Sampling and reconstruction of signals with finite rate of innovation in the presence of noise

Abstract

Abstract-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, Vetterli et al. 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.

Tomographic Approach for Parametric Estimation of Local Diffusive Sources and Application to Heat Diffusion

“…This algorithm is known to be very unstable for the case of noisy data. Therefore, we use the successful variation of the original algorithm, that is developed in [7]. Good estimates are obtained by performing the oversampling in space with N = 5(2K+1).…”

“…The roots of the annihilating filter polynomial are exactly the u k . The coefficients a k are then directly obtained using (7). From u k we find the positions p k and from a k we find the weights c k .…”

Reconstruction of Finite Rate of Innovation Signals with Model-Fitting Approach

“…However, this method becomes unstable and the accuracy of the reconstruction substantially degrades in the presence of noise. Several approaches to improve resiliency against measurement noise and model mismatch have been proposed [10], [18], [19]. In particular, there are two main approaches to remove the noise on the FRI samples that are derived from the several classes of highresolution and subspace-based methods known from spectral estimation.…”

“…In particular, there are two main approaches to remove the noise on the FRI samples that are derived from the several classes of highresolution and subspace-based methods known from spectral estimation. The first approach is based on state space parametrization of the signal subspace that reduces the denoising problem into the estimation problem of the generalized eigenvalue of matrix pencil [18], [20]. A closely related algorithm, the ESPRIT algorithm, was developed as an extension of the state space method [21], [22].…”

Sequential local FRI sampling of infinite streams of Diracs

“…Thus, measurements (3) becomes m nÈZ c m,n Ôy n ε n Õ s m d m , and alternative methods are needed to retrieve the pairs Ôt k , a k Õ. We may solve the problem by using the total least-squares method and Cadzow denoising algorithm [9] introduced for FRI in [10] or matrix pencil method [11] used for FRI in [12]. Further alternative methods can be found in [13][14][15].…”