When the density of a Poisson process is integrable various expressions published in the literature are incorrect. This is especially the case of the probability distribution of the distance between an origin and the following points of the process. The first purpose of this paper is to explain why the integrability of the density changes the situation. The second is to discuss various consequences of this fact on the probability distribution of random variables extracted from the process. Computer experiments are presented and are in excellent agreement with theoretical results. Some extensions of the same problem concerning renewal processes are discussed

Singular random signals are collections of singular random variables indexed over time. Singular random variables have continuous distribution with derivatives equal to zero almost everywhere. Such random variables do not seem interesting in signal processing. On the contrary, the paper shows that simple signals can be singular. This is especially the case of ARMA signals generated from discrete white noise with poles located inside a so-called singularity circle. The origin of singularity and its relations to a fractal structure are presented along with various simulations illustrating the theoretical results.

Time point processes can be analyzed in two different ways: by the number of points in arbitrary time intervals or by distance between points. This corresponds to two distinct physical devices: counting or time-interval measurements. We present an explicit calculation, valid for arbitrary regular processes, of the statistical properties of time intervals such as residual or life time in terms of counting probabilities. For this calculation, we show that these intervals must be considered as random variables defined by conditional distributions.

The random telegraph signal (RTS) is an ordered signal which means that the explicit expression of the moment functions requires that the time instants appearing in these moments are put in an increasing order. This property greatly complicates the calculation of higher order polyspectra, Fourier transforms of moments. This article presents an extension to the discrete-time (DT) case of a method already known in continuous time. This extension exhibits new problems which makes the calculation even more difficult. These difficulties are analyzed and the structure of trispectrum is discussed in detail.

Polyspectra are related to Fourier transforms of moment or cumulant functions of any order of random signals. They play an important role in many problems of signal analysis and processing. However, there are only a few statistical models giving explicitly the expression of polyspectra. Ordered signals are signals for which the explicit expression of the moment functions requires that the time instants appearing in these moments are put in an increasing order. There are many examples of such signals, the best known being the random telegraph signal constructed from a Poisson process. Some of these examples are presented and analyzed. The origin of the ordering structure is related with the point that real time is an oriented variable making a difference between past and future. This especially appears in Markov processes. The calculation of polyspectra is difficult because ordering is not adapted to Fourier analysis. By an appropriate grouping of various terms, the explicit expression of spectral moment functions is obtained. It shows in particular that many ordered signals present a normal density on the normal manifolds of the frequency domain and another contribution on the stationary manifold that is explicitly calculated. The analysis of the structure of this expression allows us to discuss some relationships with normal distribution, central limit theorem, and time reversibility.

Relationships between ergodicity and structures of fourth-order spectral moments are investigated. In particular it is shown that second-order ergodicity of a random process is directly related to the distribution of these moments on the normal manifolds of the frequency domain. This result is illustrated by various examples.

Optimum mean square estimation of a random variable y in terms of an observation vector x is realized by the conditional expectation value. When x and y are real and jointly normal this expectation is linear in x. This is no longer the case when x and y are complex and jointly normal and the expectation is linear in x and in its complex conjugate x*, which introduces a widely linear procedure. The purpose of the paper is to study the properties of widely linear systems for estimation and prediction. The structure of such systems is calculated and the gain in performance is analyzed. The results are applied to autoregressive signals, which introduces widely linear prediction.

This paper investigates the French system of signal processing education. At first a general description of the French system of electrical engineering education is presented. Then the particular case of signal processing studies is described by using an example of one of the French universities.>

The problem of estimating the amplitude of a signal appears in many aspects of signal processing such as, for example, amplitude modulation in communication theory. When the probability distribution of the noise is unknown, the calculation of the maximum likelihood estimator is impossible. Generally a linear estimator without bias and with minimum variance is used because it is simple to calculate and only requires the knowledge of the covariance matrix of the noise. The amplitude is estimated here with a polynomial of the observations. The coefficients of the polynomial are determined such that the estimator is unbiased with minimum variance regardless of the amplitude. This method is quite general since it only requires the knowledge of some higher-order moments of the noise.>

In digital signal processing, the two main sources of uncertainty encountered in estimation problems are model uncertainty and noise. In many instances, probabilistic information is available to partially describe these sources of uncertainty. It is shown how such information can be exploited in a broad class of set theoretic estimation problems relevant to digital signal processing. A general framework is developed to construct sets in the solution space by constraining the estimation residual based on the known component of the model to be consistent with those known properties of a so-called uncertainty process consisting of the contribution of the unknown component of the model and the noise. Specific digital signal processing applications are discussed.>

The bispectrum modelling problem using linear or quadratic filters is investigated. In the linear case, the third-order cumulant function C/sub 3/ of the output of a finite impulse response filter, driven by a white noise, is zero outside a finite domain. It is proven that this property is still valid if the filter is quadratic. The inverse problem is also considered. It is shown that, if the function C/sub 3/ is zero outside a finite domain, then the impulse response is necessarily finite in the linear case and may or may not be finite in the quadratic case. It is proven that every factorizable bispectrum can also be modeled with a quadratic filter, and an example to establish that the converse is false is given.>

