Information in the Zero Crossings of Bandpass Signals

April 1977 Telephone and Telegraph Company. Printed in Number 4 I l.S.A. Information in the Zero Crossings of Bandpass Signals By B. F. LOGAN, JR. (Manuscript received October 4, 1976) An interesting subclass of bandpass signals h is described wherein the zero crossings of h determine h within a multiplicative constant. The members may have complex zeros, but it is necessary that h should have no zeros in common with its Hilbert transform ft other than real simple zeros. It is then sufficient that the band be less than an octave in width. The subclass is shown to include full-carrier upper-sideband signals (of less than an octave bandwidth). Also it is shown that fullcarrier lower-sideband signals have only real simple zeros (for any ratio of upper and lower frequencies) and, hence, are readily identified by their zero crossings. However, under the most general conditions for uniqueness, the problem of actually recovering h from its sign changes appears to be very difficult and impractical. I. INTRODUCTION Voelcker and Requicha 1 raised the question, among others, as to when a bandpass signal h(t) might be recovered (within a multiplicative constant) from sgn h(t), that is, from its zero crossings. There are really two questions here that should be treated separately: the question of uniqueness and the question of recoverability. Recoverability implies that there is an effective (stable) way of recovering the signal from the data. Uniqueness does not always imply recoverability.