1 Error Control Codes and Their Applications in Digital World Eric Chen Computer Science Group HKr

2 2 OCR-Number (OCR-nummer ) ? A reference number links your payment to the invoice Typed OCR in-correctly ? Error detection? E24.se reported (how it is possible?) –Företagaren Thomas Hultberg fyllde i fel OCR-nummer när han skulle betala in skatten. Nu tvingas han betala in de kronorna en gång till.

3 3 Outline of the talk Analog versus Digital Why digital ? Errors and Error Effect Error Control Coding Applications and Examples My Research Results

4 4 Analog Environment Voice signal example –Microphone converts the speech to electrical signal, it is analog

5 5 Analog Communication Analog signal –The value (amplitude) varies continuously Very difficult to re-produce the analog signal at the receiver  Bad quality

6 6 Digital World Information society, information explosion –Most information is represented in the form of numbers Computer and infrastructure are digital systems Video and audio –VCD, DVD player –Digital TV –CD music –Mp3 player Information is encoded into sequences of binary digits (bits) 0 and 1

7 7 Digital Communication Digital signal –Limited number of discrete values Example– binary waveform –Binary 1 and 0 are represented by different levels of the voltage –

8 8 Analog and Digital Conversions Conversions are needed A/D converter –Sender –Convert analog signal to digital signal D/A converter –Receiver –Convert digital signal to analog signal

9 9 Advantages with digital system Immune to noises ? –Amplifier vs regenerator  Better quality for digital communication

10 Advantages with digital system (cont.) Integrated services –For audio, video, others More reliable information exchange Easy to provide secure communication Digital processing and storage Less costly implementation

11 Errors in digital communications

12 Errors and Error Effect Errors  0  1 or 1  0  Bits can be lost Error effect  downloaded programs from Internet ?  CD music ?  Internet banking services ? Errors must be detected/corrected !!!

13 Bit Error Rate BER = bit_errors / total-bits = p  p = 1/ = for optical disks  p = for a fiber link Some calculations  p = download a file of length 10 7 bits  10 bit errors Data rate at 10 Mbps  1 bit error in every 1 second !!  p = , and data rate 10 Gigabits/sec  1 bit error each 10 second !

14 Error Effect Example 1 – image Waterfall – original image

15 Error Effect Example 1 – image Waterfall – with bit error rate 1/10

16 Error Effect Example 2– English text two-frogs.txt Bit error rate 1/100, 1/1000 –two-frogs-100.txt p = 1/100 –two-frogs-1000.txt p = 1/1000 –You can still read the text ?! –Why ? Redundancy in natural language

17 two-frogs.txt A group of frogs were traveling through the woods, and two of them fell into a deep pit. When the other frogs saw how deep the pit was, they told the two frogs that they were as good as dead. The two frogs ignored the comments and tried to jump up out of the pit with all their might. The other frogs kept telling them to stop, that they were as good as dead. Finally, one of the frogs took heed to what the other frogs were saying and gave up. He fell down and died. The other frog continued to jump as hard as he could. Once again, the crowd of frogs yelled at him to stop the pain and just die. He jumped even harder and finally made it out. When he got out, the other frogs said, "Did you not hear us?" The frog explained to them that he was deaf. He thought they were encouraging him the entire time.

18 two-frogs-1000.txt (p = 1/1000) A group of frogs were traveling through the woods,!and two of them fell into a deep pit. When the other frogs saw how deep the pit was, they told the two frogs that they were as good as dead. The two frogs ignored the comments and tried to jump up out of the piv with all their might. The other frogs kept pelling them to stop, that they were as good as dead.(Finally, one of the frogs took heed to what the other frogs were saying and gave up. He fell down and died. The other frog continued to jump as hqrd as he could. Once again, the crowd of frogs yelled at him to stop the pain and just die. He jumped even harder aNd finally made it out. When he got out, the other fr/gs said, "Did you not hear us?" The frog explained to them that he was deaf. He thought they were encouraging him the entire time.

