Metzler matrix

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_{3} \cdot P_{2} \cdot P_{1} \cdot P_{0}

G_{G} = G_{3} + P_{3} \cdot G_{2} + P_{3} \cdot P_{2} \cdot G_{1} + P_{3} \cdot P_{2} \cdot P_{1} \cdot G_{0}

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 are

C_{4} = G_{0} + P_{0} \cdot C_{0}

C_{8} = G_{4} + P_{4} \cdot C_{4}

C_{12} = G_{8} + P_{8} \cdot C_{8}

C_{16} = G_{12} + P_{12} \cdot 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} \cdot C_{0}

C_{8} = G_{4} + G_{0} \cdot P_{4} + C_{0} \cdot P_{0} \cdot P_{4}

C_{12} = G_{8} + G_{4} \cdot P_{8} + G_{0} \cdot P_{4} \cdot P_{8} + C_{0} \cdot P_{0} \cdot P_{4} \cdot P_{8}

C_{16} = G_{12} + G_{8} \cdot P_{12} + G_{4} \cdot P_{8} \cdot P_{12} + G_{0} \cdot P_{4} \cdot P_{8} \cdot P_{12} + C_{0} \cdot P_{0} \cdot P_{4} \cdot P_{8} \cdot 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_{L C U} = P_{0} \cdot P_{4} \cdot P_{8} \cdot P_{12}

G_{L C U} = G_{12} + G_{8} \cdot P_{12} + G_{4} \cdot P_{8} \cdot P_{12} + G_{0} \cdot P_{4} \cdot P_{8} \cdot P_{12} + C_{0} \cdot P_{0} \cdot P_{4} \cdot P_{8} \cdot P_{12} = C_{16}

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 ( $P_{L C U}$ ) and generate ( $G_{L C U}$ ) from each LCU and the four carry outputs generated by the second-level LCU are fed into the first-level LCUs.

References

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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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↑ http://www.seas.upenn.edu/~ese171/lab/CarryLookAhead/CarryLookAheadF01.html

[1] ttp://www.seas.upenn.edu/~ese171/lab/CarryLookAhead/CarryLookAheadF01.html

[1]

Metzler matrix

Contents

4-bit adder

16-bit adder

64-bit adder

References

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Metzler matrix

4-bit adder

16-bit adder

64-bit adder

References

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