As an example of the recently introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel stochastic algorithm to reconstruct a signal with finite rate of innovation from its noisy samples. Even though variants of this problem have been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example, kernels that satisfy the Strang-Fix condition. In this paper, we consider the infinite-support Gaussian kernel, which does not satisfy the Strang-Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.
@cite_1: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Bibliography. References Grouped by Subject. Subject Index.
@cite_5: AbstractIn [1] a sampling theorem for a certain class of signals with ﬁnite rate of innovation (which includesfor example stream of Diracs) has been developed. In essence, such non band-limited signals can besampled at or above the rate of innovation. In the present paper, we consider the case of such signalswhen noise is present. Clearly, the ﬁnite rate of innovation property is lost, but if the signal-to-noise ratio(SNR) is sufﬁcient, several methods are possible to reconstruct the signal while sampling well below theNyquist rate. We thus explore the trade-offs between SNR, sampling rate, computational complexity andreconstruction quality. Applications of such methods can be found in acquisition and processing of signalsin high bandwidth communications, like ultra wide band communication [2]. I. I NTRODUCTION Band-limited functions are just an example of signals that are speciﬁed by a ﬁxed number ofsamples per unit of time, namely, if a signal x(t) is band-limited to [ ! m ;! m ], then the famoussampling theorem by Shannon [3] states that x(nT) with T < ˇ=!
@cite_13: Recently, it was shown that it is possible to develop exact sampling schemes for a large class of parametric nonbandlimited signals, namely certain signals of finite rate of innovation. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and can be reconstructed from a finite number of uniform samples. In order to prove sampling theorems, considered the case of deterministic, noiseless signals and developed algebraic methods that lead to perfect reconstruction. However, when noise is present, many of those schemes can become ill-conditioned. In this paper, we revisit the problem of sampling and reconstruction of signals with finite rate of innovation and propose improved, more robust methods that have better numerical conditioning in the presence of noise and yield more accurate reconstruction. We analyze, in detail, a signal made up of a stream of Diracs and develop algorithmic tools that will be used as a basis in all constructions. While some of the techniques have been already encountered in the spectral estimation framework, we further explore preconditioning methods that lead to improved resolution performance in the case when the signal contains closely spaced components. For classes of periodic signals, such as piecewise polynomials and nonuniform splines, we propose novel algebraic approaches that solve the sampling problem in the Laplace domain, after appropriate windowing. Building on the results for periodic signals, we extend our analysis to finite-length signals and develop schemes based on a Gaussian kernel, which avoid the problem of ill-conditioning by proper weighting of the data matrix. Our methods use structured linear systems and robust algorithmic solutions, which we show through simulation results.
Maravic and Vetterli @cite_13 and Ridolfi @cite_5 proposed and solved a related problem. Instead of modeling the noise at the , they considered the scenario where @math , the signal in question, is corrupted by additive white noise @math . Clearly, @math does not belong to the class of signals with FRI @. However, in @cite_13 , novel algebraic subspace-based approaches solve the sampling problem in the Laplace domain and these methods achieve some form of optimality. @cite_5 , various algorithms including subspace-based approaches @cite_1 ( ESPRIT and MUSIC ) as well as multidimensional search methods were used and comparisons were made. The authors concluded that, in the noisy signal case, the parameters can be recovered at a rate below that prescribed by the Shannon-Nyquist Theorem but at a factor above the critical rate.