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Reconstruction From Zero Crossings




This problem has 2 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 before, no conclusive solution has been found. Ben Logan from the Bell Laboratories wrote a paper in 1977 in the ''Bell Systems 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.


FURTHER READING

  • BF Logan, Jr. "Information in the Zero Crossings of Bandpass Signals", ''Bell System Technical. Journal'', vol. 56, pp. 487-510, April 1977



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