Equation of carry out signal in terms of delete and propagate

[shweta.bphc]

Hi,

I have been trying to derive an equaltion of carry out signal of binary full adder in terms of delete and propagate signal but not able to find one. If anyone knows the relation, please explain.

carry_out = AB + Ci(A xor B)
D = A_bar AND B_bar
P = A xor B

Thank you.

 

[ads-ee]

Using the truth tables for the P and D given that

D
  0 1
0 1 0
1 0 0

and

P
  0 1
0 0 1
1 1 0

ORing the two truth tables together

D OR P
  0 1
0 1 1
1 1 0

Then inverting the whole thing

NOT(D OR P)
  0 1
0 0 0
1 0 1

which is the same as A AND B, so carry_out can be represented by

carry_out = NOT(P OR D) OR (Ci AND P)

which is less efficient than the original carry_out = AB + Ci(A^B).