Iterative Timing Recovery

Transcription

1 Iterative Timing Recovery John R. Barry School of Electrical and Computer Engineering, Georgia Tech Atlanta, Georgia U.S.A. 0

2 Outline Timing Recovery Tutorial Problem statement TED: M&M, LMS, S-curves PLL Iterative Timing Recovery Motivation (powerful FEC) 3-way strategy Per-survivor strategy Performance comparison 1

3 The Timing Recovery Problem Receiver expects the k-th pulse to arrive at time kt: 0 T 2T 3T 4T 5T τ 4 Instead, the k-th pulse arrives at time kt + τ k. Notation: τ k is offset of k-th pulse. 2

4 Sampling Best sampling times are {kt + τ k }. r( t ) kt + τˆ k to digital detector TIMING RECOVERY Estimate {τ k } 3

5 Timing Offset Models CONSTANT τ k = τ 0 τ k TIME FREQUENCY OFFSET τ k+1 = τ k + T τ k TIME RANDOM WALK τ k+1 = τ k + N (0, σ w 2 ) τ k TIME RANDOM WALK + FREQUENCY OFFSET τ k+1 = τ k + N ( T, σ w 2 ) 4

6 The PR4 Model and Notation TIMING OFFSETS AWGN a k d k τ g(t) r( t ) = k d k g(t kt τ k ) 1 D 2 {±1} {0, ±2} SINC FUNCTION + AWGN 2T 0 2T 4T Define: d k = a k a k 2 {0, ±2} = 3-level PR4 symbol dˆ k = receiver s estimate of d k τ k = timing offset τˆ k = receiver s estimate of τ k ε k = τ k τˆ k estimate error, with std σ ε. εˆ k = receiver s estimate of ε k 5

7 ML Estimate: Trained, Constant Offset The ML estimate τˆ minimizes J( τ a) = r( t ) d i g(t it τ) dt. i 2 Exhaustive search: Try all values for τ, pick one that best represents r( t ) in MMSE sense. J 2T T 0 T 2T τˆ 6

8 ANIMATION 1 7

9 Achieves Cramer-Rao Bound r(kt + τˆ ) = d i g(kt it + τˆ τ) + n i k = s k ( ε ) + n k, where ε = τ τˆ is the estimation error. The CRB on the variance of the estimation error: σ ε T 2 σ 2 1 3σ = N E sk ( ε) 2 π 2 N ε 8

10 Implementation J 2T T 0 T 2T τˆ Gradient search: i + 1 = i µ τˆ τˆ J(τ a) τ = τ τˆi Direct calculation 1 of gradient: -- J(τ a) = r( t )g (t it τ)dt 2 τ i d i = i d i r i. Remarks: Susceptible to local minima initialize carefully. Block processing. Requires training. 9

11 Conventional Timing Recovery After each sample: Step 1. Estimate residual error, using a timing-error detector (TED) Step 2. Update τˆ, using a phase-locked loop (PLL) r( t ) r k kt + τˆ k VITERBI PLL UPDATE εˆ k T.E.D. dˆ k {0, ±2} DETECTOR d k TRAINING 10

12 LMS Timing Recovery MMSE cost function: E r k d k 2 k-th sample, r k = r(kt + k ) τˆ what we want it to be LMS approach: τˆ k + 1 = τˆ k + µ εˆ k where = εˆk r k d k. τˆ 2 τˆ = τˆk 11

13 LMS TED 1 But: -- r k d k = (r k d k ) d i g(kt it + τ) 2 τˆ = τˆ τˆ i 2 τˆ τˆk = (r k d k ) d i g (kt it + τˆ τ) = (r k d k ) d i p ( ε k ) i k i where p ( ε k ) n = g(nt ε): τ i 0 12

14 From LMS to Mueller & Müller 0 εˆk (r k d k )(d k 1 d k + 1 ) + smaller terms = r k d k 1 r k d k + 1 d k d k 1 + d k d k + 1 Independent of τ Delay second term and eliminate last two: εˆk r k d k 1 r k 1 d k Mueller & Müller (M&M) TED 13

15 0.5T S Curves 0 0.5T 0 0.5T 0.5T TIMING ERROR, ε 14 ^ ε = ε TRAINED M&M, LMS AVERAGE TED OUTPUT, E[ε ε] ^ SNR = 2 σ 2 = 10 db

16 0.5T LMS is Noisier 0 0.5T 0 0.5T 0.5T TIMING ERROR, ε 15 ^ ε = ε TRAINED M&M TRAINED LMS AVERAGE TED OUTPUT, E[ε ε] ^ ONE STD SNR = 2 σ 2 = 10 db

