Complex amplitudes tracking loop for multi-path channel estimation in OFDM systems: Synthesis and extension

version corrigée (4 corrections en rouge dans les formules par rapport à la publication de la conférence)International audienceThis study deals with pilot-aided multi-path channel estimation for orthogonal frequency division multiplexing (OFDM) systems under slow to moderate fading conditions. Some algorithms exploit the channel time-domain correlation by using Kalman filters (KFs) to track the channel multi-path complex amplitudes (CAs), assuming a primary acquisition of the delays. Recently, it was shown that less complex algorithms, based on a second-order Complex Amplitude Tracking Loop (CATL) structure and a Least-Square (LS) pilot-aided error signal, can also reach near optimal asymptotic mean-squared error (MSE) performance. The LS-CATL-based algorithms are inspired by digital Phase-Locked Loops (PLL), as well as by the "prediction-correction" principle of the KF (in steady-state mode). This paper sums up and extends our previous results for the tuning and steady-state performance of the LS-CATL algorithm: analytic formulae are given for the first-, second-, and third-order loops, usable here for the multi-path multi-carrier scenario, and adaptable to any Doppler spectrum model of wide-sense stationary channels

Simplified Random-Walk-Model-Based Kalman Filter for Slow to Moderate Fading Channel Estimation in OFDM Systems

12 pagesInternational audienceThis study deals with multi-path channel estimation for orthogonal frequency division multiplexing systems under slow to moderate fading conditions. Advanced algorithms exploit the channel time-domain correlation by using Kalman Filters (KFs) based on an approximation of the time-varying channel. Recently, it was shown that under slow to moderate fading, near optimal channel multi-path complex amplitude estimation can be obtained by using the integrated Random Walk (RW) model as the channel approximation. To reduce the complexity of the high-dimensional RW-KF for joint estimation of the multi-path complex amplitudes, we propose using a lower dimensional RW-KF that estimates the complex amplitude of each path separately. We demonstrate that this amounts to a simplification of the joint multi-path Kalman gain formulation through the Woodbury's identities. Hence, this new algorithm consists of a superposition of independent single-path single-carrier KFs, which were optimized in our previous studies. This observation allows us to adapt the optimization to the actual multi-path multi-carrier scenario, to provide analytic formulae for the mean-square error performance and the optimal tuning of the proposed estimator directly as a function of the physical parameters of the channel (Doppler frequency, Signal-to-Noise-Ratio, Power Delay Profile). These analytic formulae are given for the first-, second-, and third-order RW models used in the KF. The proposed per-path KF is shown to be as efficient as the exact KF (i.e., the joint multipath KF), and outperforms the autoregressive-model-based KFs proposed in the literature

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

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

On the study of faster-than-Nyquist multicarrier signaling based on frame theory

Multicarrier transmissions are classically based on undercomplete or exact Weyl-Heisenberg Riesz (biorthogonal or orthogonal) bases implemented thanks to oversampled filter-banks. This can be seen as a transmission below the Nyquist rate. However, when overcomplete Weyl-Heisenberg frames are used, we obtain a “faster-than-Nyquist” (FTN) system and it is theoretically impossible to recover exactly transmitted symbols using a linear receiver. Various studies have shown the interest of this high density signaling scheme as well as practical implementations based on trellis and/or iterative decoding. Nevertheless, there is still a lack of theoretical justifications with regard to pulse design in the FTN case. In this paper, we consider a linear transceiver operating over an additive white Gaussian noise channel. Using the frame theory and simulation results, we show that the mean squared error (MSE) is minimized when tight frames are used

Tracking algorithms for mobile radio channel estimation and performance analysis : applications to OFDM systems

L'estimation de canal est une tâche cruciale du récepteur dans les systèmes de communication sans fil, en particulier en cas de mobilité où les paramètres du canal varient avec le temps. Dans cette thèse, un nouvel estimateur de boucle de poursuite d'ordre 3 ( RW3-CATL), qui a une structure semblable à la PLL avec une faible complexité a été tout d'abord proposé pour estimer l'amplitude complexe du canal dans le cas mono-trajet mono-porteuse. Le lien entre un filtre de Kalman en régime asymptotique basé sur un modèle d'approximation de marche aléatoire (RW3-KF) et l'estimateur proposé est établi. Les expressions des paramètres sous-optimaux et d'EQM correspondante sont données sous forme analytiques en fonction des gains de boucle. Ensuite, les performances asymptotiques du RW3-KF ont été analysées en résolvant les équations de Riccati. L'expression analytique de la variance optimale du bruit d'état qui minimise l'EQM asymptotique a été également déduite.Channel Estimation is a crucial task of the receiver in wireless communication systems, especially in case of mobility where the channel parameters vary with time. In this thesis, a novel PLL-structured third-order tracking loop estimator (RW3-CATL) with a low complexity is firstly proposed to estimate the complex amplitude of the channel in the mono-path single-carrier scenario. The connection between a steady-state Kalman filter based on a random walk approximation model (RW3-KF) and the proposed estimator has been established. The sub-optimal parameters and the corresponding MSE of the RW3-CATL are given in closed-form expressions in function of the tracking loop parameters. Then, the asymptotic performance of the RW3-KF has been analysed by solving the Riccati equations. The closed-form expression of the optimal state noise variance which minimizes the asymptotic MSE is also derived

Algorithmes de poursuite pour l'estimation de canal radio-mobile et performances asymptotiques: applications pour les systèmes OFDM

