Iterative Timing Recovery

1. Iterative Timing Recovery Aleksandar Kavčić Division of Engineering and Applied Sciences Harvard University based on a tutorial by Barry, Kavčić, McLaughlin, Nayak & Zeng And on research by Motwani and Kavčić

2. Outline • Motivation • Timing model • Conventional timing recovery • Simple iterative timing recovery • Joint timing and intersymbol interference trellis • Soft decision algorithm • Performance results • Conclusion • Future challenge: capacity of channels with synchronization error

3. Motivation • In most communications (decoding) scenarios, we assume perfect timing recovery • This assumption breaks down, particularly at low signal-to-noise ratios (SNRs) • But, turbo-like codes work exactly at these SNRs • Need to take timing uncertainty into account

4. Xn* Xn Yn S R Channel t Perfect timing

5. Xn YL S R Channel t System Under Timing Uncertainty difference between transmitter and receiver clock basic assumption: clock mismatch always present

6. 1 -T 0 T 2T 3T t A More Realistic Case Sample instants: kT  kT+k

7. t t Properties of the timing error • Brownian Motion Process (slow varying). • Discrete samples form a Markov chain.

8. Timing recovery strategies turbo equalization (inner loop) a) c) timing recovery symbol detection timing recovery symbol detection decoding decoding   free running oscillator free running oscillator iterative timing recovery (outer loop) turbo timing/equalization b) d) turbo equalization timing recovery symbol detection joint soft timing recovery and symbol detection decoding decoding   free running oscillator free running oscillator

9. Traditional Phase Locked Loop

10. Simplest iterative timing reovery

11. Simulation results

12. Convergence speed

13. Strategy to solve the problem • Set up math model for timing error (Markov). • Build separate stationary trellis to characterize the channel and source. • Form a full trellis. • Derive an algorithm to perform the Maximum a posteriori probability (MAP) estimation of the timing offset and the input bits

14. 1 -T 0 T 2T t 3T Quantizing the Timing Offset Uniformly quantize the interval ((k-1)T, kT] to Q levels.

15. 0 θ 2θ δ -2θ -θ δ Math Model for Timing Error State Transition Diagram: State Transition Probability:

16. 1 t -T 0 T 2T 3T States for Timing Error Semi-open segment : ((k-1)T, kT]: Q 1-sample states: 1i i=1, 2, …, Q 1 deletion states: 0 1 2-sample state: 2

17. k Q = 5 T/Q 4 5 6 7 8 9 10 11 12 13 14 15 k 0 0 1 2 3 -T/Q -2T/Q -3T/Q -4T/Q -5T/Q = -T Example: timing error realization

18. 0 T 2T 3T 4T 5T 6T 7T 8T 9T 10T 0th interval 1st interval 2nd interval 3rd interval 4th interval 5th interval 6th interval 7th interval 8th interval 9th interval 10th interval 15 15 14 14 15 0 2 0 11 11 12 t 0-0 T- 1 2T- 2 3T- 3 4T- 4 5T- 5 6T- 6 7T- 7 8T- 8 9T- 9 0 11 12 13 14 15 2

19. Single trellis section 0 0 11 11 12 12 13 13 14 14 15 15 2 2

20. 1, -1 -1, -1 -1, -1 -1, 1, -1 1, 1 -1, -1 -1, -1 1, 1 1, -1 -1, -1, -1 -1, -1 -1, 1, -1 1, 1 -1, -1 -1, -1 -1, 1, 1 -1, 1 -1, 1 -1, 1 -1, 1 Source Model Second order Markov chain

21. Full states set: Full Trellis Total number of states at each time interval: Trellis length = n (block length). (note that each branch may have different number of outputs).

22. c) joint ISI-timing trellis (-1,-1,0) (-1,-1,0) … … … (-1,-1,11) (-1,-1,11) (-1,1,11) (-1,1,11) (1,-1,11) (1,-1,11) b) ISI trellis (1,1,11) (1,1,11) … … … (-1,-1) (-1,-1) (-1,-1,12) (-1,1,12) (-1,1) (-1,1) … … (1,-1) (1,-1) (1,-1,2) (1,-1,2) … (1,1) (1,1) (1,1,2) (1,1,2) Joint Trellis Example a) pulse example 1 h(t) 0 -2T -T 0 T 2T 3T -2T/5 3T/5 8T/5

23. Soft-Output Detector

24. Notation: Definition of Some Functions Definition:

25. Calculation of the Soft-outputs

26. Recursion of α(t,m,i)

27. Recursion of β(t,m,i)

28. SNR per bit (dB) known timing conventional 10 iterations 10-1 after 2 iterations after 4 iterations after 10 iterations 10-2 bit error rate 10-3 10-4 2 3 4 5 6

29. time Cycle-slip correction results true timing error timing error estimate after 1 iteration timing error estimate after 2 iterations timing error estimate after 3 iterations T 0 timing error -T -2T 1000 2000 3000 4000 5000

30. Conclusion • Conventional timing recovery fails at low SNR because it ignores the error-correction code. • Iterative timing recovery exploits the power of the code. • Performance close to perfect timing recovery • Only marginal increase in complexity compared to system that uses conventional turbo equalization/decoding

31. SNR per bit (dB) known timing conventional 10 iterations 10-1 after 2 iterations after 4 iterations after 10 iterations 10-2 bit error rate 10-3 loss due to timing error Can we compute this loss? 10-4 2 3 4 5 6

32. Open Problems • Information Theory for channels with synchronization error: • Capacity • Capacity achieving distribution • Capacity achieving codes

33. Deletion channels • Transmitted sequence x1, x2, x3, …. • Xk{ 0, 1 } • Received sequence y1, y2, y3, …. • Sequence y is a subsequence of sequence x • Symbol xk is deleted with probability 

34. Deletion channels • Some results: • Ulmann 1968, upper bounds on the capacities of deletion channels • Diggavi&Grossglauser 2002, analytic lower bounds on capacities of deletion channels • Mitzenmacher 2004, tighter analytic lower bounds

36. Numerical capacity computation methods

37. Received symbols per transmitted symbol Let K(m) denote the number of received symbols per m transmitted symbols K(m) is a random variable Asymptotically, we have A received symbols per transmitted symbol For the deletion channel,

38. Capacity per transmitted symbol compute upper bound

39. Markov sources If X is a first-order Markov source (transition matrix P), then Y is also a first-order Markov source (transition matrix Q) Prob/xt Prob/yt st-1 st st-1 st P00/0 Q00/0 0 0 0 0 P01/1 P10/0 Q01/1 Q10/0 1 1 1 1 P11/1 Q11/1

40. Trellis for Y | X Prob/y1 Prob/y2 s0 s1 s2 11 02 (1-)/1 (1-)/0 0 … (1-)/0 (1-)/0 Run a reduced-state BCJR algorithm on tis trellis to upper-bound H(Y|X) 02 03 (1-)/0 … (1-)2/1 (1-)2/0 03 14 (1-)/1 … (1-)3/1 (1-)/1 14 15 (1-)/1 … … …

42. Future research • Upper bounds for insertion/deletion channels? • Channels with non-integer timing error? • Codes? (long run-lengths are favored in deletion channels)