I N S • T A I T I G U R T O E E • G O • F E • H T T E • C F H P R O G R E S S S E R V I C E O N A N D O • L L A O E G S Y • • 1 8 8 5 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
Communication system model ECC Transmitter Source Modulator Encoder Channel Channel ECC Receiver Equalizer Sampler Decoder Discrete-time Continuous-time 1
Continuous-time discrete-time interface Discrete-time ECC to Source Modulator Encoder Continuous-time Channel Continuous-time ECC to Equalizer Sampler Decoder Discrete-time 2
Sampling: Timing recovery 0 T 2 T 3 T a 0 a 1 a 3 τ 0 τ 1 – τ 3 τ 2 TIME a 2 T – Symbol duration a 0 , a 1 , a 2 ,... – Data symbols τ 0 , τ 1 , τ 2 ,... – Timing offsets Estimate τ 0 , τ 1 , τ 2 , ... Timing Recovery Problem : 3
Timing offset models Constant offset: τ k τ = Frequency offset: τ k τ 0 k ∆ T τ k ∆ T = + = + 1 – Random walk: k τ k τ k τ 0 ∑ w k w i = + = + 1 + i 0 = where w i are i.i.d. zero-mean Gaussian random variables of σ w σ w 2 2 variance . determines the severity of the random walk. 4
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
PLL: Motivation Consider the simple case of a time-invariant offset: τ k = τ τ ˆ i Let be the current timing estimate. ε i = τ i – τ = τ – τ ˆ i ˆ i Timing error: . ε = ε i . ˆ i With a perfect timing error detector (TED), we get τ τ ε τ ˆ i ˆ i ˆ i Update: = + = + 1 τ τ αε ˆ i ˆ i ˆ i With imperfect TED: = + + 1 6
PLL-based timing recovery r k ( ) r t y ( t ) f ( t ) for further processing rcv. filter τ kT + ˆ k PLL T.E.D. UPDATE ε ˆ k τ τ αε ˆ k ˆ k ˆ k = + First-order PLL + 1 k 1 – τ τ αε β ∑ ε ˆ k ˆ k ˆ k = + + ˆ i 1 Second-order PLL + i 0 = 7
Timing Error Detector (TED) r k ( ) r t y ( t ) f ( t ) for further processing rcv. filter τ kT + ˆ k ˆ d k ε ˆ k PLL T.E.D. UPDATE ε ( ) ˆ k ˆ k 3 T - r k d ˆ k r k – d = – - - - - - - - 1 – 1 16 Mueller & Müller Timing Error Detector • TED is a decision-directed device • Usually, instantaneous hard quantization • Better decisions entail delay that destabilizes the loop 8
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
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
Problem statement We consider the following uncoded system: AWGN { } 0 N 1 – r k a k τ h ( t) LPF uncoded i.i.d. kT (uniform) The uniform samples are: N 1 ∑ – a l h ( kT – lT – τ l ) + n k , r k = l = 0 where σ 2 is the noise variance, and h ( t ) is the impulse response. Given samples { r k } and knowledge of channel model, estimate Problem: the N uncoded i.i.d. data symbols { a k } • the N timing offsets {τ k } . • 11
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
CRB, intuitively f r θ 1 ( ) f r θ 2 ( ) f r θ 1 ( ) f r θ 2 ( ) r r → fixed, unknown parameter to be estimated θ r → observations • f r θ ( ) θ Sensitivity of to changes in determines quality of estimation. • f r θ ( ) θ is narrow, for a given r , probable If s lie in a narrow range. ⇒ θ can be estimated better, i.e. , with lesser error variance. ∂ • f r θ ( ) CRB uses - - - - - - as a measure of narrowness. log ∂θ 13
