Timing Recovery at Low SNR Cramer-Rao bound, and outperforming the PLL

Transcription

1 T F T I G E O R G A I N S T I T U T E O H E O F E A L P R O G R ESS S A N D S E R V L O G Y I C E E C H N O Timing Recovery at Low SNR Cramer-Rao bound, and outperforming the PLL Aravind R. Nayak John R. Barry Steven W. McLaughlin {nayak, barry, swm}@ece.gatech.edu Georgia Institute of Technology

2 Communication system model Source ECC Encoder Modulator Transmitter Channel Channel ECC Decoder Equalizer Sampler Receiver Discrete-time Continuous-time 1

3 Continuous-time discrete-time interface Source ECC Encoder Modulator Discrete-time to Continuous-time Channel Continuous-time ECC Decoder Equalizer Sampler to Discrete-time 2

4 Sampling: Timing recovery 0 T 2T 3T a 0 a 1 a 3 τ 0 τ 1 τ 2 τ 3 TIME T Symbol duration a 0, a 1, a 2,... Data symbols τ 0, τ 1, τ 2,... Timing offsets a 2 Timing Recovery Problem: Estimate τ 0, τ 1, τ 2,... 3

5 Timing offset models Constant offset: τ k = τ Frequency offset: τ k = τ 0 + k T = τ k 1 + T Random walk: τ k 1 + = τ k + w k = τ 0 + w i i = 0 where w i are i.i.d. zero-mean Gaussian random variables of variance σ 2 w. σ2 w determines the severity of the random walk. k 4

6 Timing recovery in two stages Acquisition: Estimate τ 0 Correlation techniques Known preamble sequence at start of packet (Trained mode) Parameter τ 0 spans a large range Tracking: Keep track of τ 1, τ 2, τ 3,... Based on the phase-locked loop (PLL) Data symbols unknown (Decision-directed mode) Sufficient to track small signals τ 1 τ 0, τ 2 τ 1, τ 3 τ 2,... 5

7 PLL: Motivation Consider the simple case of a time-invariant offset: τ k = τ Let τˆi be the current timing estimate. Timing error: ε i = τ i τˆi =τ τˆi. With a perfect timing error detector (TED), we get εˆi = ε i. Update: τˆi + 1 = τˆi + εˆi = τ With imperfect TED: τˆi + 1 = τˆi + αεˆi 6

8 PLL-based timing recovery y(t) f(t) rt () r k for further processing rcv. filter kt + τˆ k PLL UPDATE εˆ k T.E.D. τˆk + 1 = τˆk + αεˆk First-order PLL k 1 τˆk + 1 = τˆk + αεˆk + β εˆi i = 0 Second-order PLL 7

9 Timing Error Detector (TED) y(t) f(t) rt () r k for further processing rcv. filter kt + τˆ k PLL UPDATE εˆ k T.E.D. dˆ k εˆk = 3T ( r k dˆ k 1 ) r k 1 dˆ k Mueller & Müller Timing Error Detector TED is a decision-directed device Usually, instantaneous hard quantization Better decisions entail delay that destabilizes the loop 8

10 Improving timing recovery Improve the quality of decisions (Approach I) Need to get around the delay induced by better decisions. Feedback from the ECC decoder and equalizer to timing recovery. Dr. Barry s presentation! Improve the timing recovery architecture (Approach II) Need to assume perfect decisions for tractability. Methods based on gradient search and projection operation. Use Cramer-Rao bound to evaluate competing methods. This presentation! 9

11 Overview: Approach II Questions: How good is the PLL-based system? Can it be improved upon? Method: Derive fundamental performance limits. Compare the PLL performance with these limits. Develop methods that outperform the PLL. 10

12 Problem statement We consider the following uncoded system: { a k } N 1 0 uncoded i.i.d. τ h(t) AWGN LPF kt (uniform) r k The uniform samples are: N 1 l = 0 r k = a l h(kt lt τ l )+n k, where σ 2 is the noise variance, and h(t) is the impulse response. Problem: Given samples {r k } and knowledge of channel model, estimate the N uncoded i.i.d. data symbols {a k } the N timing offsets {τ k }. 11

13 Cramer-Rao bound (CRB) Cramer-Rao bound answers the following question: What is the best any estimator can do? is independent of the estimator itself. is a lower bound on the error variance of any estimator. 12

