Approximate Strang-Fix: sampling infinite streams of Diracs with any kernel

In the last few years, several new methods have been developed for the sampling and the exact reconstruction of specific classes of non-bandlimited signals known as signals with finite rate of innovation (FRI). This is achieved by using adequate sampling kernels and reconstruction schemes. An important class of such kernels is the one made of functions able to reproduce exponentials. In this paper we review a new strategy for sampling these signals which is universal in that it works with any kernel. We do so by noting that meeting the exact exponential reproduction condition is too stringent a constraint, we thus allow for a controlled error in the reproduction formula in order to use the exponential reproduction idea with any kernel and develop a reconstruction method which is more robust to noise. We also present a novel method that is able to reconstruct infinite streams of Diracs, even in high noise scenarios. We sequentially process the discrete samples and output locations and amplitudes of the Diracs in real-time. In this context we also show that we can achieve a high reconstruction accuracy of 1000 Diracs for SNRs as low as 5dB.