Unlimited Sampling of Sparse Sinusoidal Mixtures

Abstract

In parallel to Shannon's sampling theorem, the recent theory of unlimited sampling yields that a bandlimited function with high dynamic range can be recovered exactly from oversampled, low dynamic range samples. In this way, the unlimited sampling methodology circumvents the dynamic range problem that limits the use of conventional analog-to-digital converters (ADCs) which are prone to clipping or saturation problem. The unlimited sampling theorem is made practicable by using a unique ADC architecture-the self-reset ADC or the SR-ADC-which resets voltage before clipping, thus producing modulo or wrapped samples. While retaining full dynamic range of the input signal, surprisingly, the sampling density prescribed by the unlimited sampling theorem is independent of the maximum recordable voltage of the new ADC and depends only on the signal bandwidth. As the corresponding problem of signal recovery from such modulo samples arises in various applications with different signal models, where the original result does not directly apply, the original paper continues to trigger research follow-ups. In this paper, we investigate the case of sampling and reconstruction of a mixture of $K$ sinusoids from such modulo samples. This problem is at the heart of spectral estimation theory and application areas include active sensing, ranging, source localization, interferometry and direction-of-arrival estimation. By relying on the SR-ADCs, we develop a method for recovery of $K$ -sparse, sum-of-sinusoids from finitely many wrapped samples, thus avoiding clipping or saturation. As our signal model is completely characterized by $K$ pairs of amplitudes and frequencies, we obtain a parametric sampling theorem; we complement it with a recovery algorithm. Numerical demonstrations validate the effectivity of our approach.