In the signal theory framework the "analytic signal” is a particular polar representation of a signal f (t). It defines an envelope (a modulus) and a phase (an argument) and these parameters are basic in communications. In this paper, we show that amplitude demodulations generally change envelopes and reduced phases. Consequently we show that recent results in laser communications about these parameters are erroneous.

Periodic nonuniform sampling of order three (PNS3) is a sampling scheme composed of three periodic sequences with the same period. It is well-known that this sampling scheme can be useful to remove aliasing. Previous studies have provided conditions on the spectrum support for exact reconstruction in the case of functions. This paper deals more generally with the best mean-square interpolation for stationary processes with any known power spectrum, from PNS3 and possibly aliasing. We show that the best estimation is based upon particular linear filters, which depend on the gap between the 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 spectral true values.

Propagation of monochromatic electro-magnetic waves in free space results in a widening of the spectral line. On the contrary, propagation preserves monochromaticity in the case of acous-tic waves. In this case, the propagation can be modelled by a linear invariant filter leading to attenuations and phase changes. Due to the Beer-Lambert law, the associated transfer function is an exponential of power functions with frequency-dependent parameters. In recent papers, we have proved that the acoustic propagation time can be modelled as a random variable following a stable probability distribution. In this paper, we show that the same model can be applied to the propagation in coaxial cables.

Periodic Nonuniform Sampling of order 2 (PNS2) is defined by two sequences with same period and some delay between them. PNS2 is known to suppress aliasing of multiband signals. Even if PNS2 reconstructs the signal by a linear combination of samples, the problem of retrieving a filtered version of the signal is more complicated. The simplest solution begins by a signal reconstruction through a sampling formula, followed by an analog filter. However new sampling formulas can solve this problem in a single stage. We give equivalent digital circuits which provide these formulas. Examples are provided, particularly when looking to retrieve subbands in communications.

Dealing with radar backscattering from trees, the Wong model is a mixing of Gaussian spectra with parameters deduced from considerations on motions of branches and leaves. Very detailed experiments by Narayanan et al. show gaps with this model. We show that autocorrelation functions by Narayanan et al are very well fitted by functions in the form exp[-|τ/τ0|α], 0 <; α ≤ 2. In this paper, we prove that the random propagation time theory explains this property. I have shown in other papers that this theory is available to study power spectra in acoustics, ultrasonics, and electromagnetics.

In acoustics, ultrasonics and in electromagnetic wave propagation, the crossed medium can be often modelled by a linear invariant filter (LIF) which acts on a wide-sense stationary process. Its complex gain follows the Beer-Lambert law i.e is in the form exp [-alphaz] where z is the thickness of the medium and alpha depends on the frequency and on the medium properties. This paper addresses a generalization for electromagnetic waves when the beam polarization has to be taken into account. In this case, we have to study the evolution of both components of the electric field (assumed orthogonal to the trajectory). We assume that each component at z is a linear function of both components at 0. New results are obtained modelling each piece of medium by four LIF. They lead to a great choice of possibilities in the medium modelling. Particular cases can be deduced from works of R. C. Jones on deterministic monochromatic light. keywords: linear filtering, polarization, Beer-Lambert law, random processes.

Many sampling formulas are available for processes in baseband (-a,a) at the Nyquist rate a/π. However signals of telecommunications have power spectra which occupate two bands or more. We know that PNS (periodic non-uniform sampling) allow an errorless reconstruction at rate smaller than the Nyquist one. For instance PNS2 can be used in the two-bands case (-a,-b)∪(b,a) at the Landau rate (a-b)/π We prove a set of formulas which are available in cases more general than the PNS2. They take into account two sampling sequences which can be periodic or not and with same mean rate or not.

The Stokes decomposition theorem deals with the electrical field E-->=X,Y of a light beam. The theorem asserts that a beam can be viewed as the sum of two differently polarized parts. This result was recently discussed for light in the frame of the unified theory of coherence. We study the general case of an electromagnetic wave which can be in radio, radar, communications, or light. We assume stationary components with any power spectrum and finite or infinite bandwidth. We show that an accurate definition of polarization and unpolarization is a key parameter which rules the set of solutions of the problem. When dealing with a ``strong definition'' of unpolarization, the problem is treated in the frame of stationary processes and linear invariant filters. When dealing with a ``weak definition'', solutions are given by elementary properties of bidimensional random variables.

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.