Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Abstract

Abstract. In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in L 2 (R) can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system.We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

Quality of Field Reconstruction in Sensor Networks

“…1 compares the asymptotic MSE evaluated through (15) (represented by solid lines and labeled by "MSE ∞ ") against the average MSE (represented by points and labeled by "MSE av "). The MSE av is obtained by generating 100 realizations of the measures as in (2), with M = 10, computing the estimates as in (3) and averaging the square error a −â 2 . The MSE is shown in the log scale plotted versus SNR m , for β = 0.2 and different values of SNR x .…”

“…Several reconstruction techniques have been proposed in the literature, which amount to the solution of a linear system (see [2], [3] and the references cited therein). A widely used technique consists in processing the measures p by means of a linear filter, B, which is an r × (2M + 1) matrix and is a function of the system parameters known at the sink.…”

“…Several efficient and fast algorithms have been proposed to numerically reconstruct or approximate a signal [2], [3]. The problem we address in this work, however, is different.…”

Performance of Linear Field Reconstruction Techniques With Noise and Uncertain Sensor Locations

“…In particular, Feichtinger and Gröchenig in [26] provide an error analysis of an iterative reconstruction algorithm taking into account round-off errors, jitters, truncation errors, and aliasing. Other theoretical works on irregular sampling can be found in [4], [5], [25]- [29], and references therein. The main differences between our work and the results that are available in the spectral analysis literature are the following:…”

“…In all cases, for any fixed , the filter matrix is deterministic. Using (4), (5), and (7), we have (12) where is the signal-to-noise ratio (SNR) on the measure. The MSE expression specialized to the different filters is given below.…”