Reconstruction of aperiodic FRI signals and estimation of the rate of innovation based on the state space method

In the absence of observation noise, it is known that it is possible to develop exact sampling schemes for a large class of parametric non-bandlimited signals, namely, certain signals of finite rate of innovation (FRI signals), either periodic or aperiodic, such as streams of Diracs, nonuniform splines or piecewise polynomials. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and they can be reconstructed from a finite number of uniform samples of the filtered signal. Unfortunately, the accuracy of such reconstruction substantially degrades when the samples are distorted by noise. For the case of periodic FRI signals, good algorithms based on the state space method have been proposed which are robust against noise. However, in the case of aperiodic signals, these algorithms may still fail to accurately reconstruct the signals due to ill-conditioning problems that often arise in the matrices involved. This paper proposes a new reconstruction method for aperiodic FRI signals that is also based on the state space method but has a considerably better numerical conditioning than previous reconstruction algorithms. This advantage is achieved by using a frequency domain formulation.