Unlimited Sampling in Phase Space

The Unlimited Sampling Framework (USF) is an alternative sampling and reconstruction protocol that allows for recovery of signals that are arbitrarily larger than the dynamic range of the analog-to-digital converter (ADC). In the USF, modulo non-linearity prior to sampling folds high dynamic range inputs that may potentially saturate a conventional ADC. In the recovery step, algorithms perform signal unfolding. Current theory backing the USF is predominantly focused towards Fourier bandlimited function classes. By generalizing the notion of bandlimitedness to a much larger class of unitary transforms, in this work, we propose Unlimited Sampling in Phase Space. The advantages are two-fold. Firstly, we generalize the USF to a much larger class of unitary transforms. Secondly, functions such us chirps are non-bandlimited in the Fourier domain but bandlimited in the phase space. This allows for practical applications that can not be handled by current, Fourier domain theory. Our main contribution in this paper is a novel modulo acquisition pipeline in phase space and a mathematically guaranteed recovery algorithm that is also backwards compatible with Fourier domain theory. Computer experiments validate our theory and reinforce its potential for new application areas such as radar and optics.