Gray code and 3-bit binary-to-Gray converter

Gray code

Gray code is an unweighted, reflected binary code in which successive numbers differ in only one bit position. Examples: 0 = 000, 1 = 001, 2 = 011, 3 = 010. It is used in shaft encoders and K-maps because only one bit changes at each step, avoiding false intermediate values.

3-bit binary to Gray code converter

Inputs: $B_2B_1B_0$ (binary, $B_2$ = MSB). Outputs: $G_2G_1G_0$.

Truth table (3-bit binary to Gray)

Simplification (K-maps)

• $G_2 = \sum m(4,5,6,7) = B_2$
• $G_1 = \sum m(2,3,4,5) = \bar B_2 B_1 + B_2\bar B_1 = B_2 \oplus B_1$
• $G_0 = \sum m(1,2,5,6) = \bar B_1 B_0 + B_1\bar B_0 = B_1 \oplus B_0$

K-map for $G_1$ (rows $B_2$, columns $B_1B_0$):

        B1B0
 B2    00  01  11  10
  0  |  0 | 0 | 1 | 1 |
  1  |  1 | 1 | 0 | 0 |

Logic circuit

 B2 ---+------------------------ G2
       |    +-------+
       +--->|       |
            |  XOR  |----------- G1
 B1 ---+--->|       |
       |    +-------+
       |    +-------+
       +--->|       |
            |  XOR  |----------- G0
 B0 ------->|       |
            +-------+

The converter needs only two XOR gates: the MSB passes straight through and each other Gray bit is the XOR of two neighbouring binary bits.