19 two-frogs-100.txt (p = 1/100) A group of frogs werd travening through phe woods, and two of them fell into a deep pit. _hen the odher frogs sAw how d%ep the pit was, they told the two frog¾ that they were as good!as dead. Vha twg frogs ignored the comments ant triEd 4o jumpáuq ouv of the pit satx all *tha4 they were as good as dead. ãinamly, one"of tha frogs took heed(to0what the otheR frogs were saying and 'ave up. He fell down and dieõ. ♂Tje ophez frog continued t⌂ jump as hard as he could. Once again, the crowd oF frogs yell%õ!at"him to stop the pain and nust die. He jumped even harder i~d finally made it out. When he got out♀ the ☻them thqt He was de!f. He thought they wÕre encouraging him uhe entire tkme.

20 add additional information, or redundancy to data added by sender, checked by receiver k data digits encoded to a codeword of n digits Code rate r = k / n kn Encoded as codeword Error Control Coding – Principle

21 Application Example– Swedish personal ID ? yy mm dd – nnnP  yy mm dd – year month day  nnn – serial number  odd– for male, even for female  P ? That is parity check digit  Used for error detection ! OCR number uses the same technique

22 Personal ID Encoding Method position ? 2×odd add 2-digits sum = = 36 take the last digit of the sum: 6 parity check digit = 10 – 6 = 4 

23 Personal ID Error Detection  ? position ? 2×odd add 2-digits sum = = 43 take the last digit of the sum: 3 parity check digit = 10 – 3 = 7  It is not equal to 4  Error in the number !

24 Given k data bits, add 1 parity check bit b….b encoded as b…b P  In the encoded word, how to get P ?? # of 1’s is even  even parity check encoding # of 1’s is odd  odd parity check encoding Example: k = 7, n = 8   Even parity, P = 1   Odd parity, P = 0 Can detect any odd number of errors Even or Odd Parity Checks

25 Hamming [7, 4] code k = 4, n = 7  Encode 4 data bits by adding 3 parity bits  Can correct any single error Encoding a b c d  a b c d x y z Where a, b, c, d are information bits x, y, z are parity check bits they are 0 or 1 Hamming Code Example

26 Given a, b, c, d. How to get x, y, z ? Place a, b, c, d in the intersections Label circles by x, y, z Parity checking rule: the sum of each circle is 0 x = a+b+c, y = a + c + d, z = b + c + d Hamming Code Example a b c d x y z

27 Given a, b, c, d. How to get x, y, z ? 0101  0101 xyz so the codeword is Hamming Code Example x y z a b c d x y z

sent  received.  Encode 0100   Compare received xyz 110  there is an error  bit d must be in error, it affects y, z  correction 0101 Hamming Code for Error Correction received reconstructed

29 Only detect errors –Using protocol to correct errors:  ACK: positive acknowledgement ( I got it)  NAK: negative acknowledgement ( sorry ) Simple, reliable, high code rate Used in data communications Error Detecting Codes senderreceiver codeword ACK/NAK

30 Detect and correct errors No feedback channel required Complicated, lower code rate Used in storage systems (computer storage, CD, DVD), and space communications Error Correcting Codes senderreceiver codeword

31 A picture of Saturn sent by Voyager 2 It would not be possible without using the error correcting code

32 Olympus Mons by Viking 1 the largest known volcano in the Solar System 27 km high, over 600km at the base, and is surrounded by a well- defined scarp that is up to 6 km high

33 Application in Compact Disc (CD) Without error correcting codes, it is not possible to have high quality music or video on the CD cross-interleave Read-Solomon code (CIRC) is used –Correct about 4000 bits burst errors 2.5 mm on disc 8 mm hole

34 More Applications ?? In any applications related to digital storage, and digital communications, you can find error control codes. Error control coding is a standard technique to detect/correct errors

35 Some of my Research Results A web database of binary quasi-cyclic codes moodle.tec.hkr.se/~chen/research/codes/searchqc2.htm A Web database of two-weight codes moodle.tec.hkr.se/~chen/research/2-weight-codes/search.php

36 References Prasad, K. V., Principles of Digital Communication Systems and Computer Networks, Charles River Media, 2004 Kularatna N., Essentials of Modern Telecommunications Systems, Artech House Incorporated, ffba /70408.pdf kel_ e ualer/BG2688.pdf

37 Questions ? The material is available at Thank you