Nele Noels

Turbo synchronization: an EM algorithm interpretation

This paper is devoted to turbo synchronization, that is to say the use of soft information to estimate parameters like carrier phase, frequency offset or timing within a turbo receiver. It is shown how maximum-likelihood estimation of those synchronization parameters can be implemented by means of the iterative expectation-maximization (EM) algorithm [A.P. Dempster, et al., 1977]. Then we show that the EM algorithm iterations can be combined with those of a turbo receiver. This leads to a general theoretical framework for turbo synchronization. The soft decision-directed ad-hoc algorithm proposed in V. Lottici and M. Luise, [2002] for carrier phase recovery turns out to be a particular instance of this implementation. The proposed mathematical framework is illustrated by simulations reported for the particular case of carrier phase estimation combined with iterative demodulation and decoding [S. ten Brink, et al., 1998].

Code-Aided Turbo Synchronization

The introduction of turbo and low-density parity-check (LDPC) codes with iterative decoding that almost attain Shannon capacity challenges the synchronization subsystems of a data modem. Fast and accurate signal synchronization has to be performed at a much lower value of signal-to-noise ratio (SNR) than in previous less efficiently coded systems. The solution to this issue is developing specific synchronization techniques that take advantage of the presence of the channel code and of the iterative nature of decoding: the so-called turbo-synchronization algorithms. The aim of this paper within this special issue devoted to the turbo principle is twofold: on the one hand, it shows how the many turbo-synchronization algorithms that have already appeared in the literature can be cast into a simple and rigorous theoretical framework. On the other hand, it shows the application of such techniques in a few simple cases, and evaluates improvement that can be obtained from them, especially in the low-SNR regime.

The Cramer-Rao bound for phase estimation from coded linearly modulated signals

In this letter, we express the Cramer-Rao bound (CRB) for carrier phase estimation from a noisy linearly modulated signal with encoded data symbols, in terms of the marginal a posteriori probabilities (APPs) of the coded symbols. For a wide range of classical codes (block codes, convolutional codes, and trellis-coded modulation), these marginal APPs can be computed efficiently by means of the Bahl-Cocke-Jelinke-Raviv (BCJR) algorithm, whereas for codes that involve interleaving (turbo codes and bit interleaved coded modulation), iterated application of the BCJR algorithm is required. Our numerical results show that when the BER of the coded system is less than about 10/sup -3/, the resulting CRB is essentially the same as when transmitting a training sequence.

True Cramer-Rao bound for timing recovery from a bandlimited linearly Modulated waveform with unknown carrier phase and frequency

This paper derives the Cramer-Rao bound (CRB) related to the estimation of the time delay of a linearly modulated bandpass signal with unknown carrier phase and frequency. We consider the following two scenarios: joint estimation of the time delay, the carrier phase, and the carrier frequency; and joint estimation of the time delay and the carrier frequency irrespective of the carrier phase. The transmit pulse is a bandlimited square-root Nyquist pulse. For each scenario, the transmitted symbols constitute either an a priori known training sequence or an unknown random data sequence. In spite of the presence of random data symbols and/or a random carrier phase, we obtain a relatively simple expression of the CRB, from which the effect of the constellation and the transmit pulse are easily derived. We show that the penalty resulting from estimating the time delay irrespective of the carrier phase decreases with increasing observation interval. However, the penalty, caused by not knowing the data symbols a priori, cannot be reduced by increasing the observation interval. Comparison of the true CRB to existing symbol synchronizer performance reveals that decision-directed timing recovery is close to optimum for moderate-to-large signal-to-noise ratios.

Effectiveness Study of Code-Aided and Non-Code-Aided ML-Based Feedback Phase Synchronizers

This paper investigates the effectiveness of a (non-)code-aided ML-based FB phase synchronizer at the low operating signal-to-noise ratio of capacity-approaching codes. We show that the performance of the code-aided synchronizer is very close to that of a data-aided synchronizer that knows all data symbols in advance. This illustrates the optimality of the code-aided synchronizer. For the non-code-aided and the data-aided synchronizer, the linearized mean square phase error (MSPE) is evaluated analytically in the case of a first order loop. We demonstrate that, the MSPE of the non-code-aided synchronizer equals that of the data-aided synchronizer when the carrier phase is essentially constant and the loop filter gain is the same for both synchronizers, but that the non-code-aided synchronizer (as compared to the data-aided synchronizer) yields a larger MSPE due to phase fluctuations. This proves that code-aided FB phase estimation outperforms non-code-aided FB phase estimation when that the phase to be estimated is time-varying.

