- Checking the correctness of a received message:
- Converting the index of
the 1 bit
locations into a
binary number
- Perform the
XOR operations
on all resulting binary numbers
- If the
result = 0 then
the messages
(most likely) contains
no error
- The decoded message consists of the received bits minus the bits at at positions: 20, 21, 22, ..., 2k
- If the
result <= length of the message then
the messages (most likely)
contains
1 error
- Change the
value of the bit at
the position given by
"result"
(computed above)
- The decoded message consists of the received bits minus the bits at at positions: 20, 21, 22, ..., 2k
- Change the
value of the bit at
the position given by
"result"
(computed above)
- If the
result > length of the message then
the messages
(most likely)
contains
multiple errors
- The receiver cannot perform
correction
- It will reject the message as corrupted
- The receiver cannot perform
correction
- Converting the index of
the 1 bit
locations into a
binary number
- Example: 101010
From the discussion above:
we know that the sender will transmit: 1011010000
- Case 1: transmission was received
with no error
- Received message = 1011010000
Outcome of the decode procedure:
Result:
- XOR result
= 0
Conclusion:
- Message has no error (which was what happened !)
- Case 2: transmission was received
with one bit error
- Received message =
1010010000
(bit position 7 was received in error)
Outcome of the decode procedure:
Result:
- XOR result
= 0111 =
7
( ≤ 10 (message length))
Conclusion:
- Message has 1 bit error at bit position 7 (which was what happened !)
(Try with a bit error in some other position !)
Case 3: transmission was received with (detectable) two bits error
- Received message =
1110010000
(Bit positions 7 and 9 were received in error)
Outcome of the decode procedure:
Result:
- XOR result =
1110 =
14
( > 10 (message length))
Conclusion:
- Message has multiple bit errors -- correcting is impossible (which was what happened !)
- Received message =
1010010000
- Fact:
- Some multiple bit errors will cause the Hamming decode procedure to believe that there was a single bit error
- Example:
- We use the same
example:
Hamming encode message: 1011010000 (original message was: 101010
- Suppose bit positions 6 and
7 were received
in error:
Received message = 1010110000
- Outcome of the decode procedure:
Result:
- XOR result =
0001
= 1
( ≤ 10 (message length))
Conclusion:
- Messages has 1 bit error
at bit position 1...
(This is wrong !!!)
- Messages has 1 bit error
at bit position 1...
- XOR result =
0001
= 1
( ≤ 10 (message length))
- We use the same
example: