Lower Bound Based on Kalman Carrier Recovery below the Information Rate of Wiener Phase Noise Channel

A new lower bound below the information rate transferred through the Additive White Gaussian Noise (AWGN) channel affected by discrete-time multiplicative Wiener’s phase noise is proposed in the paper. The proposed lower bound is based on the Kalman approach to data-aided carrier phase recovery, and is less computationally demanding than known methods based on phase quantization and trellis representation of phase memory. Simulation results show that the lower bound is close to the actual channel capacity, especially at low-to-intermediate signal-to-noise ratio

Third-Order Kalman Filter: Tuning and Steady-State Performance

4 pagesInternational audienceThis letter deals with the Kalman filter (KF) based on a third-order integrated random walk model (RW3). The resulting filter, noted as RW3-KF, is well suited to track slow time-varying parameters with strong trend behaviour. We first prove that the RW3-KF in steady-state admits an equivalent structure to the third-order digital phase-locked loops (DPLL). The approximate asymptotic mean-squared-error (MSE) is obtained by solving the Riccati equations, which is given in a closed-form expression as a function of the RW3 model parameter: the state noise variance. Then, the closed-form expression of the optimum state noise variance is derived to minimize the asymptotic MSE. Simulation results are given for the particular case where the parameter to be estimated is a Rayleigh channel coefficient with Jakes' Doppler spectrum

Soft metrics and their Performance Analysis for Optimal Data Detection in the Presence of Strong Oscillator Phase Noise

In this paper, we address the classical problem of maximum-likelihood (ML) detection of data in the presence of random phase noise. We consider a system, where the random phase noise affecting the received signal is first compensated by a tracker/estimator. Then the phase error and its statistics are used for deriving the ML detector. Specifically, we derive an ML detector based on a Gaussian assumption for the phase error probability density function (PDF). Further without making any assumptions on the phase error PDF, we show that the actual ML detector can be reformulated as a weighted sum of central moments of the phase error PDF. We present a simple approximation of this new ML rule assuming that the phase error distribution is unknown. The ML detectors derived are also the aposteriori probabilities of the transmitted symbols, and are referred to as soft metrics. Then, using the detector developed based on Gaussian phase error assumption, we derive the symbol error probability (SEP) performance and error floor analytically for arbitrary constellations. Finally we compare SEP performance of the various detectors/metrics in this work and those from literature for different signal constellations, phase noise scenarios and SNR values

Third-order Complex Amplitudes Tracking Loop for Slow Flat Fading Channel On-Line Estimation

12 pagesInternational audienceThis paper deals with channel estimation in tracking mode over a flat Rayleigh fading channel with Jakes' Doppler Spectrum. Many estimation algorithms exploit the time-domain correlation of the channel by employing a Kalman filter based on a first-order (or sometimes second-order) approximation model of the time-varying channel. However, the nature of the approximation model itself degrades the estimation performance for slow to moderate varying channel scenarios. Furthermore, the Kalman-based algorithms exhibit a certain complexity. Hence, a different model and approach has been investigated in this work to tackle all of these issues. A novel PLL-structured third-order tracking loop estimator with a low complexity is proposed. The connection between a steady-state Kalman filter based on a random walk approximation model and the proposed estimator is first established. Then, a sub-optimal mean-squared-error (MSE) is given in a closed-form expression as a function of the tracking loop parameters. The parameters that minimize this sub-optimal MSE are also given in a closed-form expression. The asymptotic MSE and Bit-Error-Ratio (BER) simulation results demonstrate that the proposed estimator outperforms the first and second order Kalman-based filters reported in literature. The robustness of the proposed estimator is also verified by a mismatch simulation

On the use of tracking loops for low-complexity multi-path channel estimation in OFDM systems

International audience—This paper treats pilot aided multi-path channel estimation with tracking loops for OFDM systems under slow to moderate fading conditions. Recent works have presented theoretical results for the tuning of second-order and third-order tracking loops in the particular context of Jakes's Doppler spectrum channel. The method for getting the loop coefficients resorted either to the use of a given constraint, which made the obtained coefficients sub-optimal, or was obtained in part by simulations. Here, we perform a global optimization of the coefficients without constraints to get the optimal coefficients, and analytical formulas are provided. One remarkable result of this optimization is that only the natural frequency depends on the transmission parameters, i.e., the channel Doppler spectrum, the power delay profile, and the noise variance. Consequently, only one parameter has to be tuned. Moreover, asymptotic performance is formulated in a more general way as a function of the 2rth moments of the Doppler spectrum (r is the loop order). Hence, all our derivations are usable for any Doppler spectrum and are not specific to Jakes's Doppler spectrum. A complete table sums up for the three orders the theoretical results of the optimal coefficients together with the asymptotic performance. The performance is also compared with that of the asymptotic Kalman filter

Complex Amplitudes Tracking Loop for multipath channel estimation in OFDM systems over slow to moderate fading

International audienceThis paper deals with multipath channel estimation for Orthogonal Frequency-Division Multiplexing systems under slow to moderate fading conditions. Most of the conventionalmethods exploit only the frequency-domain correlation by estimating the channel at pilot frequencies, and then interpolating the channel frequency response. More advanced algorithms exploit in addition the time-domain correlation, by employing Kalman filters based on the approximation of the time-varying channel. Adopting a parametric approach and assuming a primary acquisition of the path delays, channel estimators have to track the complex amplitudes of the paths. In this perspective, we propose a less complex algorithm than the Kalman methods, inspired by second-order Phase-Locked Loops. An error signal is created from the pilot-aided Least-Squares estimates of the complex amplitudes, and is integrated by the loop to carry out the final estimates. We derive closed-form expressions of the mean squared error of the algorithm and of the optimal loop coefficients versus the channel state, assuming a Rayleigh channel with Jakes'Doppler spectrum. The efficiency of our reduced complexity algorithm is demonstrated, with an asymptotic mean squared error lower than the first-order auto-regressive Kalman filters reported in the literature, and almost the same as a second-order Kalman-based algorithm

On the Equivalence between Kalman Filter at Steady State and DPLL

Fundamental results in the literature previously showed that a class of Kalman filter converges to a digital phase lock loop (DPLL) structure at the second and third order. We generalize these results at any order and give the closed-form linear relation, and its inverse, between the steady-state Kalman gains and the loop filter constants. Both relations are simple and only involve Stirling numbers of the first and second kind. This new result may help in a deeper understanding of the equivalence between Kalman filter and DPLL and be of practical interest in high dynamic scenarios

Centralized dynamics multi‐frequency GNSS carrier synchronization

In this article, we propose a new centralized multi‐frequency carrier tracking architecture using an adaptive Kalman filter to enhance the loop sensitivity and reliability of individual signal tracking in challenging signal environments. The main task of the centralized dynamics‐tracking filter is to effectively blend multiple frequency carrier phase observations in order to estimate the common geometric Doppler frequency of multiple‐frequency received signals. Conventionally, multi‐frequency signals are tracked independently with a fixed‐loop noise bandwidth tracking approach, which is suboptimal in time‐varying signal environments. A suitable collaboration in multiple‐frequency signal tracking using a centralized dynamics‐tracking loop enables robust carrier tracking even if some of the frequency channels are affected by ionospheric scintillation, carrier‐phase multipath, or interference. Additionally, computational efficiency of the multiple‐frequency tracking improves by using the proposed tracking loop architecture. Performance of the proposed multi‐frequency tracking‐loop architecture is verified with experiments using live multi‐frequency satellite signals collected from GPS Block‐IIF satellites under the influence of frequency‐selective interference signals