In acoustic, ultrasonic or electromagnetic propagation, crossed media are often modelled by linear filters with complex gains in accordance with the Beer-Lambert law. This paper addresses the problem of propagation in media where polarization has to be taken into account. Because waves are now bi-dimensional, an unique filter is not sufficient to represent the effects of the medium. We propose a model which uses four linear invariant filters, which allows to take into account exchanges between components of the field. We call it bi-filter because it has two inputs and two outputs. Such a circuit can be fitted to light devices like polarizers, rotators and compensators and to propagation in free space. We give a generalization of the Beer-Lambert law which can be reduced to the usual one in some cases and which justifies the proposed model for propagation of electromagnic beams in continuous media.

Periodic nonuniform sampling of second order (PNS2) involves two periodic sequences with the same period. This sampling scheme has been shown to remove aliasing. Moreover, under particular conditions on their spectral band (or spectral support), exact reconstruction of functions can be derived from their PNS2. This paper more generally deals with the best mean-square interpolation for stationary processes with any known power spectrum, from PNS2 and, possibly, with aliasing. We show that the best estimation is based upon particular linear filters, which depend on the gap between both sampling sequences. The mean-time error also depends on this gap. The errorless interpolation is a particular case. It requires the knowledge of the spectral support rather than the power-spectrum values.

Attenuation of ultrasonic waves is often assumed linear with respect to frequency in biological applications whereas it is considered quadratic when the propagation occurs in the atmosphere or the water. In the latter case, other studies show that a Gaussian propagation duration can explain this attenuation behavior and provide a model for the energy loss in the stationary limit. The present paper defines an equivalent random propagation duration with Cauchy distribution, which is appropriate for the propagation of ultrasound through tissue. The model adds an unobserved noise that represents the signal deterioration. In addition, the model agrees with the mode downshift in the case of a narrowband signal.

Les composants électroniques destinés à l’échantillonnage haute fréquence des signaux ont des temps de réponse susceptibles de dégrader les performances des algorithmes de reconstruction. Dans cet article, on considère un signal échantillonné par un composant modélisé par un filtre linéaire invariant activé à des instants potentiellement irréguliers et qui forment une suite vérifiant la condition de Landau (généralisation de la condition de Nyquist). On étudie les conséquences des imperfections des échantillonneurs dans la reconstruction du signal et dans l’estimation de l’enveloppe et de la phase.

Acquisition devices play an important role in digital signal processing. The possibility of a perfect reconstruction is demonstrated in regular as well as irregular sampling when the number of samples in the observation interval is high enough in function of the bandwidth of the sampled signal (length of the support of the spectrum). In the case of high sampling rates, imperfections of acquisition devices can introduce non negligible errors (when the acquisition duration of a given sample becomes not negligible in comparison with the sampling period (or mean sampling period in the case of irregular sampling). In this paper, explicit method is proposed to take into account imperfections of the sampling device in order to improve the reconstruction of the signal. The proposed method is applicable for deterministic functions and random processes in the case of regular sampling, as well as irregular sampling.

Increasing the sampling rate of Analog-to-Digital Converters (ADC) is a main challenge in many fields and especially in telecommunications. Time-Interleaved ADCs (TI-ADC) were introduced as a technical solution to reach high sampling rates by time interleaving and multiplexing several lowrate ADCs at the price of a perfect synchronization between them. Indeed, as the signal reconstruction formulas are derived under the assumption of uniform sampling, a desynchronization between the elementary ADCs must be compensated upstream with an online calibration and expensive hardware corrections of the sampling device. Based on the observation that desynchronized TI-ADCs can be effectively modeled using a Periodic Non-uniform Sampling (PNS) scheme, we develop a general method to blindly estimate the time delays involved in PNS. The proposed strategy exploits the signal stationarity properties and thus is simple and quite generalizable to other applications. Moreover, contrarily to state-ofthe-art methods, it applies to bandpass signals which is the more judicious application framework of the PNS scheme.

Increasing the sampling rate of Analog-to-Digital Converters (ADC) is a main challenge in many fields and especially in telecommunications. Time-Interleaved ADCs (TI-ADC) were introduced as a technical solution to reach high sampling rates by time-interleaving several low-rate ADCs at the price of a perfect synchronization. Indeed, as the inverse operation of DAC is derived under the assumption of uniform sampling, the desynchronization must be compensated upstream with expensive hardware corrections of the sampling device. In this paper, we propose an alternative framework for TI-ADCs based on a more flexible sampling scheme, known as Periodic Non-uniform Sampling (PNS), that takes into account the desynchronization at the reconstruction stage when the induced delay is known, thus avoiding hardware corrections. The main contribution of this paper is to propose two different strategies for desynchronization estimation, one based on auto-calibration, the other on blind estimation. The proposed method performance is evaluated in terms of signal reconstruction error.

Cet article traite du problème de l'échantillonnage non uniforme dans le cas des processus aléatoires. Une nouvelle méthode est proposée permettant d'effectuer une reconstruction exacte du signal avec une meilleure vitesse de convergence en termes de nombre d'échantillons et un filtrage linéaire directement à partir des échantillons non uniformes. Ce procédé peut être appliqué à des signaux de type passe-bas comme à des signaux de type passe-bande.

High sampling rate Analog-to-Digital Converters (ADCs) can be obtained by time-interleaving low rate (and thus low cost) ADCs into so-called Time-Interleaved ADCs (TI-ADCs). Nevertheless increasing the sampling frequency involves an increasing sensibility of the system to desynchronization between the different ADCs that leads to time-skew errors, impacting the system with non linear distortions. The estimation and compensation of these errors are considered as one of the main challenge to deal with in TI-ADCs. Some methods have been previously proposed, mainly in the field of circuits and systems, to estimate the time-skew error but they mainly involve hardware correction and they lack of flexibility, using an inflexible uniform sampling reference. In this paper, we propose to model the output of L interleaved and desynchronized ADCs with a sampling scheme called Periodic Non-uniform Sampling of order L (PNSL). This scheme has been initially proposed as an alternative to uniform sampling for aliasing cancellation, particularly in the case of bandpass signals. We use its properties here to develop a flexible on-line digital estimation and compensation method of the time delays between the desynchronized channels. The estimated delay is exploited in the PNSL reconstruction formula leading to an accurate reconstruction without hardware correction and without any need to adapt the sampling operation. Our method can be used in a simple Built-In Self-Test (BIST) strategy with the use of learning sequences and our model appears more flexible and less electronically expensive, following the principles of ”Dirty Radio Frequency” paradigm: designing imperfect analog circuits with subsequently digital corrections of these imperfections.

This paper considers the problem of non uniform sampling in the case of finite energy functions and random processes, not necessarily approaching to zero as time goes to infinity. The proposed method allows to perform exact signal reconstruction, spectral estimation or linear filtering directly from the non-uniform samples. The method can be applied to either lowpass, or bandpass signals.