A-level Computing 2009/AQA/Problem Solving, Programming, Data Representation and Practical Exercise/Fundamentals of Data Representation/Hamming code

Parity bits Gray coding Building on what you have learnt about parity bits we are now going to see a system that not only allows you to detect if the data you have been sent is incorrect, but it will allow you to correct the error. The way hamming code does this is to use multiple check digits in the same piece of sent data. Checking if correct Number the column headings Highlight the column headings that are powers of 2 (1,2,4,8), these are the parity bits Insert your data and highlight the parity bits Work your way through the parity bits 2^0 = 1 : check 1, skip 1, check 1, skip 1 ... write down whether it's odd or even parity 2^1 = 2 : check 2, skip 2, check 2, skip 2 ... write down whether it's odd or even parity 2^2 = 4 : check 4, skip 4, check 4, skip 4 ... write down whether it's odd or even parity etc.. Example: Odd Parity Hamming Code Check Let's take a look a an example of data sent with odd parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 0 0 0 0 0 1 1 1 1 insert your data 1 0 0 0 0 0 0 1 1 1 1 highlight the check bits 1 0 0 0 1 1 taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity 1 0 0 0 1 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity 0 0 0 1 taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity 1 0 0 0 taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity Note that for the check 8 skip 8 we ran out of digits, not to worry, take it as far as the bits given allow. As we can see each line is odd parity, and the sent data was supposed to be odd parity, this number is correct. 1 Exercise: Even Parity Hamming Code Question Now try this example with even parity: 10101100011 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 1 0 1 1 0 0 0 1 1 insert your data 1 0 1 0 1 1 0 0 0 1 1 highlight the check bits 1 1 1 0 0 1 taking the 1st power of 2^0 (1) check 1 skip 1 = even parity 1 0 1 1 0 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity 1 1 0 0 taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity 1 0 1 0 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity All are even parity, the data should be even parity, therefore it has been sent and received correctly 11011110010 being sent with odd parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 1 0 1 1 1 1 0 0 1 0 insert your data 1 1 0 1 1 1 1 0 0 1 0 highlight the check bits 1 0 1 1 0 0 taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity 1 1 1 1 0 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity 1 1 1 0 taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity 1 1 0 1 taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity All are odd parity, the data should be odd parity, therefore it has been sent and received correctly 00100011110 being sent with even parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 0 0 1 0 0 0 1 1 1 1 0 insert your data 0 0 1 0 0 0 1 1 1 1 0 highlight the check bits 0 1 0 1 1 0 taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity! Dodgy! 0 0 0 0 1 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity. OK 0 0 1 1 taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity! OK 0 0 1 0 taking the 4th power of 2^3 (8) check 8 skip 8 = odd parity. Dodgy! We have a mixture of odd and even parity, this means that there has been a mistake in sending this data. But where is the error? It has something to do with the lines that have odd parity! Detecting and correcting errors Number the column headings Highlight the column headings that are powers of 2 (1,2,4,8), these are the parity bits Insert your data and highlight the parity bits Work your way through the parity bits 2^0 = 1 : check 1, skip 1, check 1, skip 1 ... write down whether it's odd or even parity 2^1 = 2 : check 2, skip 2, check 2, skip 2 ... write down whether it's odd or even parity 2^2 = 4 : check 4, skip 4, check 4, skip 4 ... write down whether it's odd or even parity etc.. If there is a disparity between rows, highlight all the error data and find where it overlaps Example: Even Parity Hamming Code Check Let's take a look a an example of data sent with even parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 0 1 0 0 1 0 1 1 1 insert your data 1 0 0 1 0 0 1 0 1 1 1 highlight the check bits 1 0 0 1 1 1 taking the 1st power of 2^0 (1) check 1 skip 1 = even parity 1 0 0 0 1 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity PROBLEM! 