FRI sampling and time-varying pulses: Some theory and four short stories

Abstract

The field of signal processing is replete with exemplary problems where the measurements amount to time-delayed and amplitude scaled echoes of some template function or a pulse. When the inter-pulse spacing is favorable, something as primitive as a matched filter serves the purpose of identifying time-delays and amplitudes. When the inter-pulse spacing poses an algorithmic challenge, high-resolution methods such as finite-rate-of-innovation (FRI) may be used. However, in many practical cases of interest, the template function may be distorted due to physical properties of propagation and transmission. Such cases can not be handled well by existing signal models. Inspired by problems in spectroscopy, radar, photoacoustic imaging and ultra-wide band arrays, on which we base our case studies, in this work we take a step towards recovering spikes from time-varying pulses. To this end, we re-purpose the FRI method and extend its utility to the case of phase distorted pulses. Application of our algorithm on the above-mentioned case studies results in substantial improvement in peak-signal-to-noise ratio, thus promising interesting future directions.