17 An Interpretation of M&M Consider the complex signal r( t ) + jr(t T). Its noiseless trajectory: It passes through {0, ±2, ±2 ± 2j, ±2j} at times {kt + τ}. More often than not, in a counterclockwise direction. 16

18 ANIMATION 2 17

19 Sampling Late by 20% R k = r k + jr k 1 D k = d k + jd k 1 θ The angle between R k and D k predicts timing error: θ sinθ = Im R*D r k d k 1 r k 1 d k M&M. RD 18

20 Decision-Directed TED Replace training by decisions { dˆ k } εˆ k r k dˆ k 1 r k 1 dˆ k. Instantaneous decisions: Hard: Round r k to nearest symbol. Soft: 2sinh( 2r dˆ k= E[d k r k ] = k σ 2 ) : cosh( 2r k σ 2 ) e 2 + σ2 dˆ db 10 db 5 db r 19

21 0.5T 0 0.5T 0 0.5T 0.5T TIMING ERROR, ε 20 TRAINED M&M, LMS HARD M&M SOFT M&M AVERAGE TED OUTPUT, E[ε ε] ^ SNR = 2 σ 2 = 10 db

22 0.5T ^ AVERAGE TED OUTPUT, E[ε ε] 0 HARD SOFT 0.5T 0.5T 0 0.5T TIMING ERROR, ε SNR = 2 σ 2 = 10 db 21

23 Reliability versus Delay Three places to get decisions: r( t ) r k = r(kt + τˆ k ) TRELLIS EQUALIZER DECODER kt + τˆ k z D PLL UPDATE εˆ k T.E.D. r k D A B τˆk 1 = + α + τˆk εˆk d k D C Inherent trade-off: reliability versus delay. to get more reliable decisions requires more decoding delay D delay decreases agility to timing variations 22

24 Decision Delay Degrades Performance 5.5 RMS TIMING JITTER σ ε T (%) TRAINED M&M 1st-ORDER PLL E b N 0 = 8 db σ w T = 0.5% D = PLL GAIN, α D = 15 D = 10 D = 5 D = 0 23

25 The Instantaneous-vs-Reliable Trade-Off JITTER σ ε T 7% 6% 5% DELAY-100, PERFECT DECISIONS 4% 3% 2% 1% SNR (db) DELAY-20, PERFECT DECISIONS INSTANTANEOUS BUT UNRELIABLE DECISIONS Parameters Averaged over 40,000 bits random walk σ w T = 0.5% 1st-order M&M PLL α optimized for SNR = 10 db Delay α opt Reliability becomes more important at low SNR. 24

26 Linearized Analysis Assume εˆk = ε k + independent noise = τ k τˆ k + n k 1st order PLL, τˆ k + 1 = τˆ k + α(τ k τˆ k + n k ), is a linear system: τ k NOISE 1-st order LPF αz ( 1 α)z 1 τˆk α Ex: Random walk w k WHITE z 1 τ k WHITE 1-st order LPF αz ( 1 α)z 1 + τˆk derive optimal α to minimize σ2 ε ε k 25

27 The PLL Update 1st-order PLL: τˆ k + 1 = τˆ k + α εˆ k Already introduced using LMS Easily motivated intuitively: if εˆ k is accurate, α = 1 corrects in one step Smaller α attenuates noise at cost of slower response 2nd-order PLL: τˆ k + 1 = τˆ k + αεˆ k + β εˆ n k n = Accumulate TED output to anticipate trends P+I control Closed-loop system is second-order LPF Faster response Zero steady-state error for frequency offset 0 0 TIME 26

28 Equivalent Views of PLL Analysis: Sample at {kt + τˆ k }, where Implementation: τˆ k + 1 = τˆ k + α εˆ k + β εˆ n, εˆ k is estimate of timing error at time k. k n = A D VCO Accumulation implicit in VCO β α z 1 LOOP FILTER εˆk TED PHASE DETECTOR 27

29 Iterative Timing Recovery Motivation Powerful codes low SNR timing recovery is difficult traditional PLL approach ignores presence of code Key Questions How can timing recovery exploit code? What performance gains can be expected? Is it practical? 28

30 A Canonical Example Simplest possible channel model: {±1} alphabet, ideal ISI-free pulse shape constant timing offset τ AWGN. Add a rate-1 3 turbo code with {±1} alphabet: Rate-1 3 Turbo Code m 2048 bits S RANDOM INTERLEAVER π 1 D D D 2 1 D D D 2 mux a g(t τ) 0% xsbw AWGN r( t ) = i a i g(t it τ) + AWGN Problem: Recover message in face of unknown noise, timing offset 29