0 0 1 0 taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity PROBLEM! 1 0 0 1 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity Note that two of the lines, 2^1 and 2^2, show that an error has been detected. This means that somewhere that these lines cross over a bit has been corrupted, namely bit 6 or bit 7. If we know which one it is we can then switch it and correct the error. Look at the other checks that are in play, do any of them take part in this crossover? Looking at it, the 2^0 line also checks column 7 and it found it fine. So we are left with column 6 being the problematic one. As Hamming code is corrective, let's flip that column and we should have a correct piece of data. Another way of finding errors is to add the check digit values together, the error occurs where the check digit equals 4 and 2. Add 4 +2 = 6, the error is with the 6th digit! 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 0 1 0 1 1 0 1 1 1 insert your data 1 0 0 1 0 1 1 0 1 1 1 highlight the check bits 1 0 0 1 1 1 taking the 1st power of 2^0 (1) check 1 skip 1 = even parity 1 0 0 1 1 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity 0 1 1 0 taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity 1 0 0 1 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity The number is now even parity and correct: 10010110111 1 Exercise: Detect and Correct the error in the following Hammed Code Now try this example with even parity: 01101001011 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 0 1 1 0 1 0 0 1 0 1 1 insert your data 0 1 1 0 1 0 0 1 0 1 1 highlight the check bits 0 1 1 0 0 1 taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity PROBLEM! 0 1 1 0 0 1 taking the 2nd power of 2^1 (2) check 2 skip 2 = odd parity PROBLEM! 1 0 0 1 taking the 3rd power of 2^2 (4) check 4 skip 4 = even parity 0 1 1 0 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity The error is in the lines crossing over, that is on lines 2^0 and 2^1, but which bit is it? You'll notice that the 2^2 and 2^3 lines are correct so we can discount any bits that are covered in those lines. This leaves us with the third column. Flipping this value gives us the corrected value of: 01101001111 11111101000 sent with odd parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 1 1 1 1 1 0 1 0 0 0 insert your data 1 1 1 1 1 1 0 1 0 0 0 highlight the check bits 1 1 1 0 0 0 taking the 1st power of 2^0 (1) check 1 skip 1 = odd parity 1 1 1 1 0 0 taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity PROBLEM! 1 1 0 1 taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity 1 1 1 1 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity PROBLEM! The error is in the lines crossing over, that is on lines 2^1 and 2^3, but which bit is it? We can look at the place they cross over, bit ten, alternatively we can add the parity bit numbers together the row of parity bit 2 plus the row of parity bit: 2 + 8 = bit 10. Flipping this value gives us the corrected value of: 10111101000 00111000101 sent with even parity 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 0 0 1 1 1 0 0 0 1 0 1 insert your data 0 0 1 1 1 0 0 0 1 0 1 highlight the check bits 0 1 1 0 1 1 taking the 1st power of 2^0 (1) check 1 skip 1 = even parity 0 0 1 0 1 0 taking the 2nd power of 2^1 (2) check 2 skip 2 = even parity 1 0 0 0 taking the 3rd power of 2^2 (4) check 4 skip 4 = odd parity. PROBLEM! 