14 CRB, intuitively f( r θ 1 ) f( r θ 2 ) f( r θ 1 ) f( r θ 2 ) r r θ fixed, unknown parameter to be estimated r observations Sensitivity of f( r θ) to changes in θ determines quality of estimation. If f( r θ) is narrow, for a given r, probable θs lie in a narrow range. θ can be estimated better, i.e., with lesser error variance. CRB uses log f( r θ) as a measure of narrowness. θ 13

15 CRB for a random parameter If θ is random as opposed to being fixed and unknown, θ is characterized by a p.d.f. f ( θ) and r, θ are characterized by the joint p.d.f. f( r, θ). The measure for narrowness in this case is log f( r, θ) θ 14

16 CRB is the inverse of Fisher information For any unbiased estimator lower bounded by θˆ ( r), the estimation error covariance matrix is E[ ( θˆ ( r) θ) ( θˆ ( r) θ) T ] J 1 where J is the information matrix given by J E = log f( r, θ) log f( r, θ) θ θ T In particular, E[ ( θˆ i ( r) θ i ) 2 ] J 1 ( ii, ) 15

17 Efficient estimators An estimator that achieves the CRB is called efficient. Efficient estimators do not always exist. Fixed, unknown θ: ML is efficient log f( r θ) = 0 θ Random θ: MAP is efficient log f( r, θ) = 0 θ 16

18 CRB: lower bound on timing error variance Constant offset: σ2 σ 2 ε NE h' Frequency offset: σ2 6σ 2 ε ( N 1)N( 2N 1)E h' 17

19 CRB for a random walk The Cramer-Rao bound on the error variance of any unbiased timing estimator: E[ ( τˆk τ k ) 2 ] h f( k) where 2 η h = σ w η 2 1 is the steady state value, f( k) = tanh( ( N + 0.5) logη) 1 sinh( ( N ( k + 1) ) logη) sinh( ( N + 0.5) logη), η λ + λ 2 4 = and λ 2 2π σ 2 w = σ 2 T 2 18

20 CRB: Steady-state value 1.2% σ w T = 0.05% Parameters SNR bit =5dB N = 5000 σ ε T (%) 0.8% 0.4% h 0 0 Time 5000 Steady-state value becomes more representative as SNR and N increase. 19

21 Trained PLL away from the CRB ESTIMATION ERROR JITTER σ ε T (%) 5% 4% 3% 2% Trained PLL with α optimized CRLB 1% Time (bit periods) Parameters SNR = 5 db N = 4095 σ w T = 0.7% α = sectors Trained PLL does not achieve the steady-state CRB. 20

22 Trained PLL vs. Steady-state CRB 5% ESTIMATION ERROR JITTER σ ε T (%) 4% 3% 2% 1% α = 0.05 α = 0.03 CRB α = 0.1 7dB α = 0.2 α = 0.01 α = 0.02 α = 0.03 α = 0.05 Parameters σ w T = 0.5% 1000 trials N = SNR (db) 21

23 Outperforming the PLL: Block processing As in the random walk case, the PLL does not achieve the CRB in the constant offset and the frequency offset cases. Using Kalman filtering analysis, we can show that PLL is the optimal causal timing recovery scheme. Eliminate causality constraint to improve performance. Block processing. 22

24 Constant offset: Gradient search The trained maximum-likelihood (ML) estimator picks τ to minimize J( τˆ ; a) = r k a l hkt ( lt τˆ ) 2 k = l This minimization can be implemented using gradient descent: τˆi + 1 = τˆi µj' ( τˆi ; a) Initialization using PLL. Without training, use J( τˆ ; â) instead of J( τˆ ; a). 23

25 Parameters τ/t = π/20 Trained ML achieves CRB α = 0.01 N = % 10% RMS Timing Error σ ε / T 3% 2% 1% Decision-directed ML Trained PLL Trained ML, CRB Decision-directed PLL 0.3% 0.2% SNR (db) Two ways to improve performance over conventional PLL: * Better architecture ML for example. * Better decisions exploit error correction codes. 24

26 Frequency offset: Least squares estimation Let k = [ 01,,, N 1] T, τ = [ τ 0, τ 1, τ, N 1 ] T τˆ = [ τˆ0, τˆ1, τˆ, N 1 ] T, from PLL τˆk τ k 0 Model: τ = ( T)k + τ k Problem: Find Tˆ and τˆ0 to minimize τˆ (( T ˆ ) k + τ ˆ ) 0 2 Solution: Tˆ N kτˆk k τˆk = and N k 2 ( k) 2 τˆ0 = N ( τˆk k Tˆ ) 25

