Sampling and reconstruction of operators (original) (raw)
Related papers
Local Sampling and Approximation of Operators with Bandlimited Kohn–Nirenberg Symbols
Constructive Approximation, 2014
Recent sampling theorems allow for the recovery of operators with bandlimited Kohn-Nirenberg symbols from their response to a single discretely supported identifier signal. The available results are inherently non-local. For example, we show that in order to recover a bandlimited operator precisely, the identifier cannot decay in time nor in frequency. Moreover, a concept of local and discrete representation is missing from the theory. In this paper, we develop tools that address these shortcomings. We show that to obtain a local approximation of an operator, it is sufficient to test the operator on a truncated and mollified delta train, that is, on a compactly supported Schwarz class function. To compute the operator numerically, discrete measurements can be obtained from the response function which are localized in the sense that a local selection of the values yields a local approximation of the operator. Central to our analysis is to conceptualize the meaning of localization for operators with bandlimited Kohn-Nirenberg symbol.
2010
Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.
Irregular and multi-channel sampling of operators
Applied and Computational Harmonic Analysis, 2010
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. * Operator sampling is not simply an higher dimensional analogue of the 1-d Shannon sampling theorem. In the case of an operator acting on L 2 (R), the operator's 2-dimensional Kohn-Nirenberg symbol is to be determined from a signal defined on R. No access to sample values of the Kohn-Nirenberg symbol is given, as is the case in 2-dimensional Shannon sampling theory.
Cornerstones of Sampling of Operator Theory
Applied and Numerical Harmonic Analysis, 2015
This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called bandlimited 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 thirdnamed 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.
Operator Identification and Sampling
2010
Time–invariant communication channels are usually modelled as convolution with a fixed impulse–response function. As the name suggests, such a channel is completely determined by its action on a unit impulse. Time–varying communication channels are modelled as pseudodifferential operators or superpositions of time and frequency shifts. The function or distribution weighting those time and frequency shifts is referred to as the spreading function of the operator. We consider the question of whether such operators are identifiable, that is, whether they are completely determined by their action on a single function or distribution. It turns out that the answer is dependent on the size of the support of the spreading function, and that when the operators are identifiable, the input can be chosen as a distribution supported on an appropriately chosen grid. These results provide a sampling theory for operators that can be thought of as a generalization of the classical sampling formula f...
Sampling and recovery of bandlimited functions and applications to signal processing
Advanced Courses of Mathematical Analysis IV - Proceedings of the Fourth International School – In Memory of Professor Antonio Aizpuru Tomás, 2011
Bandlimited functions, i.e square integrable functions on R d , d ∈ N, whose Fourier transforms have bounded support, are widely used to represent signals. One problem which arises, is to find stable recovery formulae, based on evaluations of these functions at given sample points. We start with the case of equally distributed sampling points and present a method of Daubechies and DeVore to approximate bandlimited functions by quantized data. In the case that the sampling points are not equally distributed this method will fail. We are suggesting to provide a solution to this problem in the case of scattered sample points by first approximating bandlimited functions using linear combinations of shifted Gaussians. In order to be able to do so we prove the following interpolation result. Let (x j : j ∈ Z) ⊂ R be a Rieszbasis sequence. For λ > 0 and f ∈ P W , the space of square-integrable functions on R, whose Fourier transforms vanish outside of [−1, 1], there is a unique sequence (a j) ∈ 2 (Z), so that the function I λ (f)(x) := a j e −λ x−xj 2 2 , x ∈ R is continuous, square integrable, and satisfies the interpolatory conditions I λ (f)(x k) = f (x k), for all k ∈ Z. It is shown that I λ (f) converges to f in L 2 (R d) and uniformly on R, as λ → 0 + .
A generalized sampling theory without band-limiting constraints
IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 1998
We consider the problem of the reconstruction of a continuous-time function f (x) ∈ H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H = L 2 ) is considerably larger than the subspace of bandlimited functions. Second, we use a more general specification of the reconstruction subspace V (ϕ), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximationf ∈ V (ϕ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V (ϕ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filterbank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the bandlimited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.
Sampling of Band-Limited Vectors
Journal of Fourier Analysis and Applications, 2001
Given a self-adjoint, positive definite operator on a Hilbert space the concept of band-limited vectors (with a given band-width) is developed, using the spectral decomposition of that operator. By means of this concept sufficient conditions on collections of linear functionals {ϕ ν } are derived which imply that all band limited vectors in a given class are uniquely determined resp. can be reconstructed in a stable way from the set of discrete values {ϕ ν (f)}.
Sampling and Reconstruction of Multiple-Input Multiple-Output Channels
IEEE Transactions on Signal Processing, 2019
Based on the recent development of sampling and reconstruction results for slowly time-varying single-input singleoutput channel operators, we derive sampling results in the multiple-input multiple-output setting where all subchannels satisfy an underspread condition, that is, their spreading functions are supported on individual sets of small measure. At the center of our work is the extension of the single-input single-output dual tiling condition to this setting; it characterizes which periodic weighted delta trains can be used to identify a given class of multiple-input multiple-output channel operators satisfying a spreading support constraint. Building on the dual tiling condition, we compute reconstruction formulas for the operator's symbol in closed form and discuss the problem of identifying multiple-input multiple-output operators where only restrictions in size, but not on location and geometry, of the subchannel spreading supports are known.