View source for Reconstruction from zero crossings

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The problem of '''reconstruction from zero crossings''' can be stated as: given the [[zero crossing]]s of a [[continuous signal]], is it possible to [[Signal reconstruction|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. [[Benjamin F. Logan|Ben Logan]] from [[Bell Labs]] wrote an article 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{{who|date=February 2018}} are dissatisfied with the type of condition that results from his article.

According to Logan, a signal is uniquely reconstructible from its zero crossings if:
# The signal ''x''(''t'') and its [[Hilbert transform]] ''x''<sup>''t''</sup> 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]]-[[bandlimited|limited]] between some frequencies ''B'' and 2''B''.

== Further reading ==
* {{cite journal |first=B.F. |last=Logan, Jr |title=Information in the Zero Crossings of Bandpass Signals |journal=Bell System Technical Journal |volume=56 |pages=487–510 |date=April 1977 |issue=4 |doi=10.1002/j.1538-7305.1977.tb00522.x |s2cid=1636877 |url=http://languagelog.ldc.upenn.edu/myl/Logan1977.pdf}}


==References==
{{reflist}}

== External links ==
* {{cite conference |last1=Curtis |first1=S. |first2=A. |last2=Oppenheim |first3=Jae |last3=Lim  |title=Reconstruction of two-dimensional signals from threshold crossings |book-title=ICASSP'85. IEEE International Conference on Acoustics, Speech, and Signal Processing |volume=10 |pages=1057–1060 |publisher=IEEE |date=1985 |url=http://www.rle.mit.edu/dspg/documents/recon2dim_1985.pdf |doi=10.1109/ICASSP.1985.1168139 |lccn=84-62724}}

[[Category:Signal processing]]