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Hamming Code
Error Detecting and Correcting
Code
 Parity code
◦ Detect odd number of errors
◦ Cannot correct any errors
 Hamming Code
◦ Detect up to two simultaneous bit errors
◦ Correct single-bit errors
Hamming Code
 For each integer m > 2
◦ Code exists with m parity bits and 2m – m – 1
 Basically
◦ Parity bit for odd bits
◦ Parity bit for each two bits
◦ Parity bit for each four bits
◦ Parity bit for each eight bits
◦ Parity bit for each group of bits that are power of
2
 1, 2, 4, 8, 16, 32, …
Calculating Hamming Code
1. Mark all bit positions that are powers of two parity bit
1, 2, 4, 8, 16, 32, 64, etc
2. All other bit positions are for the data to be encoded
3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, etc.
3. Each parity bit calculates the parity for some of the bits in
the code word
The position of the parity bit determines the sequence of bits that it
alternately checks and skips
Position 1: check 1 bit, skip 1 bit, check 1 bit, skip 1 bit, etc.
(1,3,5,7,9,11,13,15,...)
Position 2: check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc.
(2,3,6,7,10,11,14,15,...)
Position 4: check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc.
(4,5,6,7,12,13,14,15,20,21,22,23,...)
Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits, etc.
(8-15,24-31,40-47,...)
4. Set a parity bit to 1 if the total number of ones in the
positions it checks is odd. Set a parity bit to 0 if the total
number of ones in the positions it checks is even
Hamming Code Layout
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 3 4 5 6 7 8 9 10 11
Pos 20=1 21=2 22=4 23=8
1 1 0 0 0
2 0 1 0 0
3 1 1 0 0
4 0 0 1 0
5 1 0 1 0
6 0 1 1 0
7 1 1 1 0
8 0 0 0 1
9 1 0 0 1
10 0 1 0 1
11 1 1 0 1
12 0 0 1 1
13 1 0 1 1
14 0 1 1 1
15 1 1 1 1
Even parity calculated for each bit position
P1 (1, 3, 5, 7, 9, 11, 13, 15)
P2 (2, 3, 6, 7, 10, 11, 14, 15)
P4 (4, 5, 6, 7, 12, 13, 14, 15)
P8(8, 9, 10, 11, 12, 13, 14, 15)
Hamming Code Layout
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 0 1 0 1 1 0 1 0 1 1
P1 P1 P1 P1 P1 P1 P1 P1
P2 P2 P2 P2 P2 P2 P2 P2
P4 P4 P4 P4 P4 P4 P4 P4
P8 P8 P8 P8 P8 P8 P8 P8
1 1 1 0 0 1 0 1 1 1 0 1 0 1 1
Even parity calculated for each bit position
P1 (1, 3, 5, 7, 9, 11, 13, 15)
P2 (2, 3, 6, 7, 10, 11, 14, 15)
P4 (4, 5, 6, 7, 12, 13, 14, 15)
P8(8, 9, 10, 11, 12, 13, 14, 15)
For code: 10101101011
P1 = 1
P2 = 1
P4 = 0
P8 = 1
Hamming Code (1 error)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 0 1 0 1 1 0 1 0 1 1
P1 P1 P1 P1 P1 P1 P1 P1
P2 P2 P2 P2 P2 P2 P2 P2
P4 P4 P4 P4 P4 P4 P4 P4
P8 P8 P8 P8 P8 P8 P8 P8
1 1 1 0 0 1 0 1 1 1 0 1 0 1 1
1 1 1 0 0 1 0 1 1 1 0 1 0 0 1
Even parity calculated for each bit position
