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n-bit comparator using 1-bit comparator

n-bit comparator using 1-bit comparator. Our focus is on making n-bit circuit using 1-bit building block Comparing two n-bit numbers for their equality A ----> A n-1 A n-2 …………A 0 B ----> B n-1 B n-2 ………….B 0 First, we shall compare the two numbers for equality

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n-bit comparator using 1-bit comparator

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  1. n-bit comparator using 1-bit comparator • Our focus is on making n-bit circuit using 1-bit building block • Comparing two n-bit numbers for their equality • A ----> An-1 An-2 …………A0 • B ----> Bn-1 Bn-2………….B0 • First, we shall compare the two numbers for equality • Start with MSB and compare one bit at a time • If the bit of one number is greater than the corresponding bit of the other number, then the number is greater. This becomes the solution • Else, i.e if the two bits are equal, then compare the next bit. • Repeat the last two steps

  2. Ai Bi Ei+1 Ei Basic building block for n-bit comparator • 1-bit comparator forms the building block for comparing two n-bit numbers. • E is the cascading signal • Ei+1 is the Cascading input, Ei is the cascading output • Ei+1 = 1 implies that the two numbers are not equal so far • Ei+1 = 0 implies that the two numbers are equal so far • If Ei+1 = 1, then Ei = 1 • Else, Ei = Ei+1 + (Ai XOR Bi )

  3. An-2 An-1 Bn-2 Bn-1 A0 B0 En En-1 E1 E0 Using 1-bit building blocks to make n-bit circuit • Using the building block, we can build an n-bit comparator circuit • What is the value of input En ? • The final result is, if E0 = 0, then A = B and if E0 = 1, then A != B

  4. Ai Bi (A>B)in (A>B)out (A=B)in (A=B)out (A<B)in (A<B)out 1-bit complex comparator • Next, we compare the two numbers for A > B, A = B or A < B • The basic building block for the purpose is as shown • The primary inputs are Ai and Bi • Cascading inputs are (A>B)in, (A=B)in and (A<B)in, and • Cascading outputs are (A>B)out, (A=B)out and (A<B)out • The equations can be directly written as • (A>B)out = (A>B)in + (A=B)in . (Ai.Bi’) • (A<B)out = (A<B)in + (A=B)in . (Ai’.Bi ) • (A=B)out = (A=B)in . (Ai Bi + Ai’.Bi’)

  5. Ai Bi (A>B)in (A>B)out (A<B)in (A<B)out 1-bit simplified complex comparator • The 1-bit complex comparator block is modified to have fewer cascading signals • It has two primary inputs, two cascading inputs, and two cascading outputs • The equations for the outputs are • (A>B)out = (A>B)in + (A<B)in’ . ( Ai Bi’) • (A<B)out = (A<B)in + (A>B)in’ . ( Ai’ Bi ) • If both (A>B)out and (A<B)out are 0 then the two bits are equal

  6. Ai Bi (A>B)in (A>B)out (A<B)in (A<B)out 1-bit maximizer building block • The 1-bit maximizer block is similar to the simplified complex comparator • It has two primary inputs, one primary output, two cascading inputs, and two cascading outputs • The primary output M is 1 if depending on which input is greater • The equations for the outputs are • (A>B)out = (A>B)in + (A<B)in’ . ( Ai Bi’) • (A<B)out = (A<B)in + (A>B)in’ . ( Ai’ Bi ) • M = (A>B)in . A + (A<B)in . B + (A>B)in’ . (A<B)in’ . (Ai + Bi) M

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