Synchronization of Turbo Codes Based on Online Statistics

1. Synchronization of Turbo Codes Based on Online Statistics Jian Sun and Matthew C. Valenti jian@csee.wvu.edu Wireless Communication Research Laboratory Lane Dept. of Comp. Sci. and Elec. Engr. West Virginia University May 14 2003

2. Outline • Necessity of timing synchronization • Concepts • Effective signal-to-noise ratio (SNR) • Online statistic • Algorithm to compensate coding gain loss • Over-sampling and interpolation • Joint estimation of SNR and timing offset • Simulation results and conclusions • Future work

3. Necessity of Timing Synchronization in Turbo Coded Systems • Turbo codes are attractive because of its near-Shannon-limit BER performance in very low SNR environments. • Traditional synchronization methods will fail because of low SNR. • Improper synchronization will render the turbo coded system great loss of coding gain. • Prior work done by Mielczarek and Svensson [1] • Viewing turbo decoders as SNR improvers • Post decoding combination • Two turbo decoders in parallel [1] B. Mielczarek and A. Svensson, “Timing error recovery in turbo coded systems on AWGN channels," IEEE Trans. Comm., vol. 50, pp. 1584-1592, October 2002.

4. BER Performance of Turbo Coded System with Fixed Timing Offset • Timing offset is the difference of the actual sampling time and the symbol’s ending point. • Simulations show the effect of fixed timing offset on BER performance. • Simulation settings • Following cdma2000 encoder structure • Coding rate = 1/3 • Interleaver size = 1530 • Constraint length = 4 • 10 iterations • Raised-cosine pulse shaping with roll-off factor 0.5 0 10 -1 10 τ = 0.2T -2 10 τ = 0.1T -3 10 Bit Error Rate -4 10 1 dB -5 10 τ = 0 0.25 dB -6 10 -7 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Eb/N0 (dB)

5. Effective SNR • Received signal • Effective SNR • Mean squared error of ISI Energy per symbol Additive noise Intersymbol interference Transmitted data Pulse shaping function

6. Effective SNR and Timing Offset • Simulation settings: • Uncoded BPSK signal in AWGN channel • Raised-cosine pulse shaping with roll-off factor 0.5 • The effective SNR is a function of both channel SNR and timing offset. • The shape and position of the curves changes with the channel SNR. 3 2 SNR = 2 dB 1 0 Effective SNR (dB) -1 SNR = 1 dB -2 SNR = 0 dB -3 -4 -5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Timing Offset (T)

7. Joint Estimation of SNR and Timing Offset • Effective SNR is a function of both timing offset and SNR on channel. • With multiple values of effective SNR, we can solve the equation for timing offset and SNR simultaneously. • Assuming semi-static channel, we need to estimate the effective SNR on basis of each frame without knowledge of transmitted data. • To generate multiple effective SNR estimates, multiple samples of each symbol in a frame are taken.

8. Over-Sampling and Interpolation • Decoder infrastructure { r } s Online 1 , n 1 Channel Received Estimation Joint r ( t ) Signal Matched Estimation Filter s Algorithm { r } Online 2 2 , n Channel Estimation SNR Timing Estimation Estimation Interpolated Decoded Samples Data Inter- Turbo polator Decoder

9. Interpolation Method • Linear interpolation • Information loss due to improper timing cannot be recovered • Generate satisfactory results with sufficient large number of samples per symbol • More subtle interpolation methods • Boost up noise which is dominant in the environment where turbo codes are implemented • Do not improve SNR r2 r0 r1 0 t1 t2 Linear interpolation

10. Online Statistics • Summers and Wilson’s approach [2] • The calculation does not require knowledge of the data. • The statistic itself is a random variable. • The statistic is a one-to-one map to SNR. • The statistics can be used to calculate the effective SNR associated with sample sequences • The joint estimation algorithm can be implemented with statistics directly. [2] T. A. Summers and S. G. Wilson, “SNR mismatch and online estimation in turbo decoding,” IEEE Trans. Comm., vol. 46, pp. 421-423, April 1998.

11. Joint Estimation Algorithm • Curve fitting algorithm • Minimum mean-squared error approach • Needs to store all possible curve shapes • Vulnerable to fluctuation of statistic values • Linear approximation • Solution forβ0andτ

12. Simulation Settings • Turbo encoder • Interleaver size = 1530 • Constraint length = 4 • Code rate = 1/3 • Encoder structure follows cdma2000 specifications • Semi-static AWGN channel • BPSK modulation • Raised-cosine roll-off pulse shaping with roll-off factor 0.5 • Timing offset is a random variable with uniform distribution between -0.5T and 0.5T, but it is fixed in each frame • Timing offset differs from frame to frame • Turbo decoder • 10 iterations are conducted before hard-decision is made • Linear approximation is used for joint estimation algorithm • Linear interpolation is used to regenerate signals

13. BER Performance of Turbo Coded System with 2 Samples/Symbol • About 0.2 dB loss due to linear interpolation • About 0.7 dB loss due to error in estimation of timing offset • About 0.8 dB loss due to error in joint estimation of channel SNR and timing offset. 0 10 Estimated Timing and SNR -1 Known SNR, Estimated Timing 10 Known Timing and SNR Perfect Timing -2 10 -3 10 Bit Error Rate -4 10 -5 10 -6 10 -7 10 -8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Eb/N0 (dB)

14. BER Performance of Turbo Coded System with 4 Samples/Symbol 0 10 • Almost no loss due to linear interpolation. • Less than 0.2 dB coding gain loss from error in joint estimation of channel SNR and timing offset. • Implementation with one turbo decoder and low additional complexity and latency. Estimated Timing and SNR Known Timing and SNR -1 10 Perfect Timing -2 10 -3 Bit Error Rate 10 -4 10 -5 10 -6 10 -7 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Eb/N0 (dB)

15. Possible Improvements • Simulation results indicate that BER performance can be improved without changing the decoder structure by • Using better estimators that make estimation with smaller variance; • Using better interpolators that recover more information by combining samples without loss of SNR. • Refine estimation using turbo principles. • The iterative nature of turbo decoder enables us to rebuild reliable decision using intermediate decoding results. • The intermediate decoding result can be directed back to form a loop. • The joint estimation of channel SNR and timing offset can be refined with decision-feedback.

16. Conclusions • Effective SNR can be used to evaluate loss in coding gains for turbo coded systems caused by improper timing. • With slight additional complexity, the proposed system can recover coding gains to 0.2 dB within the performance of perfect timing situation with 4 samples per symbol.