2-bit magnitude comparator

A magnitude comparator is a combinational circuit that compares two binary numbers A and B and gives three outputs: $A>B$, $A=B$ and $A<B$. Exactly one output is 1 at a time.

A 2-bit comparator compares $A = A_1A_0$ with $B = B_1B_0$ (4 inputs, 3 outputs).

Operation

1. Compare the most significant bits first. If $A_1 = 1$ and $B_1 = 0$, then $A>B$; if $A_1 = 0$ and $B_1 = 1$, then $A<B$.
2. Only if $A_1 = B_1$ are the LSBs $A_0$, $B_0$ compared in the same way.
3. $A=B$ only when both bit pairs are equal.

Equality of bit $i$ is given by an X-NOR gate:

$$x_i = A_iB_i + \overline{A_i}\,\overline{B_i} = \overline{A_i \oplus B_i}$$

Truth table

Output expressions

$$\begin{aligned} (A=B) &= x_1x_0 \\ (A>B) &= A_1\overline{B_1} + x_1A_0\overline{B_0} \\ (A<B) &= \overline{A_1}B_1 + x_1\overline{A_0}B_0 \end{aligned}$$

(The same results come from 4-variable K-maps of the table, e.g. $A>B = A_1\overline{B_1} + A_0\overline{B_1}\,\overline{B_0} + A_1A_0\overline{B_0}$.)

Circuit diagram

A1,B1 ─[XNOR]── x1        A0,B0 ─[XNOR]── x0

x1, x0 ───────────[AND]────────────── A=B

A1, B1' ──────────[AND]──┐
                         ├─[OR]────── A>B
x1, A0, B0' ──────[AND]──┘

A1', B1 ──────────[AND]──┐
                         ├─[OR]────── A<B
x1, A0', B0 ──────[AND]──┘

Gates: 2 X-NOR gates for $x_1$, $x_0$; one AND for $A=B$; two ANDs and an OR for each of $A>B$ and $A<B$ (inverters give the complemented bits).

Example: $A = 10$, $B = 11$: $A_1 = B_1$ so $x_1 = 1$; $\overline{A_0}B_0 = 1$, so $A<B = 1$, and the other outputs are 0.