31 Iterative ML Timing Recovery The ML estimator with training minimizes: J( τ a) = r( t ) a i g(t it τ) dt i 2 Without training, the ML estimator minimizes E a [J( τ a)]. An EM-like approach: τˆ TIMING ESTIMATOR SYMBOL ESTIMATOR { ã k } Useful in concept, but overstates complexity. For example, the timing estimator might itself be iterative: τˆ i + 1 = τˆ i µj ( τˆ i { ã ). i } 30

32 A Reduced-Complexity Approach Collapse three loops to a single loop. τˆ Initialize 0 Iterate for i = 0, 1, 2, decode component 1 decode component 2 update timing estimate, i + 1 = i µj ( i { }) interpolate end τˆ τˆ τˆ ã i As a benchmark, an iterative receiver that ignores the presence of FEC will ( i) replace the pair of decoders by ã k = tanh (( r( kt + τˆi )) σ 2 ). 31

33 Results 5% 4% 3% Parameters RMS TIMING ERROR, σ ε T τ = 0.123T, AWGN channel Rate-1 3 Turbo Code K = 2048 message bits N = 6150 coded bits S = 16 random interleaver, length inner per outer iteration Averaged over 180 trials σ ε 2 = E[( ) 2 ] τˆ τ 2% 1% 0.5% 0.4% 0.3% EXPLOIT FEC Trained ML, Cramer-Rao Bound IGNORE FEC 4 db 0.2% E b N 0 (db) 32

34 New Model: Random Walk and ISI Equivalent equalized readback waveform: r(t) = k a k h(t kt τ k ) + AWGN. System diagram: m k LDPC ENCODE c k TIMING OFFSET 1 a k 1 D 2 τ PRECODER PR4 h(t) AWGN r(t) Rate-8 9 (4095, 3640) irregular LDPC code with node-degree distribution polynomials: h(t) = g sinc (t) g sinc (t 2T) 2T 0 2T 4T λ ( x) = x x x x 7 bit ρ ( x) = x x 30 check 33

35 Conventional Turbo Equalizer + PLL PR trellis r(t) r k BCJR kt + τˆk apriori + PLL + BIT NODES CHECK NODES λ k = LLR(c k ) final decisions 34

36 Performance of Conventional Approach 1 WORD-ERROR RATE σ w T = 1% (α = 0.055) σ w T = 0.5% (α = 0.04) KNOWN TIMING Parameters Known Timing: 10 5 Conventional: 25 5 max words SNR per bit (db) Big penalty as σ w T increases: cycle slips. 35

37 Cycle-Slip Example T ESTIMATE OF TIMING OFFSET ( ) τˆk 0 T CYCLE SLIP ACTUAL τ k ESTIMATE τˆk T Parameters SNR bit = 5.0 db σ w T = 1.0% α = T TIME k (in bit periods) 36

38 Nested Loops Iterate between PLL and turbo equalizer, which in turn iterates between BCJR and LDPC decoder, which in turn iterates between bit nodes and check nodes. { τˆk } TURBO EQUALIZER { LLR( c k )} LDPC DECODER r(t) PLL BCJR EQ BIT NODES CHECK NODES { LLR( c k )} { LLR( d k )} 37

39 A Decoder-Centric View FROM BCJR PLL & RESAMPLE & BCJR BIT NODES CHECK NODES 38

40 Iterative Timing Recovery and TEQ r(t) kt + τˆk r k INTERPOLATE BCJR apriori + PLL + BIT NODES CHECK NODES PLL d k λ k final decisions 39

41 Automatic Cycle-Slip Correction Iterative receiver automatically corrects for cycle slips: ESTIMATE OF TIMING OFFSET ( ) τˆk T 0 T ACTUAL τ N = T Parameters 1.5T SNR bit = 5.0 db σ w T = 1.0% α = N 5 iterations TIME k (in bit periods) 40

42 3 Approaches to Timing Recovery Conventional: 3-Way Iterative Timing Recovery: TURBO EQ TIMING RECOVERY EQ DECODE EQUALIZATION TIMING RECOVERY & SAMPLING Per-Survivor Iterative Timing Recovery: DECODER PS-BCJR DECODER 41

43 Per-Survivor Processing [Raheli, Polydoros, Tzou 91-95] A general framework for estimating Markov process with unknown parameters and independent noise Basic idea: Add a separate estimator to each survivor of Viterbi algorithm Has been applied to channel identification, adaptive sequence detection, carrier recovery Application to timing recovery [Kovintaveat et al., ISCAS 2003]: Start with traditional Viterbi algorithm on PR trellis Run a separate PLL on each survivor, based on its decision history Motivations: ➊ PLL is fully trained whenever correct path is chosen! ➋ Can avoid decision delay altogether 42

