Gray Code

Objectives

1. Understand gray code.

Notes

• Reference Wikipedia
• Gray code is named after Frank Gray
• The code is used for
  • Error correction in communications
  • Analog to digital conversion.
  • Genetic algorithms
  • K-Maps
  • And any time we want to count, but only change one bit at a time.
• It is "reflective code" because the top half and bottom half of tables are reflected.
  • Two bit gray code

    Value	1	0
    0	0	0
    1	0	1
    2	1	1
    3	1	0

  • Note bit 1: goes 0011
  • Bit 0: 01 is reflected to 10
  • Note that each number, only changes by one bit.
• Let's try three bit.

  • Value	2	1	0
    0	0	0	0
    1	0	0	1
    2	0	1	1
    3	0	1	0
    4	1	1	0
    5	1	1	1
    6	1	0	1
    7	1	0	0

• Build the table for 4 bit gray code.

  • Value	3	2	1	0
    0	0	0	0	0
    1	0	0	0	1
    2	0	0	1	1
    3	0	0	1	0
    4	0	1	1	0
    5	0	1	1	1
    6	0	1	0	1
    7	0	1	0	0
    8	1	1	0	0
    9	1	1	0	1
    10	1	1	1	1
    11	1	1	1	0
    12	1	0	1	0
    13	1	0	1	1
    14	1	0	0	1
    15	1	0	0	0

• Conversion from binary to gray code:
  • To convert from binary
    • The MSB is the same
      • G_msb = B_msb
    • Every other bit xor with the previous binary bit.
      • G_n-1 = B_n ⊕ B_n-1

    • Convert 13 to gray code
      11012 =
          G3 = B3 = 1
          G2 = B3 ⊕ B2
             = 1  ⊕ 1 = 0
          G1 = B2 ⊕ B1
             = 1  ⊕ 0 = 1
          G0 = B1 ⊕ B0
             = 0  ⊕ 1 = 1
             = 1011gray
      
      Convert 10 to gray code
      10102 
          G3 = B3 = 1
          G2 = B3 ⊕ B2
             = 1  ⊕ 0 = 1
          G1 = B2 ⊕ B1
             = 0  ⊕ 1 = 1
          G0 = B1 ⊕ B0
             = 1  ⊕ 0 = 1
             = 1111gray 

  • 
• Convert from Gray to Binary
  • B_MSB = G_MSB
  • B_n-1 = B_n ⊕ G_n-1

• Convert 1001gray to binary
  1001gray =
      B3 = G3 = 1
      B2 = B3 ⊕ G2
         = 1  ⊕ 0 = 1
      B1 = B2 ⊕ G1
         = 1  ⊕ 0 = 1
      B0 = B1 ⊕ G0
         = 1  ⊕ 1 = 0
         =  11102 

• 
• the digital file