In this paper, we study the problem of noise with regard to the perfect reconstruction of non-bandlimited signals, the class of signals having a finite number of degrees of freedom per unit time. The finite rate of innovation (FRI) method provides a means of recovering a non-bandlimited signal through using of appropriate kernels. In the presence of noise, however, the reconstruction function of this scheme may become ill-conditioned. Further, the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity. In this paper, to obtain improved noise robustness, we propose the matrix pencil (MP) method for sample signal reconstruction, which is based on principal component analysis (PCA). Through the selection of an adaptive eigenvalue, a non-bandlimited signal can be perfectly reconstructed via a stable solution of the Yule-Walker equation. The proposed method can obtain a high signal-to-noise-ratio (SNR) for the reconstruction results. Herein, the method is applied to certain non-bandlimited signals, such as a stream of Diracs and nonuniform splines. The simulation results demonstrate that the MP and PCA are more effective than the FRI method in suppressing noise. The FRI method can be used in many applications, including those related to bioimaging, radar, and ultrasound imaging.

We propose a maximum likelihood estimation approach for the recovery of continuously-defined sparse signals from noisy measurements, in particular periodic sequences of Diracs, derivatives of Diracs and piecewise polynomials. The conventional approach for this problem is based on least-squares (a.k.a. annihilating filter method) and Cadzow denoising. It requires more measurements than the number of unknown parameters and mistakenly splits the derivatives of Diracs into several Diracs at different positions. Moreover, Cadzow denoising does not guarantee any optimality. The proposed approach based on maximum likelihood estimation solves all of these problems. Since the corresponding log-likelihood function is non-convex, we exploit the stochastic method called particle swarm optimization (PSO) to find the global solution. Simulation results confirm the effectiveness of the proposed approach, for a reasonable computational cost.

We address a problem of sampling and reconstructing periodic piecewise polynomials based on the theory for signals with a finite rate of innovation (FRI signals) from samples acquired by a sinc kernel. This problem was discussed in a previous paper. There was, however, an error in a condition about the sinc kernel. Further, even though the signal is represented by parameters, these explicit values are not obtained. Hence, in this paper, we provide a correct condition for the sinc kernel and show the procedure. The point is that, though a periodic piecewise polynomial of degree R is defined as a signal mapped to a periodic stream of differentiated Diracs by R + 1 time differentiation, the mapping is not one-to-one. Therefore, to recover the stream is not sufficient to reconstruct the original signal. To solve this problem, we use the average of the target signal, which is available because of the sinc sampling. Simulation results show the correctness of our reconstruction procedure. We also show a sampling theorem for FRI signals with derivatives of a generic known function.