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_28: Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, nonuniform splines or piecewise polynomials, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. Recently, it was shown that, by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation, it is possible to reconstruct such signals uniquely . These sampling schemes, however, use kernels with infinite support, and this leads to complex and potentially unstable reconstruction algorithms. In this paper, we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying Strang-Fix conditions, exponential splines and functions with rational Fourier transform. This last class of kernels is quite general and includes, for instance, any linear electric circuit. We, thus, show with an example how to estimate a signal of finite rate of innovation at the output of an RC circuit. The case of noisy measurements is also analyzed, and we present a novel algorithm that reduces the effect of noise by oversampling
@cite_22: Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the Nth-order decay of the L sub 2 -approximation error as a function of the knot spacing T.
@cite_3: 
Subsequently, Dragotti @cite_28 examined acquisition of the same signals with an eye toward implementability of the sampling kernel. Instead of using the sinc and Gaussian kernels (which do not have compact support), the authors limited the choice of kernels to functions satisfying the Strang--Fix conditions @cite_3 ( e.g. splines and scaling functions), exponential splines @cite_22 and functions with rational Fourier transforms. They combined the moment-sampling and annihilating filter approaches to solve for the parameters. In our work, however, we will continue to use the Gaussian as our sampling kernel. We believe that, even though the Gaussian has infinite support, it can be well approximated by its truncated version. Hence, we can still draw insights from the analysis of using Gaussian filters and the subsequent reconstruction of the signal from its noisy samples @math . More importantly, unlike with previous approaches, the sampling kernel plays no fundamental role in the reconstruction algorithm. We use the Gaussian kernel because of its prominence in earlier work and the intuitiveness of its information spreading properties.