The true Cramer-Rao bound for carrier frequency estimation from a PSK signal

This paper considers the Cramer-Rao bound (CRB) related to estimating the carrier frequency of a noisy phase-shift keying signal. The following scenarios are discussed: 1) carrier frequency estimation irrespective of the carrier phase, based on either known or random data and 2) joint carrier phase and frequency estimation, based on either known or random data. Ideal symbol timing is assumed. We compare the results obtained from a (commonly used) simplified observation model against those resulting from the correct model. Because of the presence of nuisance parameters (random data and/or random carrier phase), the analytical computation of the corresponding CRBs is often not feasible. Here we present results that are based upon a combined analytical/numerical approach. Our results show that the choice of the observation model has essentially no effect on the CRBs at moderate and high signal-to-noise ratios. We also show that of the two scenarios considered, joint frequency and phase estimation yields the smaller CRB; the penalty resulting from frequency estimation, irrespective of the carrier phase, decreases with increasing observation interval.

Performance Analysis of ML-Based Feedback Carrier Phase Synchronizers for Coded Signals

This paper considers carrier phase recovery in transmission systems with an iteratively decodable error-control code [turbo codes, low-density parity check (LDPC) codes], whose large coding gains enable reliable communication at very low signal-to-noise ratio (SNR). We compare three types of feedback phase synchronizers, which are all based upon the maximum-likelihood (ML) estimation principle: a data-aided (DA) synchronizer, a non-code-aided (NCA) synchronizer, and an iterative code-aided (CA) synchronizer. We introduce a blockwise forward-backward recursive phase estimator, and we show that the mean-square phase error (MSPE) of the NCA synchronizer equals that of the DA synchronizer when the carrier phase is constant and the loop filter gain is the same for both synchronizers. When the signal is affected by phase noise, the NCA synchronizer (as compared with the DA synchronizer) yields a larger MSPE due to phase fluctuations. We also show that, at the normal operating SNR of the considered code, the performance of the CA synchronizer is very close to that of a DA synchronizer that knows all transmitted symbols in advance

Iterative carrier phase synchronization for low-density parity-check coded systems

In this paper, we consider the effect of a carrier phase offset on the performance of a low-density parity-check (LDPC) coded QAM modulated system. We investigate an ML-based carrier phase synchronization algorithm that makes use of the posterior probabilities of the data symbols, provided by the iterative decoder. The resulting carrier phase synchronizer is an extension, to LDPC coded systems, of the iterative phase estimator for turbo coded systems, presented in V.Lottici and M.Louse [2002], and has negligible BER performance degradation as compared to the ideally synchronized system.

Block-processing soft-input soft-output demodulator for coded PSK using DCT-based phase noise estimation

This paper considers the detection of coded phaseshift keying signals subjected to additive white Gaussian noise and oscillator phase noise. We propose a detector that partitions the received frame into smaller blocks and models the unknown phasor variations over each block as a truncated discrete cosine transform (DCT) expansion. Detection and decoding are iteratively performed between a soft-input soft-output (SISO) demodulator, a SISO demapper, and a SISO decoder based on the sum-product algorithm and the factor graph framework, whereas the expectation-maximization algorithm is used in the demodulator for the DCT coefficients estimation. The resulting demodulator is shown to have an excellent performance/complexity tradeoff and to be well-suited for parallel processing on multiple cores.

The true Cramer-Rao bound for phase-independent carrier frequency estimation from a PSK signal

This contribution considers the Cramer-Rao bound (CRB) related to phase-independent carrier frequency estimation from a noisy PSK signal. Instead of estimating the frequency jointly with the carrier phase, we treat the phase as a nuisance parameter. Ideal symbol timing is assumed. Both cases of known data (training sequence) and random data are considered. We show that frequency estimation irrespective of the carrier phase yields a larger CRB than does joint frequency and phase estimation; the penalty resulting from the former strategy vanishes with increasing observation interval.