27 Least Squares away from CRB Parameters N = packets T/T unif[0, 0.005] τ 0 /T ~ unif[0, 0.1] α optimized RMS Estimation Error in T / T Decision-Directed Trained CRB SNR (db) RMS Estimation Error in τ 0 / T 10 1 Decision-Directed Trained CRB SNR (db) * Trained MM + PLL + LS about 2 db away from the CRB Gradient descent? 26

28 Gradient descent not suitable Given uniform samples { r k }, pick Tˆ and τˆ0 to minimize J( Tˆ, τˆ0 ; a) = r k a l hkt ( lt l Tˆ τˆ0 ) 2 k l 1500 J ( Tˆ, τˆ0 ) J( Tˆ, τˆ0 ) parabolic in τˆ Tˆ T Gradient descent sensitive to initialization proceeds along greatest gradient: rattling in the bowl 27

29 Levenberg-Marquardt method Gradient descent: Moves along the direction of greatest gradient, Long, narrow valley this is not a good idea. Newton s method: Makes parabolic approximation, Directly computes the location of the minimum, Efficacy depends on how good the parabolic approximation is. LM combines these two estimates using a weight factor λ. 28

30 Levenberg-Marquardt (LM) method rt () τˆ ŵ 0 PLL LS Initialization Uniform Sampler ŷ Compute Update ŵ Levenberg-Marquardt The update box * updates the estimate w i w i+1, * increases λ if error increased; decreases λ if error decreased. 29

31 Trained LM achieves CRB Parameters N = packets T/T unif[0, 0.005] τ 0 /T ~ unif[0, 0.1] α optimized RMS Estimation Error in T / T Trained LM CRB SNR (db) RMS Estimation Error in τ 0 / T Trained LM CRB SNR (db) * Trained MM + PLL + LS + LM method achieves the CRB. 30

32 Random walk: Linearization and Projection N-dimensional estimation problem, ML estimation prohibitively complex. Instead: Linearize the PLL-based system, Apply projection operator. 31

33 Linear Gaussian model from PLL TED equation: Define: εˆk = ε k + n k = τ k τˆk + n k y k = τˆk + εˆk Therefore, we get the following linear Gaussian model: y k = τ k + n k Output y k is the sum of the PLL and the TED outputs. Validity of model depends on linearity of TED characteristics. τˆk is an estimate based on previous observations (a priori). y k is based on previous and present observsations (a posteriori). 32

34 MAP estimator For the linear Gaussian model, the MAP estimator is τˆmap y 2 1 ( ) = ( K τ + σ ni) Kτ y where y is the vector of a posteriori observations is the covariance matrix of the timing offset vector K τ 2 σ n is the variance of the noise n k τ 33

35 MAP estimator: Performance Timing Error Variance 10 3 CRB Trained MAP Trained PLL Parameters N = packets random walk model σ w /T = 0.33% first order PLL MM TED α optimized SNR (db) 5.5 db gain over PLL. 1.5 db away from CRB. CRB not attainable with the timing model chosen. (The a posteriori density f ( θ r) needs to be Gaussian, which is not the case here.) Gap partly due to loss due to linearization of the TED characteristics. 34

36 MAP estimator: Reduced-complexity Implementation MAP estimator takes the form of a matrix operation: τˆmap y 2 1 ( ) = ( K τ + σ ni) Kτ y Using the structure of the matrices involved, we can rewrite this as where is a convolution matrix, A 2 τˆmap ( y) A 1 A 2 y implemented as a time-invariant filter, is diagonal matrix with different diagonal entries, A 1 implemented as time-varying scaling of the filter output. 35

37 Summary Conventional timing recovery based on the PLL. Cramer-Rao bound gives a bound on performance of any timing estimator. Derived the CRB for different timing offset models. PLL does not achieve the CRB. With constant offset, gradient descent achieves the CRB. With frequency offset, the Levenberg-Marquardt method achieves the CRB. With a random walk, the MAP estimator significantly outperforms the CRB. (Caveat: With a random walk, the CRB is not achievable.) 36

38 Questions? Thank you! 37