On reconstruction of bandlimited signals from purely timing information

Abstract

The well known Logan’s theorem asserts that, under some assumptions, a one octave bandpass signal is recoverable, modulo a multiplicative constant, from its zero crossings only. However, such recovery is numerically problematic and the theorem is not applicable to general bandlimited signals. In this paper, we demonstrate that the additional timing information sufficient for recovery of general band limited signals can be provided in the form of the zero crossings of several of its derivatives and propose novel numerically robust algorithms for such timing extraction and for signal reconstruction, both with high fidelity. We tested the proposed algorithms extensively, with both synthetic and audio signals. In particular, we numerically demonstrate that the timing of the zero crossings of a typical speech signal and of its first two derivatives are not sufficient for a numerically robust recovery of such a signal, but that a robust recovery becomes possible by adding the timing of the zero crossings of the third order derivative.