Comparators

A digital logic component that compares two binary numbers.

Equality Comparator

Outputs whether the two numbers are equal or not.

Magnitude Comparator

Outputs whether the two numbers are equal or less than or greater than.

1-bit Magnitude Comparator

Suppose the comparator takes in $A$ and $B$. And the output $G$ denotes whether $A$ is greater than $B$.

$G = A\overline{B}$

$E = AB + \overline{A}\cdot\overline{B}$

$L = \overline{A} B$

n-bit Magnitude Comparator

Let $x_i = A_i B_i + \overline{A_i}\cdot \overline{B_i} = \overline{A_i \oplus B_i}$. $x_i$ denotes if $A_i = B_i$.

$$G = \sum_{i=0}^{n-1} \Bigg(\bigg(\prod_{j=i+1}^{n-1}{x_j}\bigg) A_i\overline{B_i}\Bigg)$$ $$E = \prod_{i=0}^{n-1} {x_i}$$ $$L = \sum_{i=0}^{n-1} \Bigg(\bigg(\prod_{j=i+1}^{n-1}{x_j}\bigg) \overline{A_i}B_i\Bigg)$$