Carrier and Clock Recovery in (Turbo-) Coded Systems: Cramér-Rao Bound and Synchronizer Performance

N Noels,M Moeneclaey,H Steendam

doi:10.1155/asp.2005.972

Abstract

In this paper, we derive the Cramer-Rao bound (CRB) for joint carrier phase, carrier frequency, and timing estimation from a noisy linearly modulated signal with encoded data symbols. We obtain a closed-form expression for the CRB in terms of the marginal a posteriori probabilities of the coded symbols, allowing efficient numerical evaluation of the CRB for a wide range of coded systems by means of the BCJR algorithm. Simulation results are presented for a rate 1/2 turbo code combined with QPSK mapping. We point out that the synchronization parameters for the coded system are essentially decoupled. We find that, at the normal (i.e., low) operating SNR of the turbo-coded system, the true CRB for coded transmission is (i) essentially the same as the modified CRB and (ii) considerably smaller than the true CRB for uncoded transmission. Comparison of actual synchronizer performance with the CRB for turbo-coded QPSK reveals that a code-aware soft-decision-directed synchronizer can perform very closely to this CRB, whereas code-unaware estimators such as the conventional non-data-aided algorithm are substantially worse; when operating on coded signals, the performance of the latter synchronizers is still limited by the CRB for uncoded transmission.

Highlights

The impressive performance of turbo receivers implicitly assumes perfect synchronization, that is, the carrier phase, frequency offset, and time delay must be recovered accurately before data detection
A common approach to judge the performance of parameter estimators is to compare their resulting mean square error (MSE) with the Cramer-Rao bound (CRB), which is a fundamental lower bound on the error variance of unbiased estimators [1]
Our results point out that, at the normal operating SNR of the turbo code, (i) the CRB is essentially the same as the modified CRB (MCRB), (ii) the CRB is significantly smaller than the CRB for uncoded transmission, and (iii) the CRB is a tight lower bound on the MSE resulting from the joint synchronization and turbo-decoding scheme the authors proposed in [9]

Summary

INTRODUCTION

The impressive performance of turbo receivers implicitly assumes perfect synchronization, that is, the carrier phase, frequency offset, and time delay must be recovered accurately before data detection. We obtain a closed-form expression for the CRB in terms of the marginal APPs, allowing the numerical evaluation of the bound for a wide range of coded systems, including schemes with iterative detection (turbo schemes). This CRB is evaluated for rate 1/2 turbo-coded QPSK, and compared to (i) the MCRB, (ii) the CRB for uncoded (Turbo-) Coded Systems: CRB and Synchronizer Performance transmission, and (iii) the MSE of some practical synchronizers. Our results point out that, at the normal operating SNR of the turbo code, (i) the CRB is essentially the same as the MCRB, (ii) the CRB is significantly smaller than the CRB for uncoded transmission, and (iii) the CRB is a tight lower bound on the MSE resulting from the joint synchronization and turbo-decoding scheme the authors proposed in [9]

PROBLEM FORMULATION

DERIVATION OF THE CRB

NUMERICAL RESULTS AND DISCUSSION

ACTUAL ESTIMATOR PERFORMANCE

CONCLUSION