44 Per-Survivor BCJR? Motivation: Exploit PSP concept in iterative receiver: PS-BCJR DECODER Problem: BCJR algorithm has no survivors. Proposal: Add depth-one survivor for purposes of timing recovery only. The result is PS-BCJR Start with traditional BCJR algorithm on PR trellis Embed timing-recovery process inside Run multiple PLL s in parallel, one for each survivor 43

45 PS-BCJR Branch Metric Key: Each node p {0,1,2,3} in trellis at time k has its own τ k (p), an estimate of the timing offset τ k. r k (p) = r k (kt + τ k (p)), corresponding sample The branch metrics depend on samples of the starting state: 0 [, ] 1 [+, ] τ k (0) τ k (1) γ k (2,1) = exp{ r k (2) d (2,1) 2 (2σ 2 ) + a (2,1) λ k a 2} 2 [,+] τ k (2) 3 [+,+] τ k (3) 44

46 Per-Survivor BCJR: Forward Recursion Associate with each node p {0,1,2,3} at time k the following: forward metric α k ( p ) predecessor π k ( p ) forward timing offset estimate τ k ( p ) α k+1 (1) = = 66 π k+1 (1) = argmax 0,2 {7 3, 9 5} = 2 τ k+1 (1) = τ k ( 2 ) + µ(r k (2)d ( π k (2),2) r k 1 (π k (2))d (2,1) ) r(kt + τ k (2)) r((k 1)T + τ k 1 (π k (2))) Notation complicated, but idea is simple: update blue node timing using M&M PLL driven by the samples & inputs corresponding to blue branches. 45

47 Backward Recursion Associate with each node p {0,1,2,3} at time k the following backward metric β k ( p ) successor σ k ( p ) backward timing offset estimate τ k b (p) β k (1) = = 26 σ k (1) = argmax 2,3 {8 1, 9 2} = 3 τ b b k (1) = τ k+1 (3) + µ(r k+1 (σ k+1 (3))d (1,3) r k (3)d ( 3,σ k+1 (3)) ) r((k+1)t + τ b k+2(σk+1 (3))) r(kt + τ b k+1 (3)) Again, update blue node timing using backward M&M PLL driven by samples & inputs corresponding to blue branches

48 Compare Forward/Backward Timing Backward timing estimates can exploit knowledge of forward estimates. Option 1: Ignore forward estimates during backward pass. Option 2: Average backward with forward estimate whenever they differ by more than some threshold (say 0.1T) in absolute value. 47

49 New Encoder PRECODER m k 1 D D 3 D 4 1 D D 4 PUNCTURE π S-RANDOM INTERLEAVER (s = 16) c k 1 1 D 2 a k RATE-8 9 LENGTH 4095 RSC ENCODER 48

50 Compare 3 Systems Conventional: TURBO EQ 3-Way Iterative Timing Recovery: Complexity = #IT (PLL + BCJR + DEC) TIMING RECOVERY EQ DECODE EQUALIZATION TIMING RECOVERY & SAMPLING Per-Survivor Iterative Timing Recovery: Complexity = #IT (8 PLL + BCJR + DEC) DECODER PSP-BCJR DECODER 49

51 Moderate Random Walk (σ w T = 0.5%) SNR PER BIT, E b N 0 (db) 50 CONVENTIONAL TIMING RECOVERY ITERATION 3-WAY ITERATIVE TIMING RECOVERY BER (50, 100) PS (50) (TRAINED PLL) (KNOWN TIMING)

52 Severe Random Walk (σ w T = 1%) BER SNR PER BIT, E b N 0 (db) 51

53 Example: PSP Corrects Quickly 2.5T σ w T = 1% 2T TIMING Time (in bit ESTIMATE periods) 1.5T T 0.5T ACTUALActual τ τ 5 PSP 5 PSP 2 2 PSP PSP 1 NON-PSP, 50 ITERATIONS ITERATIVE I CTR (50) TIMING RECOVERY 0.5T TIME Timing (in BIT Estimate PERIODS) 52

54 Convergence Rate: (σ w T = 1%) 1 CONVENTIONAL SECTOR-ERROR RATE ITERATIVE 3-WAY ITERATIVE PS (TRAINED PLL) σ w T = 1% E b N 0 = 5 db (KNOWN TIMING) NUMBER OF ITERATIONS 53

55 How Long do Cycle Slips Persist? Pr[ SLIP AT ITERATION k SLIP ] ITERATIVE PS σ w T = 1% E b N 0 = 5 db ITERATIVE 3-WAY NUMBER OF ITERATIONS, k 54

56 Summary Powerful codes permit low SNR conventional strategies fail exploiting code is critical Problem is solvable We described two strategies for iterative timing recovery Embed timing recovery inside turbo equalizer Automatically corrects for cycle slips Challenges remaining complexity close gap to known timing 55