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:

4-bit adder with Carry Look Ahead (CLA)

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:

$P_{G}=P_{3}\cdot P_{2}\cdot P_{1}\cdot P_{0}$

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

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:

$P_{LCU}=P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}$

$G_{LCU}=G_{12}+G_{8}\cdot P_{12}+G_{4}\cdot P_{8}\cdot P_{12}+G_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}+C_{0}\cdot P_{0}\cdot P_{4}\cdot P_{8}\cdot P_{12}=C_{16}$

16-bit adder with 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.

64-bit adders with a second-level LCU

References

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