Thèse financée par la bourse du ministère du 01-oct-2010 au 30-sep-2013Channel Estimation is a crucial task of the receiver in wireless communication systems, especially in case of mobility where the channel parameters vary with time. In this thesis, a novel third-order tracking loop estimator (RW3-CATL), which is similar to the phase-locked loop (PLL) with low complexity structure was first proposed to estimate complex amplitude of the channel in the single-path single-carrier scenario. The connection between a steady-state Kalman filter based on a random walk approximation model (RW3-KF) and the proposed estimator is established. The sub-optimal parameters and the corresponding MSE of the RW3-CATL are given in closed-form expressions in function of the tracking loop parameters. The sub-optimal parameters and the corresponding MSE of the RW3-CATL are given in closed-form expressions in function of the tracking loop parameters. Then, the asymptotic performance of the RW3-KF has been analysed by solving the Riccati equations. The closed-form expression of the optimal state noise variance which minimises the asymptotic MSE is also derived. For multi-carrier multi-path systems such as OFDM systems, the RW3-CATL is extended to a vector loop (RW3-LS-CATL). This loop retains the structure as the RW3-CATL in the single-path single-carrier scenario, but a suitable vector error signal is re-designed from an LS estimator of the path complex amplitudes, based on the current OFDM symbol pilots and the a priori knowledge of the path delays. In this scenario, the Kalman filter, RW-KF, jointly estimates the complex amplitudes of the paths. It has a high complexity , and we do not have closed-form formulas for the parameter setting. To reduce the complexity , we propose a dimension-reduced filter obtained from an approximation of the original Kalman filter by exploiting the Woodbury identities, which deals with the complex amplitude of each path separately. The optimum parameter settings of this filter is deducted from the single-path single-carrier case. The performance of the low complexity filter, obtained by simulation, is almost the same as the original Kalman filter.L'estimation de canal est une tâche cruciale du récepteur dans les systèmes de communication sans fil, en particulier en cas de mobilité où les paramètres du canal varient avec le temps. Dans cette thèse, un nouvel estimateur de boucle de poursuite d'ordre 3 (RW3-CATL), qui a une structure semblable à la boucle à verrouillage de phase (PLL) avec une faible complexité a été tout d'abord proposé pour estimer l'amplitude complexe du canal dans le cas mono-trajet mono-porteuse. Le lien entre un filtre de Kalman en régime asymptotique basé sur un modèle d'approximation de marche aléatoire (RW3-KF) et l'estimateur proposé est établi. Les expressions des paramètres sous-optimaux et d'EQM correspondante sont données sous forme analytiques en fonction des gains de boucle. Ensuite, les performances asymptotiques du RW3-KF ont été analysées en résolvant les équations de Riccati. L'expression analytique de la variance optimale du bruit d'état qui minimise l'EQM asymptotique a été également déduite. Pour les systèmes multi-trajet multi-porteuses, tels que les systèmes OFDM, la boucle RW3-CATL est étendue à la structure vectorielle (RW3-LS-CATL). Cette boucle conserve la même structure qu'en mono-trajet mono-porteuse, mais un signal d'erreur vectoriel adéquat est redéfini à partir d'un estimateur LS des amplitudes complexes des trajets, basé sur les pilotes du symbole OFDM courant et sur la connaissance a priori des retards des trajets. Par ailleurs, pour ce scénario, le filtre de Kalman, RW-KF estime conjointement les amplitudes complexes des trajets. Il présente une forte complexité, et nous n'avons pas de formules analytiques pour le régler. Pour réduire la complexité, nous proposons un filtre de dimension réduite, obtenu par une approximation du filtre original à l'aide des identités de Woodbury. Ce filtre revient à estimer l'amplitude complexe de chaque trajet de manière séparée. Le réglage optimal des paramètres est déduit du cas mono-trajet mono-porteuse. Les performances de ce filtre à complexité réduite, obtenues par simulation, sont quasiment les mêmes que celles du filtre de Kalman original

Complex amplitudes tracking loop for multi-path channel estimation in OFDM systems: Synthesis and extension

version corrigée (4 corrections en rouge dans les formules par rapport à la publication de la conférence)International audienceThis study deals with pilot-aided multi-path channel estimation for orthogonal frequency division multiplexing (OFDM) systems under slow to moderate fading conditions. Some algorithms exploit the channel time-domain correlation by using Kalman filters (KFs) to track the channel multi-path complex amplitudes (CAs), assuming a primary acquisition of the delays. Recently, it was shown that less complex algorithms, based on a second-order Complex Amplitude Tracking Loop (CATL) structure and a Least-Square (LS) pilot-aided error signal, can also reach near optimal asymptotic mean-squared error (MSE) performance. The LS-CATL-based algorithms are inspired by digital Phase-Locked Loops (PLL), as well as by the "prediction-correction" principle of the KF (in steady-state mode). This paper sums up and extends our previous results for the tuning and steady-state performance of the LS-CATL algorithm: analytic formulae are given for the first-, second-, and third-order loops, usable here for the multi-path multi-carrier scenario, and adaptable to any Doppler spectrum model of wide-sense stationary channels

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

International audienceThis paper deals with channel estimation over a flat fading Rayleigh 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 of the time-varying channel. In the low-variation channel scenario, generally speaking, a well-chosen higher order estimator can perform better than a lower order one (Ros et al., [1] [2]). Based on this fact, we propose a third-order tracking loop estimator inspired by the principle of the phase-locked loop (PLL). The proposed estimator has a less complex structure compared to the Kalman based estimators. In addition, the mean-squared-error (MSE) of the proposed estimator is studied, as well as the parameter optimization with the aim of minimizing the MSE. The closed form expression of the optimal MSE is given and validates the interest of our approach