”;
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 −
| Gray Code | Binary Code | ||||
|---|---|---|---|---|---|
| G2 | G1 | G0 | B2 | B1 | B0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 0 | 1 |
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.
”;