P1 (1, 3, 5, 7, 9, 11, 13, 15)
P2 (2, 3, 6, 7, 10, 11, 14, 15)
P4 (4, 5, 6, 7, 12, 13, 14, 15)
P8(8, 9, 10, 11, 12, 13, 14, 15)
A S
P1 = 1 1
P2 = 1 0
P4 = 0 1
P8 = 1 0
Xmit
Rec
2, 4, and 8 incorrect
2 + 4 + 8 = 14
A = Actual Parity
S = Should be
Parity
Hamming Code (2 errors)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 0 1 0 1 1 0 1 0 1 1
P1 P1 P1 P1 P1 P1 P1 P1
P2 P2 P2 P2 P2 P2 P2 P2
P4 P4 P4 P4 P4 P4 P4 P4
P8 P8 P8 P8 P8 P8 P8 P8
1 1 1 0 0 1 0 1 1 1 0 1 0 1 1
1 1 1 0 0 1 0 1 0 1 0 1 0 0 1
Even parity calculated for each bit position
P1 (1, 3, 5, 7, 9, 11, 13, 15)
P2 (2, 3, 6, 7, 10, 11, 14, 15)
P4 (4, 5, 6, 7, 12, 13, 14, 15)
P8(8, 9, 10, 11, 12, 13, 14, 15)
A S
P1 = 1 0
P2 = 1 0
P4 = 0 1
P8 = 1 1
Xmit
Rec
1,2, and 4 incorrect
1 + 2 + 4 = 7
Fix 7, will still have parity problem
A = Actual Parity
S = Should be
Parity
Example
 A byte of data: 10011010
 Create the data word, leaving spaces for the parity bits:
_ _ 1 _ 0 0 1 _ 1 0 1 0
 Calculate the parity for each parity bit (a ? represents the bit
position being set):
◦ Position 1 checks bits 1,3,5,7,9,11:
? _ 1 _ 0 0 1 _ 1 0 1 0. Even parity so set position 1 to a 0:
0 _ 1 _ 0 0 1 _ 1 0 1 0
◦ Position 2 checks bits 2,3,6,7,10,11:
0 ? 1 _ 0 0 1 _ 1 0 1 0. Odd parity so set position 2 to a 1:
0 1 1 _ 0 0 1 _ 1 0 1 0
◦ Position 4 checks bits 4,5,6,7,12:
0 1 1 ? 0 0 1 _ 1 0 1 0. Odd parity so set position 4 to a 1:
0 1 1 1 0 0 1 _ 1 0 1 0
◦ Position 8 checks bits 8,9,10,11,12:
0 1 1 1 0 0 1 ? 1 0 1 0. Even parity so set position 8 to a 0:
0 1 1 1 0 0 1 0 1 0 1 0
◦ Code word: 011100101010
Finding and Fixing a Bad Bit
123456789012
 Transmitted word: 011100101010
 Received word: 011100101110
 Calculate the parity bits from received
word
SB A
◦ P1 (1, 3, 5, 7, 9, 11) = 0 0
1
◦ P2 (2, 3, 6, 7, 10, 11) = 0 1
2
◦ P4 (4, 5, 6, 7, 12) = 1 1 4
◦ P8 (8, 9, 10, 11, 12) = 1 0
8
 2 + 8 = 10
Finding and Fixing a Bad Bit
123456789012
 Transmitted word:
 Received word: 010101100011
 Calculate the parity bits from received
word
SB A
◦ P1 (1, 3, 5, 7, 9, 11) = 0 0
◦ P2 (2, 3, 6, 7, 10, 11) = 0 1
◦ P4 (4, 5, 6, 7, 12) = 0 1
◦ P8 (8, 9, 10, 11, 12) = 0 0
 Bit positions 2 & 4 are incorrect
2 + 4 = 6
 Correct code is 010100100011
Finding and Fixing a Bad Bit
123456789012
 Transmitted word:
 Received word: 111110001100
 Calculate the parity bits from received
word
SB A
◦ P1 (1, 3, 5, 7, 9, 11) = 0 1
◦ P2 (2, 3, 6, 7, 10, 11) = 1 1
◦ P4 (4, 5, 6, 7, 12) = 0 1
◦ P8 (8, 9, 10, 11, 12) = 0 0
 Bit positions 1 & 4 are incorrect
1 + 4 = 5
 Correct code is 111100001100
Finding and Fixing a Bad Bit
123456789012
 Transmitted word:
 Received word: 000010001010
 Calculate the parity bits from received
word
SB A
