Introduction to Data Communication: the discrete channel model

1. Introduction to Data Communication: the discrete channel model A.J. Han Vinck University of Essen April 2005

2. content • communication model • transmission model • MAP-ML receiver • burst error model • Interleaving: block-convolutional • several models

3. The communication model k K‘ data reduction/ compression data protection source n K‘ k Message construction decoder sink

4. Point-to-point transmitter channel receiver modem physical modem message  bits bits  message Signal generator Signal processor

5. transmission model (OSI) Data Link Control Data Link Control Transmission of reliable packets Physical Physical Unreliable trans-mission of bits link

6. transmission channel model input xiP(y|xi) output y transition probabilities • memoryless: • output only on input • input and output alphabet finite

7. binary symmetric channel model (BSC) • 1-p • 0 0 • p • 1 • 1-p Error Source e yi = xi e xi + Output Input E is the binary error sequence s.t. P(1) = 1-P(0) = p Xi is the binary information sequence for message i Y is the binary output sequence

8. Error probability (MAP) Suppose decision is message i for a received vector Y then, the probability of a correct decision = P( Xi transmitted | Y received ) Hence, decide i that maximizes P( Xi transmitted | Y received ) (Maximum Aposteriori Probability, MAP)

9. Maximum Likelihood (ML) receiver find i that maximizes P( Xi | Y ) = P( Xi , Y ) / P( Y ) = P( Y |Xi ) P ( Xi ) / P( Y ) for equally likely Xi this is equivalent to find maximum P( Y | Xi )

10. example For p = 0.1 and X1 = ( 0 0 ); P( X1 = 1/3 ) X2 = ( 1 1 ); P( X0 = 2/3) Give your MAP and ML decision for Y = ( 0 1 )

11. Something to think about message  bits message  compression  protection of bits MPEG, JPEG, etc Error correction bits  message correction of incorrect bits  decompression  message Compression reduces bit rate Protection increases bit rate

12. Bit protection • Obtained by Error Control Codes (ECC) • Forward Error Correction (FEC) • Error Detection and feedback (ARQ) • Performance depends on error statistics! • Error models are very important

13. Error control code with rate k/n Code book Code word in receive message estimate 2k decoder channel Code book contains all processing n There are 2k code words of length n

14. example Transmit: 0 0 0 or 1 1 1 How many errors can we correct? How many errors can we detect? Transmit: A = 00000; B = 01011; C = 10101; D = 11110 How many errors can we correct? How many errors can we detect? What is the difference?

15. A simple error detection method 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 Fill row wise Transmit column wise RESULT: any burst of length L can be detected L 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 row parity What happens with bursts of length larger than L?

16. Modeling: binary transmission test sequence error sequence channel Problem: 0 1 0 0 0 1 0 1 0 0 0 0 1 1 1 ••• Determination of burst and guard space burst guard burst

17. modeling How to model scratches on a CD? Answer is important for the design of ECC

18. Density increases sensitivity Blue Laser CD DVD

19. Modeling: networking packet - 1-error causes retransmission - long packets always have an error - short packets with ECC give lower efficiency Ack/Nack Suppose that a packet arrives correctly with probability Q. What is then the throughput as a funtion of Q?

20. burst error model Random error channel; outputs independent P(0) = 1- P(1); Error Source Burst error channel; outputs dependent P(0 | state = bad ) = P(1|state = bad ) = 1/2; P(0 | state = good ) = 1 - P(1|state = good ) = 0.999 Error Source State info: good or bad transition probability Pgb Pbb Pgg good bad Pbg

21. question Pgb Pbb Pgg good bad Pbg P(0 | state = bad ) = P(1|state = bad ) = 1/2; P(0 | state = good ) = 1 - P(1|state = good ) = 0.99 What is the average for P(0) for: Pgg = 0.9, Pgb = 0.1; Pbg = 0.99, Pbb = 0.01 ? Indicate how you can you extend the model?