A class of nonlinear filter, the p-th order complex transversal Volterra filter (TVF), is proposed. Its output is represented by a specific inner product of two vectors, which allows a unique formulation of problems such as the nonlinear detection, using the deflection criterion, and the nonlinear estimation in a mean square (MS) sense, by p-th order complex TV filtering, is introduced. It is shown that the p-th order complex TVF which maximizes the deflection is also the one which minimizes the distance, in a MS sense, to the likelihood ratio. The calculation of the optimal p-th order complex TVF requires the knowledge of the statistical properties of the data up to the 2p-th order.>

The problem of decentralized detection appears when various sensors are used simultaneously. In the centralized vision all the information arriving at each sensor is sent to the fusion center, while in the decentralized a local processing is realized and the global procedure needs a fusion of the signals coming from the sensors. The authors study the problem by imposing two constraints: the first is that the local procedure reduces the information to a scalar quantity and the second is that the fusion rule is only an addition of the signals coming from each sensor. With these constraints, the problem can be stated in a geometrical context. Using the projection theorem, the authors state the general equation giving the local optimum procedure. This equation is very difficult to solve. The results in some particular cases, for example when the noise at each of the sensors are independent, are presented.>

Results for an arbitrary linear-quadratic (LQ) array structure are presented. For this purpose, knowledge of the statistical properties of the noise up to the fourth order is needed. Unfortunately, in most situations of practical interest the latter information is not available a priori and must be estimated. Three adaptive algorithms which are extensions of the well-known sample matrix inversion (SMI), recursive matrix inversion (RMI), and Frost algorithms are then developed for the real-time computation of the optimal LQ array processor. The study of the numerical complexity of these algorithms is discussed.>

Linear-quadratic filters are a special example of Volterra filters that are limited to the second order. It is shown that all the results recently published which are valid in the linear-quadratic case can be extended with the appropriate notations to Volterra filters of arbitrary order. Particularly, the optimum Volterra filter giving the maximum of the deflection for detecting a signal in noise is wholly calculated. In addition, several geometrical properties of optimal Volterra filters are investigated by introducing appropriate scalar products. In particular, the concept of space orthogonal to the signal and the noise alone reference (NAR) property are introduced, allowing a decomposition of the optimal filter that exhibits a relation between detection and estimation. Extensions to the infinite case and relations with the likelihood ratio are also investigated.>

Routh's table for determining the zero location of a polynomial with respect to the imaginary axis is generalized and presented without distinction between the real and complex cases. The new table is then transposed to the z-domain. It is shown that this transposition gives, in particular, the classical tables computed by processes based on the three-term recursion of polynomials. This generalization is used to solve the problem of classical singularities appearing in the study of the zero location of a polynomial with respect to the imaginary axis or to the unit circle. It also makes clear the connections existing between the classical methods.>

In many detection systems separated sensors and a central processor are used. Because of constraints on communications, all the information from the sensors cannot be sent to the central processor. This makes the perception of the system less accurate and degrades a performance. Previous studies have considered the case in which a detection procedure is realized at each sensor. A novel approach is presented within the framework of quantization, and the two procedures are compared in a Gaussian noise environment. It is shown that the quantized structure achieves better performance. The optimization of quantized systems according to the Neyman-Pearson criterion is shown to be difficult. Using results concerning time quantization, the optimum system for the deflection criterion is considered.>

The authors present the optimal linear-quadratic array processor for detection without the Gaussian assumption. For this purpose, a knowledge of the statistical properties of the noise up to the fourth order is needed. Using an appropriate scalar product, the optimal array is calculated and its performance discussed in a current and practical case of non-Gaussian jammers and fourth-order white noise.>

If an autoregressive digital filter of nth order is represented by a point a/sub n/ of the n-dimensional space, the set of all stable filters defines a region S in this space called the stability domain. For n>or=3, the geometry of S is complicated compared to that of its representation in the n-dimensional space of the reflection coefficients introduced by the lattice representation of stable filters. If nth-order filters are considered, it is shown that for n>or=3, S is not convex and the expression of both the radius of the greatest sphere included in S and that of the smallest sphere containing S are established.>

The author presents the principle of a linear-quadratic array. To calculate the optimal processor, it is necessary to know the statistical properties of the signals up to the fourth order. Using an appropriate scalar product the optimum array is calculated, and the improvement in performance is discussed in a particular case of white noise.>

Vector quantization is a procedure widely used in communications. The problem has been solved for the basic detection situation using the locally optimal procedure and with some constraints on the quantization scheme. It is stated in its widest sense, which consists of finding the partition of the observation space and the values associated with it in such a way that the detection performance is optimal. It is solved completely with the deflection criterion, and the solution introduces the concept of quantization by a likelihood ratio procedure.>

The problem of linear-quadratic systems for detection has long been solved by assuming the deflection criterion and Gaussian noise. It is shown here that the Gaussian assumption can be removed, and a complete solution is presented for an arbitrary probability distribution with finite fourth-order moments. The optimal solution can always be obtained by solving a linear system of equations. Some properties of the optimal systems are developed for particular examples of nonGaussian noise. It is shown that there is a strong relationship between linear-quadratic optimal detection and optimal estimation, which extends results known for the purely linear case.>

When dealing with uncertainties, the robust approach, using the games theory, assures a minimal level of performance. In this paper, we address the problem of robust signal detection in coloured unknown noise. The detection criterion we use is the local deflection (or relative efficiency). The minimax approach shows that the robust filter corresponds to the Gaussian density function which gives the matched filter.

