Electronic – a Carry Propagate Adder

I thought about seeing what my brain could regurgitate of long ago explnatins, as opposed to commin sense observations, BUT here's a page which does it all very well indeed.

They explain a "nice" 'trick'.

• Two's compelment overflow detection can be achieved by XORing the carry-in and the carry-out bo=its of the leftmost full adder.

Their diagram. See above ref for detailed comment:

The worst case scenario for a Ripple-Carry Adder (RCA) is when the LSB generates a carry out, and the carry ripples through the entire adder from bit 0 to bit (N - 1). An example pattern would be 00000001 + 11111111. In adder terminology, bits 7-1 are "Propagators", and bit 0 is a "Generator". The critical path is from the carry-out of the LSB to the carry-out of the MSB, and every adder is in the critical path.

The idea behind a Carry-Skip Adder (CSA) is to reduce the length of this critical path by giving the carry path a shortcut if all bits in a block would propagate a carry. A block-wide propagate signal is fairly easy to compute, and each block can calculate its own propagate signal simultaneously. So the worst case is still the same scenario, but what happens looks a bit different.

Lets say we still have the same problem of 0000......001 + 0111.....111. The first block will calculate a carry in the first bit, and will propagate the carry through bits 1, 2, and 3. At this point, the first block carry-out signal is valid. The propagate select signals are already valid, since it is 2-3 gate delays and the carry signal is 4 gate delays. The carry-in multiplexer for bits 8-11 gets the carry signal from the carry-out of bit 3 since bits 4-7 would propagate a carry. Note that this takes 1 gate delay, while a normal RCA would take 4 gate delays. Each block will add 1 gate delay to the carry signal.

If the MSB killed carry propagation, then that would cause the last CSA block to ripple carry the input, which would take another 4 gate delays. This setup of a LSB generate and a MSB kill is the new worst case. The source of the critical path is the same between the RCA and CSA, but the critical path is different.

If an arbitrary block generated a carry by itself, the carry will always propagate to the next block. However, if the second block generates a carry itself, or kills the carry, than that is the end of the critical path. If the second block propagates the carry, then we see the advantage of the CSA architecture.

Also, when the term "critical path" is used, it generally implies that you are considering a set of inputs that will cause the worst-case delay. Your scenarios that you are providing give "ugly" cases that may have large delay, but it isn't the largest delay.