Digital comparator

An XNOR gate is a basic comparator, because its output is "1" only if its two input bits are equal.

The analog equivalent of digital comparator is the voltage comparator. Many microcontrollers have analog comparators on some of their inputs that can be read or trigger an interrupt.

Implementation

Consider two 4-bit binary numbers A and B so

${\displaystyle A=A_3{A}_2{A}_1{A}_0}$

${\displaystyle B=B_3{B}_2{B}_1{B}_0}$

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.,

${\displaystyle A_3=B_3}$, ${\displaystyle A_2=B_2}$, ${\displaystyle A_1=B_1}$ and ${\displaystyle A_0=B_0}$

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

${\displaystyle x_i=A_i{B}_i+{\overline{A}}_i{\overline{B}}_i}$ we can also replace it by XNOR gate in digital electronics.

${\displaystyle x_i}$ is 1 only if ${\displaystyle A_i}$ and ${\displaystyle B_i}$ are equal.

For the equality of A and B, all ${\displaystyle x_i}$ 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

${\displaystyle (A=B)=x_3{x}_2{x}_1{x}_0}$

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 A>B.

This sequential comparison can be expressed logically as:

${\displaystyle (A>B)=A_3{\overline{B}}_3+x_3{A}_2{\overline{B}}_2+x_3{x}_2{A}_1{\overline{B}}_1+x_3{x}_2{x}_1{A}_0{\overline{B}}_0}$

${\displaystyle (A<B)={\overline{A}}_3{B}_3+x_3{\overline{A}}_2{B}_2+x_3{x}_2{\overline{A}}_1{B}_1+x_3{x}_2{x}_1{\overline{A}}_0{B}_0}$

(A>B) and (A < B) are output binary variables, which are equal to 1 when A>B or A<B respectively.