08 Comparator
Comparator/Magnitude checker¶
used to check if a binary number is less than/equal to/greater than another binary no
1 bit comparator¶
| x | y | G | E | L |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 0 |
\(L = x'y, E = x \odot y, G = xy'\)
2 Bit Comparator¶
| A_1 | A_0 | B_1 | B_0 | G | E | L |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 | ||
| 0 | 0 | 0 | 1 | 1 | ||
| 0 | 0 | 1 | 0 | 1 | ||
| 0 | 0 | 1 | 1 | 1 | ||
| 0 | 1 | 0 | 0 | 1 | ||
| 0 | 1 | 0 | 1 | 1 | ||
| 0 | 1 | 1 | 0 | 1 | ||
| 0 | 1 | 1 | 1 | 1 | ||
| 1 | 0 | 0 | 0 | 1 | ||
| 1 | 0 | 0 | 1 | 1 | ||
| 1 | 0 | 1 | 0 | 1 | ||
| 1 | 0 | 1 | 1 | 1 | ||
| 1 | 1 | 0 | 0 | 1 | ||
| 1 | 1 | 0 | 1 | 1 | ||
| 1 | 1 | 1 | 0 | 1 | ||
| 1 | 1 | 1 | 1 | 1 |
- \(G = A_1 {B_1}' + A_0 A_1 {B_0}' + A_0 {B_0}' {B_1}'\)
- \(E = (A_1 \odot B_1) (A_2 \odot B_2)\)
- \(L = {A_1}' B_1 + {A_0}' {A_1}' B_0 + {A_0}' B_0 B_1\)
3 Bit Comparator¶
- E
- \(x_2 = A_2 \odot B_2 \quad (A_2 = B_2)\)
- when \(A_2 = 0, B_2 = 0, A_2 = 1, B_2 = 1\)
- \(x_1 = A_1 \odot B_1 \quad (A_1 = B_1)\)
- \(x_0 = A_0 \odot B_0 \quad (A_0 = B_0)\)
- \(E = x_2 \cdot x_1 \cdot x_0 = (A_2 \odot B_2) \cdot (A_1 \odot B_1) \cdot (A_0 \odot B_0)\)
- L
- if \(A_2 < B_2\)
- \(A_2 = 0, B_2 = 1 \implies {A_2}' B_2\)
- if \(A_2 = B_2, A_1 < B_1\)
- \(x_2\) and\(A_1 = 0, B_1 = 1 \implies {A_1}' B_1\)
- \(x_2 \cdot {A_1}' B_1\)
- if \(A_2 = B_2, A_1 = B_1, A_0 < B_0\)
- \(x_2, x_1\) and\(A_0 = 0, B_0 = 1 \implies {A_0}' {B_0}\)
- \(x_2 \cdot x_1 \cdot {A_0} B_0\)
- \(\therefore, L = {A_2}' B_2 + x_2 {A_1}' B_1 + x_2 x_1 {A_0}' B_0\)
- G
- \(G = A_2 {B_2}' + x_2 A_1 {B_1}' + x_2 x_1 A_0 {B_0}'\)
4 bit comparator¶
\(E = x_3 x_2 x_1 x_0\)
Diagram¶
