Digital comparator

Implementation

Consider two 4-bit binary numbers A and B so

A=A_3A_2A_1A_0

B=B_3B_2B_1B_0

Here each subscript represents one of the digits in the numbers.

;Equality The binary numbers A and B will be equal if all the pairs of significant digits of both numbers are equal, i.e.,

A_3=B_3, A_2=B_2, A_1=B_1 and A_0=B_0

Since the numbers are binary, the digits are either 0 or 1 and the boolean function for equality of any two digits A_i and B_i can be expressed as

x_i= A_i \cdot B_i + \overline{A_i} \cdot \overline{B_i} we can also replace it by XNOR gate in digital electronics.

x_i is 1 only if A_i and B_i are equal.

For the equality of A and B, all x_i variables (for i=0,1,2,3) must be 1.

So the equality condition of A and B can be implemented using the AND operation as

(A=B) = x_3 \cdot x_2 \cdot x_1 \cdot x_0

The binary variable (A=B) is 1 only if all pairs of digits of the two numbers are equal.

;Inequality

In order to manually determine the greater of two binary numbers, we inspect the relative magnitudes of pairs of significant digits, starting from the most significant bit, gradually proceeding towards lower significant bits until an inequality is found. When an inequality is found, if the corresponding bit of A is 1 and that of B is 0 then we conclude that AB.

This sequential comparison can be expressed logically as:

(AB)=A_3 \cdot \overline{B_3} + x_3 \cdot A_2 \cdot \overline{B_2} + x_3 \cdot x_2 \cdot A_1 \cdot \overline{B_1} + x_3 \cdot x_2 \cdot x_1 \cdot A_0 \cdot \overline{B_0}

(A

(A B) and (A B or A

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References

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