1. ECE 6332, Spring, 2014 Wireless Communication Zhu Han Department of Electrical and Computer Engineering Class 18 March. 26 th, 2014

2. Outline Chapter 8 –FEC Basics –Line Code

3. Automatic Repeat-reQuest (ARQ) Alice and Bob on their cell phones –Both Alice and Bob are talking What if Alice couldn’t understand Bob? –Bob asks Alice to repeat what she said What if Bob hasn’t heard Alice for a while? –Is Alice just being quiet? –Or, have Bob and Alice lost reception? –How long should Bob just keep on talking? –Maybe Alice should periodically say “uh huh” –… or Bob should ask “Can you hear me now?”

4. ARQ Acknowledgments from receiver –Positive: “okay” or “ACK” –Negative: “please repeat that” or “NACK” Timeout by the sender (“stop and wait”) –Don’t wait indefinitely without receiving some response –… whether a positive or a negative acknowledgment Retransmission by the sender –After receiving a “NACK” from the receiver –After receiving no feedback from the receiver

5. Error Correcting Codes Adding redundancy to the original message To detect and correct errors Crucial when it’s impossible to resend the message (interplanetary communications, storage..) and when the channel is very noisy (wireless communication) Message = [1 1 1 1] Noise = [0 0 1 0] Message = [1 1 0 1]

6. Types of Error Correcting Codes Repetition Code Linear Block Code, e.g. Hamming Cyclic Code, e.g. CRC BCH and RS Code Convolutional Code –Tradition, Viterbi Decoding –Turbo Code –LDPC Code Coded Modulation –TCM –BICM

7. Repetition Code Simple Example: reduce the capacity by 3 Simple Example: reduce the capacity by 3 Recovered state

8. Parity Check Add one bit so that xor of all bit is zero –Send, correction, miss –Add vertically or horizontally Applications: ASCII, Serial port transmission

9. ISDN Number ISBN 10 –a modulus 11 with weights 10 to 2, using X instead of 10 where ten would occur as a check digitmodulus –ISBN 0-306-40615-2 ISBN 13 –Calculating an ISBN 13 check digit requires that each of the first twelve digits of the 13- digit ISBN be multiplied alternately by 1 or 3. Next, take the sum modulo 10 of these products. This result is subtracted from 10.check digitmodulo –ISBN 978-0-306-40615-7.

10. Hammings Solution A type of Linear Block Code Encoding: H(7,4) Multiple Checksums Message=[a b c d] r= (a+b+d) mod 2 s= (a+b+c) mod 2 t= (b+c+d) mod 2 Code=[r s a t b c d] Coding rate: 4/7 –Smaller, more redundancy, the better protection. –Difference between detection and correction Message=[1 0 1 0] r=(1+0+0) mod 2 =1 s=(1+0+1) mod 2 =0 t=(0+1+0) mod 2 =1 Code=[ 1 0 1 1 0 1 0 ]

11. Error Detection Ability 100,000 iterations Add Errors to (7,4) data No repeat randoms Measure Error Detection Error Detection One Error: 100% Two Errors: 100% Three Errors: 83.43% Four Errors: 79.76% Stochastic Simulation: Results:

12. How it works: 3 dots Only 3 possible words Distance Increment = 1 One Excluded State (red) It is really a checksum. Single Error Detection No error correction ABC ABC AC Two valid code words (blue) This is a graphic representation of the “Hamming Distance”

13. Hamming Distance Definition: –The number of elements that need to be changed (corrupted) to turn one codeword into another. The hamming distance from: –[0101] to [0110] is 2 bits –[1011101] to [1001001] is 2 bits –“butter” to “ladder” is 4 characters –“roses” to “toned” is 3 characters

14. Another Dot The code space is now 4. The hamming distance is still 1. Allows: Error DETECTION for Hamming Distance = 1. Error CORRECTION for Hamming Distance =1 For Hamming distances greater than 1 an error gives a false correction.

15. Even More Dots Allows: Error DETECTION for Hamming Distance = 2. Error CORRECTION for Hamming Distance =1. For Hamming distances greater than 2 an error gives a false correction. For Hamming distance of 2 there is an error detected, but it can not be corrected.

16. Multi-dimensional Codes Code Space: 2-dimensional 5 element states Circle packing makes more efficient use of the code-space

17. Cannon Balls http://wikisource.org/wiki/Cannonball_stacking http://mathworld.wolfram.com/SpherePacking.html Efficient Circle packing is the same as efficient 2-d code spacing Efficient Sphere packing is the same as efficient 3-d code spacing Efficient n-dimensional sphere packing is the same as n-code spacing

18. Example Visualization of eight code words in a 6-typle space

19. Another Example: Encoding we multiply this matrix But why? You can verify that: To encode our message By our message Hamming[1 0 0 0]=[1 0 0 0 0 1 1] Hamming[0 1 0 0]=[0 1 0 0 1 0 1] Hamming[0 0 1 0]=[0 0 1 0 1 1 0] Hamming[0 0 0 1]=[0 0 0 1 1 1 1] Where multiplication is the logical AND And addition is the logical XOR

20. Example: Add noise If our message is Message = [0 1 1 0] Our Multiplying yields Code = [0 1 1 0 0 1 1] Lets add an error, so Pick a digit to mutate Code => [0 1 0 0 0 1 1]

21. Example: Testing the message We receive the erroneous string: Code = [0 1 0 0 0 1 1] We test it: Decoder*Code T =[0 1 1] And indeed it has an error The matrix used to decode is: To test if a code is valid: Does Decoder*Code T =[0 0 0] –Yes means its valid –No means it has error/s

22. Example: Repairing the message To repair the code we find the collumn in the decoder matrix whose elements are the row results of the test vector We then change We trim our received code by 3 elements and we have our original message. [0 1 1 0 0 1 1] => [0 1 1 0] Decoder*codeT is [ 0 1 1] This is the third element of our code Our repaired code is [0 1 1 0 0 1 1]

23. Coding Gain Coding Rate R=k/n, k, no. of message symbol, n overall symbol Word SNR and bit SNR For a coding scheme, the coding gain at a given bit error probability is defined as the difference between the energy per information bit required by the coding scheme to achieve the given bit error probability and that by uncoded transmission.

24. ECE 4371 Fall 2008 Coding Gain Example

25. Encoder/Decoder of Linear Code Encoder: just xor gates Decoder: Syndrome

26. Interleaving Arrange data in a non-contiguous way in order to increase performancedatacontiguous Interleaving is mainly used in data communication, multimedia file formats, radio transmission (for example in satellites) or by ADSLmultimediafile formats radiotransmissionsatellitesADSL Protect the transmission against burst errorsburst errors Example –Without interleaving –With interleaving

27. ARQ, FEC, HEC ARQ Forward Error Correction (error correct coding) Hybrid Error Correction txrx Error detection code ACK/NACK txrx Error correction code txrx Error detection/ Correction code ACK/NACK