Conversion of Gray Code to Binaryn

Data Structure and Algorithms Mathematics Digital Electronics

The reflected binary code or Gray code is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). Gray codes are very useful in the normal sequence of binary numbers generated by the hardware that may cause an error or ambiguity during the transition from one number to the next. So, the Gray code can eliminate this problem easily since only one bit changes its value during any transition between two numbers.

Conversion of Gray to Binary Code

Gray codes are used in rotary and optical encoders, Karnaugh maps, and error detection. The hamming distance of two neighbours Gray codes is always 1 and also first Gray code and last Gray code also has Hamming distance is always 1, so it is also called Cyclic codes. You can convert a Gray code to Binary number using two methods.

Using Karnaugh (K) - map −

You can construct Gray codes using other methods but they may not be performed in parallel like given above method. For example, 3 bit Gray codes can be contracted using K-map which is given as following below:

Decimal	Gray Code	Binary
0	000	000
1	001	001
2	011	010
3	010	011
4	110	100
5	111	101
6	101	110
7	100	111

Using Exclusive-Or (⊕) operation −

This is very simple method to get Binary number from Gray code. These are following steps for n-bit binary numbers −

The Most Significant Bit (MSB) of the binary code is always equal to the MSB of the given binary number.
Other bits of the output binary code can be obtained by checking gray code bit at that index. If current gray code bit is 0, then copy previous binary code bit, else copy invert of previous binary code bit.

For example, for 3-bit binary number, let Binary digits are b₂ , b₁ , b₀, where b₂ is the most significant bit (MSB) and b₀ is the least significant bit (LSB) of Binary. Gray code digits are g₂ , g₁ , g₀, where g₂ is the most significant bit (MSB) and g₀ is the least significant bit (LSB) of Gray code.

Gray Code g₂g₁g₀	Binary b₂b₁b₀
000	000
001	001
011	010
010	011
110	100
111	101
101	110
100	111

Therefore, you solve boolean expression using k-map, you will get b₂=g₂, b₁=g₁⊕g₂, and b₀=g₀⊕g₁⊕g₂.

Similarly, you can convert n-bit (b_nb_(n-1)...b₂b₁b₀) Binary number into Gray code (g_ng_(n-1)...g₂g₁g₀). For least significant bit (LSB) b_n=g_n, b_(n-1)=g_(n-1)⊕g_n, …. b₁=g₁⊕g₂⊕g₃...⊕g_n and b₀=g₀⊕g₁⊕g₂⊕g₃...⊕g_n.