 Gray Code

Gray Code
This is a variable weighted code and is cyclic, which means that it is arranged so that every transition from one value to the next value involves only one bit change. The gray code is sometimes referred to as reflected binary, because the first eight values compare with those of the last 8 values, but in reverse order.

 Decimal Binary Gray 0 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100 8 1000 1100 9 1001 1101 10 1010 1111 11 1011 1110 12 1100 1010 13 1101 1011 14 1110 1001 15 1111 1000

The gray code is often used in mechanical applications such as shaft encoders.
Modulo 2 Arithmetic
This is binary addition with the carry ignored.
Converting Gray Code to Binary
A.  Write down the number in gray code
B.  The most significant bit of the binary number is the most significant bit of the gray code
C.  Add (using modulo 2) the next significant bit of the binary number to the next significant bit of the gray coded number to obtain the next binary bit
D.  Repeat step C till all bits of the gray coded number have been added modulo 2
the resultant number is the binary equivalent of the gray number

Example, convert 1101101 in gray code to binary

Gray         Binary
1.           1101101
2.           1101101 1 copy down the msb
3.           1101101 10 1 modulo2 1 = 0
4.           1101101 100 0 modulo2 0 = 0
3/4         1101101 1001 0 modulo2 1 = 1
3/4         1101101 10010 1 modulo2 1 = 0
3/4         1101101 100100 0 modulo2 0 = 0
3/4         1101101 1001001 0 modulo2 1 = 1

Converting Binary to Gray
A. write down the number in binary code
B. the most significant bit of the gray number is the most significant bit of the binary code
C. add (using modulo 2) the next significant bit of the binary number to the next significant bit of the binary number to obtain the next gray coded bit
D. repeat step C till all bits of the binary coded number have been added modulo 2
the resultant number is the gray coded equivalent of the binary number

Example, convert 1001001 in binary code to gray code

Binary       Gray
1.           1001001
2.           1001001 1 copy down the msb
3.           1001001 11 1 modulo2 0 = 1
4.           1001001 110 0 modulo2 0 = 0
3/4         1001001 1101 0 modulo2 1 = 1
3/4         1001001 11011 1 modulo2 0 = 1
3/4         1001001 110110 0 modulo2 0 = 0
3/4         1001001 1101101 0 modulo2 1 = 1 Excess 3 Gray Code
In many applications, it is desirable to have a code that is BCD as well as unit distance. A unit distance code derives its name from the fact that there is only one bit change between two consecutive numbers. The excess 3 gray code is such a code, the values for zero and nine differ in only 1 bit, and so do all values for successive numbers.
Outputs from linear devices or angular encoders may be coded in excess 3 gray code to obtain multi-digit BCD numbers.

Decimal Excess 3 Gray
0            0010
1            0110
2            0111
3            0101
4            0100
5            1100
6            1101
7            1111
8            1110
9            1010

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