# Code Converters – Binary to/from Gray Code

**Prerequisite –** Number System and base conversions

Gray Code system is a binary number system in which every successive pair of numbers differs in only one bit. It is used in applications in which the normal sequence of binary numbers generated by the hardware may produce an error or ambiguity during the transition from one number to the next.

For example, the states of a system may change from 3(011) to 4(100) as- 011 â€” 001 â€” 101 â€” 100. Therefore there is a high chance of a wrong state being read while the system changes from the initial state to the final state.

This could have serious consequences for the machine using the information. The Gray code eliminates this problem since only one bit changes its value during any transition between two numbers.

### Converting Binary to Gray Code –

Let be the bits representing the binary numbers, where is the LSB and is the MSB, and

Let be the bits representing the gray code of the binary numbers, where is the LSB and is the MSB.

The truth table for the conversion is-

To find the corresponding digital circuit, we will use the K-Map technique for each of the gray code bits as output with all of the binary bits as input.

K-map for –

K-map for –

K-map for –

K-map for –

Corresponding minimized boolean expressions for gray code bits –

The corresponding digital circuit –

### Generalized Boolean Expression for conversion of Binary to Gray Code –

Boolean expression for conversion of binary to gray code for n-bit :

G_{ n } = B_{ n }

G_{ n-1 } = B_{ n } XOR B_{ n-1 } : :

G_{ 1 } = B_{ 2 } XOR B_{ 1 }

### Converting Gray Code to Binary –

Converting gray code back to binary can be done in a similar manner.

Let be the bits representing the binary numbers, where is the LSB and is the MSB, and

Let be the bits representing the gray code of the binary numbers, where is the LSB and is the MSB.

Truth table-

Using K-map to get back the binary bits from the gray code –

K-map for –

K-map for –

K-map for –

K-map for –

Corresponding Boolean expressions –

Corresponding digital circuit –

### Generalized Boolean Expression for conversion of Binary to Gray Code –

Boolean expression for conversion of gray to binary code for n-bit :

B_{ n } = G_{ n }

B_{ n-1 } = B_{ n } XOR G_{ n-1 } = G_{ n } XOR G_{ n-1 } : :

B_{ 1 } = B_{ 2 } XOR G_{ 1 } = G_{ n } XOR ………… XOR G_{ 1 }

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