◦ P1 (1, 3, 5, 7, 9, 11) = 1 0
◦ P2 (2, 3, 6, 7, 10, 11) = 1 0
◦ P4 (4, 5, 6, 7, 12) = 1 0
◦ P8 (8, 9, 10, 11, 12) = 0 0
 Bit positions 1, 2 & 4 are incorrect
1 + 2 + 4 = 7
 Correct code is: 000010101010

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3288940.ppt

  • 2. Error Detecting and Correcting Code  Parity code ◦ Detect odd number of errors ◦ Cannot correct any errors  Hamming Code ◦ Detect up to two simultaneous bit errors ◦ Correct single-bit errors
  • 3. Hamming Code  For each integer m > 2 ◦ Code exists with m parity bits and 2m – m – 1  Basically ◦ Parity bit for odd bits ◦ Parity bit for each two bits ◦ Parity bit for each four bits ◦ Parity bit for each eight bits ◦ Parity bit for each group of bits that are power of 2  1, 2, 4, 8, 16, 32, …
  • 4. Calculating Hamming Code 1. Mark all bit positions that are powers of two parity bit 1, 2, 4, 8, 16, 32, 64, etc 2. All other bit positions are for the data to be encoded 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, etc. 3. Each parity bit calculates the parity for some of the bits in the code word The position of the parity bit determines the sequence of bits that it alternately checks and skips Position 1: check 1 bit, skip 1 bit, check 1 bit, skip 1 bit, etc. (1,3,5,7,9,11,13,15,...) Position 2: check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc. (2,3,6,7,10,11,14,15,...) Position 4: check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc. (4,5,6,7,12,13,14,15,20,21,22,23,...) Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits, etc. (8-15,24-31,40-47,...) 4. Set a parity bit to 1 if the total number of ones in the positions it checks is odd. Set a parity bit to 0 if the total number of ones in the positions it checks is even
  • 5. Hamming Code Layout 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 Pos 20=1 21=2 22=4 23=8 1 1 0 0 0 2 0 1 0 0 3 1 1 0 0 4 0 0 1 0 5 1 0 1 0 6 0 1 1 0 7 1 1 1 0 8 0 0 0 1 9 1 0 0 1 10 0 1 0 1 11 1 1 0 1 12 0 0 1 1 13 1 0 1 1 14 0 1 1 1 15 1 1 1 1 Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15)
  • 6. Hamming Code Layout 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 1 0 1 1 0 1 0 1 1 P1 P1 P1 P1 P1 P1 P1 P1 P2 P2 P2 P2 P2 P2 P2 P2 P4 P4 P4 P4 P4 P4 P4 P4 P8 P8 P8 P8 P8 P8 P8 P8 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15) For code: 10101101011 P1 = 1 P2 = 1 P4 = 0 P8 = 1
  • 7. Hamming Code (1 error) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 1 0 1 1 0 1 0 1 1 P1 P1 P1 P1 P1 P1 P1 P1 P2 P2 P2 P2 P2 P2 P2 P2 P4 P4 P4 P4 P4 P4 P4 P4 P8 P8 P8 P8 P8 P8 P8 P8 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15) A S P1 = 1 1 P2 = 1 0 P4 = 0 1 P8 = 1 0 Xmit Rec 2, 4, and 8 incorrect 2 + 4 + 8 = 14 A = Actual Parity S = Should be Parity
