋅ P _{4} ⋅ P _{8} ⋅ P _{12} C _{4} corresponds to the carry input into the second CLA; C _{8} to the third CLA; C _{12} to the fourth CLA; and C _{16} to overflow carry bit.

In addition, the LCU can calculate its own propagate and generate:

P _{LCU} = P _{0} ⋅ P _{4} ⋅ P _{8} ⋅ P _{12} G _{LCU} = G _{12} + G _{8} ⋅ P _{12} + G _{4} ⋅ P _{8} ⋅ P _{12} + G _{0} ⋅ P _{4} ⋅ P _{8} ⋅ P _{12} C _{16} = G _{LCU} + C _{0} ⋅ P _{LCU}

64-bit adder

By combining 4 CLAs and an LCU together creates a 16-bit adder.Four of these units can be combined to form a 64-bit adder.An additional (second-level) LCU is needed that accepts the propagate (

P _{LCU} ) and generate (G _{LCU} ) from each LCU and the four carry outputs generated by the second-level LCU are fed into the first-level LCUs.

References

• Book: Katz, Randy . Randy Katz

. Randy Katz . Contemporary Logic Design . The Benjamin/Cummings Publishing Company . 1994 . 0-8053-2703-7 . 249–256 . registration .

• Book: Vahid, Frank . Digital Design . limited . John Wiley and Sons Publishers . 2006 . 0-470-04437-3 . 296–316.

Notes and References

1. Web site: Carry Look Ahead Adder . 2011-10-07 . dead . https://web.archive.org/web/20110925185302/http://www.seas.upenn.edu/~ese171/lab/CarryLookAhead/CarryLookAheadF01.html . 2011-09-25 .