0 0 1 1 taking the 4th power of 2^3 (8) check 8 skip 8 = even parity There is only one line with an error, the line of parity bit 4. This means that the error is in bit 4,5,6 or 7. Parity bit lines 1 and 2 imply that bits 5,6,7 are all fine, leaving us with the error in bit 4. Alternatively, as the error only occurs on parity bit line 4, then we know the error is with bit 4! Flipping this value gives us the corrected value of: 00111001101 Applying hamming code Number the column headings Highlight the column headings that are powers of 2 (1,2,4,8), these are the parity bits Insert your data into the bits that aren't parity bits Work your way through the parity bits looking at 2^0 : check 1, skip 1, check 1, skip 1 ... write the parity bit in column 1 to make the bits the desired parity looking at 2^1 : check 2, skip 2, check 2, skip 2 ... write the parity bit in column 2 to make the bits the desired parity looking at 2^2 : check 4, skip 4, check 4, skip 4 ... write the parity bit in column 4 to make the bits the desired parity ... Example: Applying Hamming code to an ASCII character We are going to take a look at sending the ASCII letter e: 1100101 with odd parity. This data would then be ready to send 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 1 0 ? 0 1 0 ? 1 ? ? insert your data in columns that aren't parity bits 1 0 0 0 1 ? taking the 1st power of 2^0 (1) check 1 skip 1 then work out the digit that is needed to go into the parity bit to apply odd parity ? = 1 1 1 0 1 1 ? taking the 2nd power of 2^1 (2) check 2 skip 2 then work out the digit that is needed to go into the parity bit to apply odd parity ? = 1 0 1 0 ? taking the 3rd power of 2^2 (4) check 4 skip 4, then work out the digit that is needed to go into the parity bit to apply odd parity ? = 0 1 1 0 ? taking the 4th power of 2^3 (8) check 8 skip 8, then work out the digit that is needed to go into the parity bit to apply odd parity ? = 1 We have now worked out the odd parity Hammed number ready for sending: 11010100111 1 Exercise: Applying Hamming code to an ASCII character Apply even parity hamming code so we can transmit the ASCII character 'D' (1000100): 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 0 ? 0 1 0 ? 0 ? ? insert your data in columns that aren't parity bits 1 0 0 0 0 ? taking the 1st power of 2^0 (1) check 1 skip 1 then work out the digit that is needed to go into column one to apply even parity ? = 1 1 0 0 1 0 ? taking the 2nd power of 2^1 (2) check 2 skip 2 then work out the digit that is needed to go into column one to apply even parity ? = 0 0 1 0 ? taking the 3rd power of 2^2 (4) check 4 skip 4, then work out the digit that is needed to go into column one to apply even parity ? = 1 1 0 0 ? taking the 4th power of 2^3 (8) check 8 skip 8, then work out the digit that is needed to go into column one to apply even parity ? = 1 We have now worked out the even parity Hammed number ready for sending: 10010101001 Apply even parity hamming code so we can transmit the ASCII character 'G': ASCII 'G' = 1000111 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 1 0 0 ? 0 1 1 ? 1 ? ? insert your data in columns that aren't parity bits 1 0 0 1 1 ? taking the 1st power of 2^0 (1) check 1 skip 1 then work out the digit that is needed to go into column one to apply even parity ? = 1 1 0 0 1 1 ? taking the 2nd power of 2^1 (2) check 2 skip 2 then work out the digit that is needed to go into column one to apply even parity ? = 1 0 1 1 ? taking the 3rd power of 2^2 (4) check 4 skip 4, then work out the digit that is needed to go into column one to apply even parity ? = 0 1 0 0 ? taking the 4th power of 2^3 (8) check 8 skip 8, then work out the digit that is needed to go into column one to apply even parity ? = 1 We have now worked out the even parity Hammed number ready for sending: 10010110111 Apply odd parity hamming code so we can transmit the denary value 9: 9 = 0001001 11 10 09 08 07 06 05 04 03 02 01 number the columns and highlight the powers of 2 0 0 0 ? 1 0 0 ? 1 ? ? insert your data in columns that aren't parity bits 0 0 1 0 1 ? taking the 1st power of 2^0 (1) check 1 skip 1 then work out the digit that is needed to go into column one to apply odd parity ? = 1 0 0 1 0 1 ? taking the 2nd power of 2^1 (2) check 2 skip 2 then work out the digit that is needed to go into column one to apply odd parity ? = 1 1 0 0 ? taking the 3rd power of 2^2 (4) check 4 skip 4, then work out the digit that is needed to go into column one to apply odd parity ? = 0 0 0 0 ? taking the 4th power of 2^3 (8) check 8 skip 8, then work out the digit that is needed to go into column one to apply odd parity ? = 1 We have now worked out the even parity Hammed number ready for sending: 00011000111