Division of Engineering and Applied Sciences DIMACS-04 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

Division of Engineering and Applied Sciences slide 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

Division of Engineering and Applied Sciences slide 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

Division of Engineering and Applied Sciences slide 4 t Channel XnXn YnYn S R Xn*Xn* Perfect timing

Division of Engineering and Applied Sciences slide 5 System Under Timing Uncertainty t Channel XnXn YLYL S R difference between transmitter and receiver clock basic assumption: clock mismatch always present

Division of Engineering and Applied Sciences slide 6 A More Realistic Case 0-TT2T3Tt 1 Sample instants: kT  kT+  k

Division of Engineering and Applied Sciences slide 7 Properties of the timing error Brownian Motion Process (slow varying). Discrete samples form a Markov chain. t tt

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

Division of Engineering and Applied Sciences slide 9 Traditional Phase Locked Loop

Division of Engineering and Applied Sciences slide 10 Simplest iterative timing reovery

Division of Engineering and Applied Sciences slide 11 Simulation results

Division of Engineering and Applied Sciences slide 12 Convergence speed

Division of Engineering and Applied Sciences slide 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

Division of Engineering and Applied Sciences slide 14 Quantizing the Timing Offset Uniformly quantize the interval ((k-1)T, kT] to Q levels. 0-TT2T 3T t 1

Division of Engineering and Applied Sciences slide 15 Math Model for Timing Error State Transition Diagram: State Transition Probability: 0θ2θ -2θ-θ δ δ

Division of Engineering and Applied Sciences slide 16 States for Timing Error 0-TT2T3T t 1 Semi-open segment : ((k-1)T, kT]: Q 1-sample states: 1 i i=1, 2, …, Q 1 deletion states: sample state: 2

Division of Engineering and Applied Sciences slide 17 Example: timing error realization kk k T/Q -T/Q -2T/Q -3T/Q -4T/Q -5T/Q = -T Q = 5

0T2T3T4T5T6T7T8T9T10T 0-  0 3T-  3 2T-  2 5T-  5 4T-  4 T-  1 6T-  6 7T-  7 8T-  8 9T-  9 t 0th interval 1st interval 2nd interval 3rd interval 4th interval 5th interval 8th interval 6th interval 7th interval 10th interval 9th interval

Division of Engineering and Applied Sciences slide 19 Single trellis section

Division of Engineering and Applied Sciences slide 20 Source Model Second order Markov chain -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

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

(-1,-1) (-1,1) (1,-1) (1,1) (-1,-1) (-1,1) (1,-1) (1,1) b) ISI trellis (-1,-1,0) (-1,-1,1 1 ) (-1,1,1 1 ) (1,-1,1 1 ) (1,1,1 1 ) (1,-1,2) (1,1,2) (-1,-1,1 2 ) (-1,-1,0) (-1,-1,1 1 ) (-1,1,1 1 ) (1,-1,1 1 ) (1,1,1 1 ) (1,-1,2) (1,1,2) (-1,1,1 2 ) … … … … … … … … … c) joint ISI-timing trellis -2T0-TT2T3T 1 0 h(t)h(t) 3T/5-2T/58T/5 a) pulse example Joint Trellis Example

Division of Engineering and Applied Sciences slide 23 Soft-Output Detector

Division of Engineering and Applied Sciences slide 24 Definition of Some Functions Notation: Definition:

Division of Engineering and Applied Sciences slide 25 Calculation of the Soft-outputs

Division of Engineering and Applied Sciences slide 26 Recursion of α(t,m,i)

Division of Engineering and Applied Sciences slide 27 Recursion of β(t,m,i)

Division of Engineering and Applied Sciences slide known timing after 10 iterations after 2 iterations conventional 10 iterations after 4 iterations bit error rate SNR per bit (dB)

Division of Engineering and Applied Sciences slide 29 Cycle-slip correction results T -T 0 T true timing error timing error estimate after 1 iteration timing error estimate after 2 iterations timing error estimate after 3 iterations time timing error

Division of Engineering and Applied Sciences slide 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

Division of Engineering and Applied Sciences slide known timing after 10 iterations after 2 iterations conventional 10 iterations after 4 iterations bit error rate SNR per bit (dB) loss due to timing error Can we compute this loss?

Division of Engineering and Applied Sciences slide 32 Open Problems Information Theory for channels with synchronization error: –Capacity –Capacity achieving distribution –Capacity achieving codes

Division of Engineering and Applied Sciences slide 33 Deletion channels Transmitted sequence x 1, x 2, x 3, …. –X k  { 0, 1 } Received sequence y 1, y 2, y 3, …. –Sequence y is a subsequence of sequence x Symbol x k is deleted with probability 

Division of Engineering and Applied Sciences slide 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

Division of Engineering and Applied Sciences slide 35

Division of Engineering and Applied Sciences slide 36 Numerical capacity computation methods

Division of Engineering and Applied Sciences slide 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,

Division of Engineering and Applied Sciences slide 38 Capacity per transmitted symbol upper bound compute

Division of Engineering and Applied Sciences slide 39 Markov sources P 00 /0 P 11 /1 P 01 /1P 10 /0 s t-1 stst Prob/ x t Q 00 /0 Q 11 /1 Q 01 /1Q 10 /0 s t-1 stst Prob/ y t If X is a first-order Markov source (transition matrix P), then Y is also a first-order Markov source (transition matrix Q)

Division of Engineering and Applied Sciences slide 40 Trellis for Y | X (1-  )/1 s0s0 s1s1 Prob/ y 1 s2s2 (1-  )  /0 (1-  )  2 /0 (1-  )  3 / (1-  )/0 (1-  )  /0 (1-  )  2 /1 (1-  )/0 (1-  )/1 (1-  )  /1 Prob/ y 2 …… … … … … Run a reduced-state BCJR algorithm on tis trellis to upper-bound H(Y|X)

Division of Engineering and Applied Sciences slide 41

Division of Engineering and Applied Sciences slide 42 Future research Upper bounds for insertion/deletion channels? Channels with non-integer timing error? Codes? (long run-lengths are favored in deletion channels)