CRB for a random parameter θ If is random as opposed to being fixed and unknown, • θ f θ ( ) is characterized by a p.d.f. and • θ f r θ ( , ) r , are characterized by the joint p.d.f. . The measure for narrowness in this case is ∂ f r θ ( , ) - - - - - - log ∂θ 14
CRB is the inverse of Fisher information θ ( ) ˆ r For any unbiased estimator , the estimation error covariance matrix is lower bounded by T J 1 – [ ( θ ( ) θ ) θ ( ( ) θ ) ] ≥ ˆ r ˆ r E – – where J is the information matrix given by ∂ ∂ T f r θ ( , ) f r θ ( , ) J E = - - - - - - log - - - - - - log ∂θ ∂θ In particular, 2 J 1 – [ ( θ ( ) θ i ) ] ≥ ( , ) ˆ i r E i i – 15
Efficient estimators • An estimator that achieves the CRB is called efficient. • Efficient estimators do not always exist. Fixed, unknown θ : ML is efficient ∂ f r θ ( ) - - - - - - 0 log = ∂θ Random θ : MAP is efficient ∂ f r θ ( , ) 0 - - - - - - log = ∂θ 16
CRB: lower bound on timing error variance Constant offset: σ 2 σ ε ≥ 2 - - - - - - - - - - - - - - NE h ' Frequency offset: 6 σ 2 σ ε ≥ 2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( ) N 2 N ( ) E h ' N 1 1 – – 17
CRB for a random walk The Cramer-Rao bound on the error variance of any unbiased timing estimator: [ ( τ τ k ) 2 ] ≥ ⋅ ( ) E ˆ k h f k – where η 2 σ w h = - - - - - - - - - - - - - - - is the steady state value, η 2 1 – ( ( ( ) ) η ) N 0.5 2 k 1 sinh + – + log ( ) ( ( ) η ) 1 f k N 0.5 = tanh + log – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - , ( ( ) η ) N 0.5 sinh + log 2 σ w λ 2 2 π 2 λ 4 + – η λ 2 1 = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - and = + - - - - - - - - - – - - - - - - - - - - - - - . 2 3 σ 2 T 2 18
CRB: Steady-state value Parameters SNR bit = 5 dB σ w ⁄ T = 0.05% N = 5000 1.2% σ ε ⁄ T (%) h 0.8% 0.4% 0 Time 0 5000 Steady-state value becomes more representative as SNR and N increase. 19
Trained PLL away from the CRB 5% ESTIMATION ERROR JITTER σ ε ⁄ T (%) Trained PLL with α optimized 4% Parameters SNR = 5 dB CRLB 3% N = 4095 σ w ⁄ T = 0.7% α = 0.03 10000 sectors 2% 1% 0 2050 4100 Time (bit periods) Trained PLL does not achieve the steady-state CRB. 20
Trained PLL vs. Steady-state CRB 5% α = 0.03 α = 0.05 α = 0.1 α = 0.2 ESTIMATION ERROR JITTER σ ε ⁄ T (%) 4% α = 0.01 Parameters 3% σ w ⁄ T = 0.5% α = 0.02 1000 trials N = 500 7 dB α = 0.03 CRB 2% α = 0.05 1% 0 0 10 20 30 SNR (dB) 21
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
Constant offset: Gradient search τ The trained maximum-likelihood (ML) estimator picks to minimize ∞ 2 J τ ( ) ∑ ∑ ( τ ) ˆ a r k a l h kT lT ˆ ; = – – – ∞ l k = – This minimization can be implemented using gradient descent: τ τ µ J ' τ ( ) ˆ i ˆ i ˆ i a = – ; + 1 • Initialization using PLL. • J τ ( ) J τ ( ) ˆ a ˆ a Without training, use ˆ instead of . ; ; 23
Parameters τ/ T = π/ 20 Trained ML achieves CRB α = 0.01 N = 5000 20% 10% Decision-directed ML σ ε / T Decision-directed PLL RMS Timing Error 3% 2% Trained PLL 1% Trained ML, CRB 0.3% 0.2% -8 -3 2 7 SNR (dB) Two ways to improve performance over conventional PLL: * Better architecture – ML for example. * Better decisions – exploit error correction codes. 24
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