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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 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?]} 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*^{t}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 frequencies
*B*and 2*B*.

## Further reading

- B. F. Logan, Jr. "Information in the Zero Crossings of Bandpass Signals",
*Bell System Technical Journal*, vol. 56, pp. 487–510, April 1977.