lookahead carry unit

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:

:$P_G = P_3 \cdot P_2 \cdot P_1 \cdot P_0$
:$G_G = G_3 + P_3 \cdot G_2 + P_3 \cdot P_2 \cdot G_1 + P_3 \cdot P_2 \cdot P_1 \cdot G_0$

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

:$C_ = G_0 + P_0 \cdot C_0$
:$C_ = G_ + P_ \cdot C_$
:$C_ = G_ + P_ \cdot C_$
:$C_ = G_ + P_ \cdot C_$

Substituting $C_$ into $C_$, then $C_$ into $C_$, then $C_$ into $C_$ yields the expanded equations:

:$C_ = G_0 + P_0 \cdot C_0$
:$C_ = G_4 + G_0 \cdot P_4 + C_0 \cdot P_0 \cdot P_4$
:$C_ = G_8 + G_4 \cdot P_8 + G_0 \cdot P_4 \cdot P_8 + C_0 \cdot P_0 \cdot P_4 \cdot P_8$
:$C_ = G_ + G_8 \cdot P_ + G_4 \cdot P_8 \cdot P_ + G_0 \cdot P_4 \cdot P_8 \cdot P_ + C_0 \cdot P_0 \cdot P_4 \cdot P_8 \cdot P_$

$C_$ corresponds to the carry input into the second CLA; $C_$ to the third CLA; $C_$ to the fourth CLA; and $C_$ to overflow carry bit.

In addition, the LCU can calculate its own propagate and generate:
:$P_ = P_0 \cdot P_4 \cdot P_8 \cdot P_$
:$G_ = G_ + G_8 \cdot P_ + G_4 \cdot P_8 \cdot P_ + G_0 \cdot P_4 \cdot P_8 \cdot P_$
:$C_ = G_ + C_0 \cdot P_$

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 ($P_$) and generate ($G_$) 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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* {{cite book , last=Vahid , first=Frank , title=Digital Design , url=https://archive.org/details/digitaldesign00vahi , url-access=limited , publisher=John Wiley and Sons Publishers , year=2006 , isbn=0-470-04437-3 , page
296
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Digital circuits
Adders (electronics)

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