Carry-lookahead Adder - Theory of Operation

Theory of Operation

A ripple-carry adder works in the same way as pencil-and-paper methods of addition. Starting at the rightmost (least significant) digit position, the two corresponding digits are added and a result obtained. It is also possible that there may be a carry out of this digit position (for example, in pencil-and-paper methods, "9+5=4, carry 1"). Accordingly all digit positions other than the rightmost need to take into account the possibility of having to add an extra 1, from a carry that has come in from the next position to the right.

This means that no digit position can have an absolutely final value until it has been established whether or not a carry is coming in from the right. Moreover, if the sum without a carry is 9 (in pencil-and-paper methods) or 1 (in binary arithmetic), it is not even possible to tell whether or not a given digit position is going to pass on a carry to the position on its left. At worst, when a whole sequence of sums comes to ...99999999... (in decimal) or ...11111111... (in binary), nothing can be deduced at all until the value of the carry coming in from the right is known, and that carry is then propagated to the left, one step at a time, as each digit position evaluated "9+1=0, carry 1" or "1+1=0, carry 1". It is the "rippling" of the carry from right to left that gives a ripple-carry adder its name, and its slowness. When adding 32-bit integers, for instance, allowance has to be made for the possibility that a carry could have to ripple through every one of the 32 one-bit adders.

Carry lookahead depends on two things:

1. Calculating, for each digit position, whether that position is going to propagate a carry if one comes in from the right.
2. Combining these calculated values to be able to deduce quickly whether, for each group of digits, that group is going to propagate a carry that comes in from the right.

Supposing that groups of 4 digits are chosen. Then the sequence of events goes something like this:

1. All 1-bit adders calculate their results. Simultaneously, the lookahead units perform their calculations.
2. Suppose that a carry arises in a particular group. Within at most 3 gate delays, that carry will emerge at the left-hand end of the group and start propagating through the group to its left.
3. If that carry is going to propagate all the way through the next group, the lookahead unit will already have deduced this. Accordingly, before the carry emerges from the next group the lookahead unit is immediately (within 1 gate delay) able to tell the next group to the left that it is going to receive a carry - and, at the same time, to tell the next lookahead unit to the left that a carry is on its way.

The net effect is that the carries start by propagating slowly through each 4-bit group, just as in a ripple-carry system, but then move 4 times as fast, leaping from one lookahead carry unit to the next. Finally, within each group that receives a carry, the carry propagates slowly within the digits in that group.

The more bits in a group, the more complex the lookahead carry logic becomes, and the more time is spent on the "slow roads" in each group rather than on the "fast road" between the groups (provided by the lookahead carry logic). On the other hand, the fewer bits there are in a group, the more groups have to be traversed to get from one end of a number to the other, and the less acceleration is obtained as a result.

Deciding the group size to be governed by lookahead carry logic requires a detailed analysis of gate and propagation delays for the particular technology being used.

It is possible to have more than one level of lookahead carry logic, and this is in fact usually done. Each lookahead carry unit already produces a signal saying "if a carry comes in from the right, I will propagate it to the left", and those signals can be combined so that each group of (let us say) four lookahead carry units becomes part of a "supergroup" governing a total of 16 bits of the numbers being added. The "supergroup" lookahead carry logic will be able to say whether a carry entering the supergroup will be propagated all the way through it, and using this information, it is able to propagate carries from right to left 16 times as fast as a naive ripple carry. With this kind of two-level implementation, a carry may first propagate through the "slow road" of individual adders, then, on reaching the left-hand end of its group, propagate through the "fast road" of 4-bit lookahead carry logic, then, on reaching the left-hand end of its supergroup, propagate through the "superfast road" of 16-bit lookahead carry logic.

Again, the group sizes to be chosen depend on the exact details of how fast signals propagate within logic gates and from one logic gate to another.

For very large numbers (hundreds or even thousands of bits) lookahead carry logic does not become any more complex, because more layers of supergroups and supersupergroups can be added as necessary. The increase in the number of gates is also moderate: if all the group sizes are 4, one would end up with one third as many lookahead carry units as there are adders. However, the "slow roads" on the way to the faster levels begin to impose a drag on the whole system (for instance, a 256-bit adder could have up to 24 gate delays in its carry processing), and the mere physical transmission of signals from one end of a long number to the other begins to be a problem. At these sizes carry-save adders are preferable, since they spend no time on carry propagation at all.