22. Interleaving: block Channel models are difficult to derive: - burst definition ? (a burst starts and ends with a 1) - random (?) and burst errors ? for practical reasons: convert burst into random error read in row wise transmit column wise 1 0 0 1 1 0 1 0 0 1 1 0 000 0 0 1 1 0 1 0 0 1 1

23. received power time Reception after fading channel Example (from Timo Korhonen, Helsinki) • In fading channels received data can experience burst errors that destroy large number of consecutive bits. This is harmful for channel coding • Interleaving distributes burst errors along data stream • A problem of interleaving is introduced extra delay • Example below shows block interleaving: Received interleaved data: 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 Block deinterleaving : Recovered data: 1 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 1

24. A1 A1 A1 A1 A1 A2 B1 B1 A2 A2 A3 A3 C1 A3 C1 A2 A2 B1 B1 B1 B2 B2 B2 B2 B2 B3 B3 B3 C2 C2 A3 C1 C1 A3 C1 C2 B3 C2 B3 C2 C3 C3 C3 C3 C3 example • Consider the code C = { 000, 111 } • A burst error of length 3 can not be corrected. • Let us use a block interleaver 3X3 2 errors Interleaver Deinterleaver 1 error 1 error 1 error

25. De-Interleaving: block read in column wise this row contains 1 error 1 0 0 1 1 0 1 0 0 1 1 e e e e e e 1 1 0 1 0 0 1 1 read out row wise

26. Interleaving: convolutional input sequence 0 input sequence 1 delay of b elements  input sequence m-1 delay of (m-1)b elements Example:b = 5, m = 3 in out

27. Interleaving: destroys memory bursty Message interleaver channel interleaver -1 message encoder decoder „random error“ Note: interleaving brings encoding and decoding delay Homework: compare the block and convolutional interleaving w.r.t. delay

28. Middleton type of burst channel model 0 1 0 1 Transition probability P(0) channel 1 channel 2 Select channel k with probability Q(k) … channel k has transition probability p(k)

29. Impulsive Noise Classification (a) Single transient model • Parameters of single transient : • peakamplitude - pseudo frequency f0 =1/T0- damping factor- duration- Interarrival Time Measurements carried out by France Telecom in a house during 40 h 2 classes of pulses (on 1644 pulses) : single transient and burst

30. the Z-channel Application in optical communications 0 1 0 (light on) 1 (light off) x y p 1-p P( x = 0 ) = 1 - P( x = 1) =P0

31. the erasure channel Application: cdma detection, disk arrays 1-e e e 1-e 0 1 0 E 1 Disk 1 Disk 2 x y Disk 3 Known position of error Disk 4 Disk 5 P( x = 0) = 1 – P( x = 1 ) = P0

32. From Gaussian to binary to erasure e + + xi = +/- yi = xi+ e + - - Output Input - + + + E E E - - - +

33. A Simple code • For low packet loss rates (e.g. 5%), sending duplicates is expensive (wastes bandwidth) • XOR code • XOR a group of data pkts together to produce repair pkt • Transmit data + XOR: can recover 1 lost pkt     10101 00111 11100 11000 10110     10101 10110 11100 11000 00111

34. Channel with insertions and deletions • Bad synchronization or clock recovery at receiver: • insertion ••• 1 0 0 0 1 1 1 0 0 1 0 •••  •••1 0 0 1 0 1 1 1 0 0 1 0 ••• • deletion ••• 1 0 0 0 1 1 1 0 0 1 0 •••  •••1 0 0 1 1 1 0 0 1 0 ••• Problem: finding start and end of messages

35. Channel with insertions and deletions • Due to errors in bit pattern flag = 1 1 1 1 1 0, avoid 1 1 1 1 1 in frame ••• 0 1 1 1 1 1 0 0 1 1 0 1 •••  ••• 0 1 1 1 1 0 1 0 0 1 1 0 1 ••• ••• 0 1 1 0 1 0 1 0 0 1 1 0 1 •••  insertion ••• 0 1 1 1 0 0 0 0 1 1 0 1 •••  ••• 0 1 1 1 1 0 0 0 1 1 0 1 ••• ••• 0 1 1 1 1 0 0 1 1 0 1 •••  deletion

36. Channels with interference • Example (optical channel) Error probability depends on symbols in neighboring slots

37. Channels with memory (ex: recording) • Example: Yi = Xi + Xi-1 Xi  { +1, -1 } Xi Xi-1 Yi  { +2, 0, -2 }

38. tasks • Construct a probability transformer from uniform to Gaussian • Give an overview of burst error models, statistics of important parameters