Gray Code to Binary Converter

A gray code-to-binary converter is a digital circuit that can translate a gray code into an equivalent pure binary code. Thus, a gray code to binary converter takes a gray code as input and gives a pure binary code as output.

The truth table of a 3-bit gray code to binary code converter is given below −

Let us obtain the Boolean expression for the binary output bits. For this, we will simplify the truth table using the K-map technique.

K-Map for Binary Bit B₀

The K-map simplification for the binary output bit B₀ is shown in the following figure.

The Boolean expression for the binary bit B₀ will be,

$$\mathrm{B_{0} \: = \: \overline{G_{2}} \: \overline{G_{1}} \: G_{0} \: + \: \overline{G_{2}} \: G_{1} \: \overline{G_{0}} \: + \: G_{2} \: \overline{G_{1}} \: \overline{G_{0}}\: + \: G_{2} \: G_{1} \: G_{0}}$$

We can further simplify this expression as follows,

$$\mathrm{\Rightarrow \: B_{0} \: = \: \overline{G_{2}} \: (\overline{G_{1}} \: G_{0} \: + \: G_{1} \: \overline{G_{0}}) \: + \: G_{2} \: (\overline{G_{1}} \: \overline{G_{0}}\: + \: G_{1} \: G_{0})}$$

$$\mathrm{\Rightarrow \: B_{0} \: = \: \overline{G_{2}} \: ( G_{0} \: \oplus \: G_{1}) \: + \: G_{2} \: \overline{(G_{0} \: \oplus \: G_{1})}}$$

$$\mathrm{B_{0} \: = \: G_{0} \: \oplus \: G_{1} \: \oplus \: G_{2}}$$

This is the simplified expression for the binary bit B₀.

K-Map for Binary Bit B₁

The K-map simplification for the binary output B₁ is shown below.

The Boolean expression for the binary bit B₁ is,

$$\mathrm{B_{1} \: = \: G_{2} \: \overline{G_{1}} \: + \: \overline{G_{2}} \: G_{1} \: = \: G_{1} \: \oplus \: G_{2}}$$

K-Map for Binary Bit B₂

The following figure shows the K-map simplification for the binary bit B₂.

From this K-Map, we obtain the following Boolean expression −

$$\mathrm{B_{2} \: = \: G_{2}}$$

The logic circuit implementation of this 3-bit gray to binary code converter is shown in the following figure.

This logic circuit can translate a 3-bit gray code into an equivalent 3-bit binary code. We can also follow the same procedure to implement a gray code to binary code converter for any number of bits.