Cornerstones of Sampling of Operator Theory

This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains

Sampling and reconstruction of operators

We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.Comment: Submitted to IEEE Transactions on Information Theor

Density Criteria for the Identification of Linear Time-Varying Systems

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delays and Doppler shifts. We prove that stable identifiability is possible if the upper uniform Beurling density of the delay-Doppler support set is strictly smaller than 1/2 and stable identifiability is impossible for densities strictly larger than 1/2. The proof of this density theorem reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Finally, we introduce a subspace method for solving the system identification problem at hand.Comment: IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, June 201

Measurement of time--varying Multiple--Input Multiple--Output Channels

We derive a criterion on the measurability / identifiability of Multiple--Input Multiple--Output (MIMO) channels based on the size of the so-called spreading support of its subchannels. Novel MIMO transmission techniques provide high-capacity communication channels in time-varying environments and exact knowledge of the transmission channel operator is of key importance when trying to transmit information at a rate close to channel capacity

An Inverse Problem for Localization Operators

A classical result of time-frequency analysis, obtained by I. Daubechies in 1988, states that the eigenfunctions of a time-frequency localization operator with circular localization domain and Gaussian analysis window are the Hermite functions. In this contribution, a converse of Daubechies' theorem is proved. More precisely, it is shown that, for simply connected localization domains, if one of the eigenfunctions of a time-frequency localization operator with Gaussian window is a Hermite function, then its localization domain is a disc. The general problem of obtaining, from some knowledge of its eigenfunctions, information about the symbol of a time-frequency localization operator, is denoted as the inverse problem, and the problem studied by Daubechies as the direct problem of time-frequency analysis. Here, we also solve the corresponding problem for wavelet localization, providing the inverse problem analogue of the direct problem studied by Daubechies and Paul.Comment: 18 pages, 1 figur

Irregular and multi--channel sampling of operators

The classical sampling theorem for bandlimited functions has recently been generalized to apply to so-called bandlimited operators, that is, to operators with band-limited Kohn-Nirenberg symbols. Here, we discuss operator sampling versions of two of the most central extensions to the classical sampling theorem. In irregular operator sampling, the sampling set is not periodic with uniform distance. In multi-channel operator sampling, we obtain complete information on an operator by multiple operator sampling outputs

Estimation of Overspread Scattering Functions

In many radar scenarios, the radar target or the medium is assumed to possess randomly varying parts. The properties of a target are described by a random process known as the spreading function. Its second order statistics under the WSSUS assumption are given by the scattering function. Recent developments in operator sampling theory suggest novel channel sounding procedures that allow for the determination of the spreading function given complete statistical knowledge of the operator echo from a single sounding by a weighted pulse train. We construct and analyze a novel estimator for the scattering function based on these findings. Our results apply whenever the scattering function is supported on a compact subset of the time-frequency plane. We do not make any restrictions either on the geometry of this support set, or on its area. Our estimator can be seen as a generalization of an averaged periodogram estimator for the case of a non-rectangular geometry of the support set of the scattering function