Reconstruction from zero crossings
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The problem of reconstruction from zero crossings can be stated as: given the zero crossings of a continuous signal, is it possible to reconstruct the signal (to within a constant factor)? Worded differently, what are the conditions under which a signal can be reconstructed from its zero crossings?
This problem has two parts. Firstly proving that there is a unique reconstruction of the signal from the zero crossings and secondly how to actually go about reconstructing the signal. Though there have been quite a few attempts, no conclusive solution has yet been found. Ben Logan from Bell Labs wrote a paper in 1977 in the Bell System Technical Journal giving some criteria under which unique reconstruction is possible. Though this has been a major step towards the solution, many people are dissatisfied with the type of condition which results from his paper.
According to Logan a signal is uniquely reconstructible from its zero crossings if:
- The signal x(t) and its Hilbert transform xt have no zeros in common with each other.
- The frequency domain representation of the signal is at most 1 octave long, in other words, it is bandpass-limited between some B and 2B.
- BF Logan, Jr. "Information in the Zero Crossings of Bandpass Signals", Bell System Technical Journal, vol. 56, pp. 487-510, April 1977