Error Detection and Correction

1
Error Detection and Correction

• and its application in CDs

2
Parity Check Codes

• k information bits are taken by an encoder
• r parity bits are added to the information bits
• A block of n bits, nkr, is produced
• The n-bit block is called the codeword
• This code is called an (n,k) block code
• The code rate is Rk/n

3
Parity Check Codes

• The decoder applies the same encoding rules to
  the received codeword and looks for any error in
  the parity check
• If an error occurs then if possible, algorithms
  are used with the received and calculated parity
  check bits to try and recreate the original
  codeword
• The information bits from the corrected codeword
  are then returned to the user, or if the codeword
  cannot be corrected a re-transmission will have
  to occur
• If an error occurs which changes one codeword
  into another valid codeword then the error will
  go undetected

4
Hamming

• Richard Hamming noticed the problem of flipping
  two bits in single parity check bit codes.
• He described it as the distance in the codeNow
  it is known as the Hamming distance
• Single parity bit codes have a Hamming distance
  of 2 the minimum number of bit flips required
  to produce another valid codeword
• Hamming was interested in maximising the distance
  and the code rate as much as possible
• In the 1940s he developed better codes where
  parity bits overlapped

5
Single Parity Check Bit

• This is the simplest error detection code
• 2 types of parity even and odd

Even parity 00101101 even so add parity bit 0
to keep it even 001011010
Odd parity 00101101 even so add parity bit 1
to make it odd 001011011

• The decoder adds up all the received bits to
  check the parity
• If the check detects an error then the data will
  be requested again

6
Single Parity Check Bit

• Can only detect an odd number of errors
• If an even number of bit flips occur then the
  parity of the received codeword will remain the
  same
• The probability of a larger number of bit flips
  is less than a smaller number of flips
• This check is best suited for reliable storage
  and reliable communication lines

7
Single Parity Check Bit

• A demodulator converts data transmitted over an
  analogue line back into a digital signal
• Hard-decision decoding each bit is either 0 or
  1
• Soft-decision decoding calculation of error on
  received data
• Wagner coding soft-decision decoding used with
  a single parity check bit for single bit error
  correction
• A bit-quality measurement is done on received
  data and the bit with the lowest bit-quality
  measurement is flipped to correct the parity
• This saves the cost of re-transmitting the data

8
Product Codes

• A product code provides a greater amount of error
  detection and correction than the single parity
  bit code

• The information bits are stored in a 2d array
• Block codes are applied to the rows and columns
• The block codes are applied to the check bits to
  produce an overall check bit

1
1
0
0
0
0

• This also works with higher dimensional arrays

9
Product Codes

• All single bit errors can be corrected

0
0
1
1
10
Product Codes

• All single bit errors can be corrected
• All two bit errors can be detected but not
  corrected

11
Product Codes

• All single bit errors can be corrected
• All two bit errors can be detected but not
  corrected
• All three bit errors can be detected. Some can be
  corrected, but others may be wrongly corrected
• This again shows the importance of probability

1
0
0
1
0
12
Product Codes

• Soft-decision decoding allows further error
  correction
• Rank decoding bits are ranked in order of
  reliability

• Look at the highest ranked undecoded bit which is
  not flagged
• If both parity checks are correct, decode the bit
  as received
• If both parity checks fail, flag the bit
• If only one parity check is correct then decode
  the bit as received if its rank is greater than
  the lowest ranked undecoded bit in its column and
  row. Flag the lowest bit in both the row and
  column if it is not already

2 1
1 1
9 1

• If there is only one undecoded bit in the row or
  column then decode it to force correct parity
• When all remaining undecoded bits are flagged,
  decode the highest ranked flagged bit and do step
  2

0
6 0
5 0
3 1
5 0
1
4 0
8 1
7 0
4 0
1
13
Reed-Solomon Codes

• A type of block code denoted as RS(n,k)
• Designed to operate on a block of symbols
• The encoder takes k information symbols and 2t
  check symbols to produce a n-symbol block, nk2t
• An error in a symbol one or more bits are wrong
• Half as many symbol errors as there are check
  symbols, t, can be corrected
• RS(255,223) can have (255-223)/216 symbol errors
  corrected
• The number of symbols which can be corrected does
  not depend on the number of bit errors in a
  symbol
• Reed-Solomon codes are very good at correcting
  bursts of errors

14
CD Error Correction

• Cross-Interleaved Reed-Solomon Coding (CIRC) is
  used
• The data is split into 8-bit symbols
• The encoder uses a RS(28,24) code to make a frame
• All the frames are interleaved

15
Interleaving Example

• No interleaving
• burst error on a frame cannot be corrected
• causes data loss
• Interleaving
• interleaving spreads out data across other frames
• burst error occurs on an interleaved frame
• de-interleaving shows the error is now spread
  across different frames
• the burst error can now be corrected

110 100 011
000 100 011
000 101 001
16
CD Error Correction

• Cross-Interleaved Reed-Solomon Coding (CIRC) is
  used
• The data is split into 8-bit symbols
• The encoder uses a RS(28,24) code to make a frame
• All the frames are interleaved
• Frames encoded again using a RS(32,28) code
• The frames are interleaved again in a different
  pattern
• The original frame is now spread over 109 frames
• 436/3213.625 frames can be corrected
• RS(28,24) C2 level of encoding
• corrects physical disk errors
• RS(32,28) C1 level of encoding
• corrects errors due to fingerprints and scratches

17
Bibliography

• Arnold M MichelsonError-control techniques for
  digital communicationWiley, c1985
• Michael PurserIntroduction to error-correcting
  codesArtech House, c1995
• Professor David JoynerReed-Solomon Codes and CD
  Encodinghttp//web.usna.navy.mil/wdj/reed-sol.ht
  mApril 24th 2002