Linear prediction and interpolation are widely used in signal processing. Usually they are obtained by linear filters of finite memory, and the prediction and interpolation errors decrease when this memory increases. If the asymptotic values of these errors are nul, the signal is singular. In this paper a numerical method to compute the prediction and interpolation errors for signals with a given power spectrum is presented. A large number of results obtained with this method are discussed, especially the difference between regular and singular signals.

As is well-known, the introduction of causality conditions in Wiener filtering problems completely changes their solution. A constraint such as causality can be presented as a particular reduction of the observation space, and the constrained filter can always be obtained by projection onto this space. However, it is sometimes simpler to use an indirect method which gives the impulse response of the constrained filter by an appropriate modification of the unconstrained response. This method is presented and applied to many examples. In particular, the structure of constrained prediction filters is analyzed, and it is shown that the constrained innovation can be expressed in terms of the unconstrained one by an appropriate filter.

Microscopic correlation signals are discrete-time signals whose autocorrelation function R (k) vanishes for all lags except O. This property is usually obtained by using pseudo-random sequences such as white noise. We show that the impulse response of a phase filter can be considered as an infinite signal with the microscopic correlation property. In this paper, characteristics of such signals are studied. Approximation by finite duration signals is investigated and this truncation is shown to have less effect on the autocorrelation function than truncation of other pseudo-random sequences.

We consider the prediction problem of a stochastic signal when particular constraints are imposed on the impulse response of the filter. The problem can be solved by using the concept of constrained z-transform and some particular examples of application of the method are described.

A real autoregressive signal is the output of a real autoregressive filter whose input is a real white noise. The same definition can be used for complex signals and systems, but the second order statistics of a complex white noise are not completely defined by its correlation function as in the real case. We present the consequences of this fact for autoregressive complex signals. In particular we show that the linear predictor of such signals is not necessarily a finite moving average filter.

In order to regulate the false alarm probability of some detection systems it is necessary to estimate the power of the noise. But this estimation must have the Noise Alone Reference property, which means that there is no bias of the estimate in presence of signal. We present a new NAR estimation method. The idea is to use an inverse filter. At the output of this filter the signals appear only in some very short time intervals and can be eliminated by an appropriate procedure. Computer simulations show the good performance of this method.

Quadratic filters are the simplest non linear time-invariant systems and correspond to the second term of the Volterra expansion. Such filters are completely defined by their kernel which is a symmetric finite or infinite square matrix. Some examples are presented. The harmonic representation of such filters is discussed and also the calculation of the statistical properties of the output in the case of a Gaussian input. Finally some problems of implementation of such filters are discussed.

By introducing an appropriate representation of the observation, detection problems may be interpreted in terms of estimation. The case of the detection of a deterministic signal in Gaussian noise is associated with two orthogonal subspaces: the first is the signal subspace which is generally one dimensional and the second is called a reference noise alone (RNA) space because it contains only the noise component and no signal. The detection problem can then be solved in the signal subspace, while the use of the RNA space is reduced to the estimation of the noise in the signal subspace. This decomposition leads to a very simple interpretation of singular detection, even in the non-Gaussian case, in terms of perfect estimation. The method is also extended to multiple signal detection problems and to some special cases of detection of random signals.

In this paper we consider the problem of estimation and detection of Weak optical signals formulated as a problem in point process theory. Indeed at very low light intensity the only information available from an intensity detector is the random distribution of time instants\{t_i \}at which photons of the field are absorbed and photoelectrons emitted. There is also a noise due to thermoelectrons or to a background optical field. Starting from the statistical properties of the point process\{t_i\}, we formulate for various kinds of optical fields the problem of estimation of the light intensity of a modulated beam. We show that in some cases the number of photoelectrons is a sufficient statistic. In general, theoretical results are complex and we formulate the problem in the case of linear estimation, which is solved by means of the resolvent of an integral equation using the covariance function of the field. Some detection problems are also considered.

This correspondence discusses the comparison between the class of spherically invariant processes and a particular class of Gaussian compound processes. We give a simple expression for the probability distribution and calculate some expectation values. The comparison shows that spherically invariant processes are slightly more general.

We will examine the problem of optimum filtering when signals and noises are incoming on different channels. The filtering is defined by a rectangular matrix of filters and the non-linear operation in the signal processing is the general quadratic system. After the calculation of the output signal-to-noise ratio, we discuss the problem of optimum filtering. In this paper we consider only the solution in some particular cases as, for example, the correlator. In the second part, we discuss some properties of the optimum linear filters and we show in the third how the problem of directivity synthesis in omnidirectional noise is a particular case of optimum filtering.