Bernard C. Levy

A generalized state-space for singular systems

Systems of the form E\dot{x}=Ax + Bu, y=Cx , with E singular, are studied. Of particular interest are the impulsive modes that may appear in the free-response of such systems when arbitrary initial conditions are permitted, modes that are associated with natural system frequencies at infinity. A generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes. Allowable equivalence transformations of such singular systems are specified. The present framework is shown to overcome several difficulties inherent in other treatments of singular systems, and to extend, in a natural and satisfying way, many results previously known only for regular state-space systems.

New results in 2-D systems theory, part II: 2-D state-space models—Realization and the notions of controllability, observability, and minimality

In this part, a comparison between the different state-space models is presented. We discuss proper definitions of state, controllability and observability and their relations to minimality of 2-D systems. We also present new circuit realizations and 2-D digital filter hardware implementation of 2-D transfer functions.

Combining and updating of local estimates and regional maps along sets of one-dimensional tracks

In this paper we consider the problem of combining and updating estimates that may have been generated in a distributed fashion or may represent estimates, generated at different times, of the same process sample path. The first of these cases has applications in decentralized estimation, while the second has applications in updating maps of spatially-distributed random quantities given measurements along several tracks. The method of solution for the second problem uses the result of the first, and the similarity in the formulation and solution of these problems emphasizes the conceptual similarity between many problems in decentralized control and in the analysis of random fields.

Differential Methods in Inverse Scattering

This paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium. By writing the medium equations in the form of a two-component system describing the interaction of rightward and leftward propagating waves, the causality of the propagation phenomena is exploited in order to identify the medium layer by layer. The recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by Schur in order to test for the boundedness of functions analytic inside the unit circle. It recovers the local reflectivity function of the medium. Using similar ideas, some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function.The differential inverse scattering methods turn out to be very efficient since, in some sense, they let the medium perform the inversion by itself and thus...

Blind Calibration of Timing Offsets for Four-Channel Time-Interleaved ADCs

In this paper, we describe a blind calibration method for timing mismatches in a four-channel time-interleaved analog-to-digital converter (ADC). The proposed method requires that the input signal should be slightly oversampled. This ensures that there exists a frequency band around the zero frequency where the Fourier transforms of the four ADC subchannels contain only three alias components, instead of four. Then the matrix power spectral density (PSD) of the ADC subchannels is rank deficient over this frequency band. Accordingly, when the timing offsets are known, we can construct a filter bank that nulls the vector signal at the ADC outputs. We employ a parametrization of this filter bank to develop an adaptive null steering algorithm for estimating the ADC timing offsets. The null steering filter bank employs seven fixed finite-impulse response filters and three unknown timing offset parameters which are estimated by using an adaptive stochastic gradient technique. A convergence analysis is presented for the blind calibration method. Numerical simulations for a bandlimited white noise input and for inputs containing several sinusoidal components demonstrate the effectiveness of the proposed technique

HERMITIAN SOLUTIONS OF THE EQUATION X = Q + NX-1N

We consider the matrix equation X = Q + NX−1N∗. Its Hermitian solutions are parametrized in terms of the generalized Lagrangian eigenspaces of a certain matrix pencil. We show that the equation admits both a largest and a smallest solution. The largest solution corresponds to the unique positive definite solution. The smallest solution is the unique negative definite solution if and only if N is nonsingular. If N is singular, no negative definite solution exists. An interesting relation between the given equation and a standard algebraic Riccati equation of Kalman filtering theory is also obtained. Finally, we present an algorithm which converges to the positive definite solution for a wide range of initial conditions.

Modeling and estimation of discrete-time Gaussian reciprocal processes

Discrete-time Gaussian reciprocal processes are characterized in terms of a second-order two-point boundary-value nearest-neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics. This second-order model is the analog for reciprocal processes of the standard first-order state-space models for Markov processes. The model is used to obtain a solution to the smoothing problem for reciprocal processes. The resulting smoother obeys second-order equations whose structure is similar to that of the Kalman filter for Gauss-Markov processes. It is shown that the smoothing error is itself a reciprocal process. >

Adaptive blind calibration of timing offset and gain mismatch for two-channel time-interleaved ADCs

In this paper, we describe a blind calibration method for gain and timing mismatches in a two-channel time-interleaved low-pass analog-to-digital converters (ADC). The method requires that the input signal should be slightly oversampled. This ensures that there exists a frequency band around the zero frequency where the Fourier transforms of the ADC subchannels are alias free. Low-pass filtering the ADC subchannels to this alias-free band reduces the blind calibration problem to a conventional gain and time delay estimation problem for an unknown signal in noise. An adaptive filtering structure with three fixed FIR filters and two adaptive gain and delay parameters is employed to achieve the calibration. A convergence analysis is presented for the blind calibration technique. Numerical simulations for a bandlimited white noise input and for inputs containing several sinusoidal components demonstrate the effectiveness of the proposed method

Robustness and modeling error characterization

The results on robustness theory presented here are extensions of those given in [1]. The basic innovation in these new results is that they utilize minimal additional information about the structure of the modeling error as well as its magnitude to assess the robustness of feedback systems for which robustness tests based on the magnitude of modeling error alone are inconclusive.

Linear estimation of boundary value stochastic processes-- Part I: The role and construction of complementary models

This paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by Weinert and Desai in their important paper [1]. Specifically, the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions. A major contribution of this paper is an application of Greens identity in deriving a differential operator representation of the estimator. To clarify the development and to illustrate the range of applications of our approach, two brief examples are provided: one is a 1-D discrete two-point boundary value process and the other is a 2-D process governed by Poissons equation on the unit disk.