  • 8. Hamming Code (2 errors) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 0 1 0 1 1 0 1 0 1 1 P1 P1 P1 P1 P1 P1 P1 P1 P2 P2 P2 P2 P2 P2 P2 P2 P4 P4 P4 P4 P4 P4 P4 P4 P8 P8 P8 P8 P8 P8 P8 P8 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 Even parity calculated for each bit position P1 (1, 3, 5, 7, 9, 11, 13, 15) P2 (2, 3, 6, 7, 10, 11, 14, 15) P4 (4, 5, 6, 7, 12, 13, 14, 15) P8(8, 9, 10, 11, 12, 13, 14, 15) A S P1 = 1 0 P2 = 1 0 P4 = 0 1 P8 = 1 1 Xmit Rec 1,2, and 4 incorrect 1 + 2 + 4 = 7 Fix 7, will still have parity problem A = Actual Parity S = Should be Parity
  • 9. Example  A byte of data: 10011010  Create the data word, leaving spaces for the parity bits: _ _ 1 _ 0 0 1 _ 1 0 1 0  Calculate the parity for each parity bit (a ? represents the bit position being set): ◦ Position 1 checks bits 1,3,5,7,9,11: ? _ 1 _ 0 0 1 _ 1 0 1 0. Even parity so set position 1 to a 0: 0 _ 1 _ 0 0 1 _ 1 0 1 0 ◦ Position 2 checks bits 2,3,6,7,10,11: 0 ? 1 _ 0 0 1 _ 1 0 1 0. Odd parity so set position 2 to a 1: 0 1 1 _ 0 0 1 _ 1 0 1 0 ◦ Position 4 checks bits 4,5,6,7,12: 0 1 1 ? 0 0 1 _ 1 0 1 0. Odd parity so set position 4 to a 1: 0 1 1 1 0 0 1 _ 1 0 1 0 ◦ Position 8 checks bits 8,9,10,11,12: 0 1 1 1 0 0 1 ? 1 0 1 0. Even parity so set position 8 to a 0: 0 1 1 1 0 0 1 0 1 0 1 0 ◦ Code word: 011100101010
  • 10. Finding and Fixing a Bad Bit 123456789012  Transmitted word: 011100101010  Received word: 011100101110  Calculate the parity bits from received word SB A ◦ P1 (1, 3, 5, 7, 9, 11) = 0 0 1 ◦ P2 (2, 3, 6, 7, 10, 11) = 0 1 2 ◦ P4 (4, 5, 6, 7, 12) = 1 1 4 ◦ P8 (8, 9, 10, 11, 12) = 1 0 8  2 + 8 = 10
  • 11. Finding and Fixing a Bad Bit 123456789012  Transmitted word:  Received word: 010101100011  Calculate the parity bits from received word SB A ◦ P1 (1, 3, 5, 7, 9, 11) = 0 0 ◦ P2 (2, 3, 6, 7, 10, 11) = 0 1 ◦ P4 (4, 5, 6, 7, 12) = 0 1 ◦ P8 (8, 9, 10, 11, 12) = 0 0  Bit positions 2 & 4 are incorrect 2 + 4 = 6  Correct code is 010100100011
  • 12. Finding and Fixing a Bad Bit 123456789012  Transmitted word:  Received word: 111110001100  Calculate the parity bits from received word SB A ◦ P1 (1, 3, 5, 7, 9, 11) = 0 1 ◦ P2 (2, 3, 6, 7, 10, 11) = 1 1 ◦ P4 (4, 5, 6, 7, 12) = 0 1 ◦ P8 (8, 9, 10, 11, 12) = 0 0  Bit positions 1 & 4 are incorrect 1 + 4 = 5  Correct code is 111100001100
  • 13. Finding and Fixing a Bad Bit 123456789012  Transmitted word:  Received word: 000010001010  Calculate the parity bits from received word SB A ◦ P1 (1, 3, 5, 7, 9, 11) = 1 0 ◦ P2 (2, 3, 6, 7, 10, 11) = 1 0 ◦ P4 (4, 5, 6, 7, 12) = 1 0 ◦ P8 (8, 9, 10, 11, 12) = 0 0  Bit positions 1, 2 & 4 are incorrect 1 + 2 + 4 = 7  Correct code is: 000010101010