Reconstruction from zero crossings
Encyclopedia
The problem of reconstruction from zero crossings can be stated as: given the zero crossing
s 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 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:
Zero crossing
Zero-crossing is a commonly used term in electronics, mathematics, and image processing. In mathematical terms, a "zero-crossing" is a point where the sign of a function changes Zero-crossing is a commonly used term in electronics, mathematics, and image processing. In mathematical terms, a...
s of a continuous
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be 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 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 transformHilbert transformIn mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
xt have no zeros in common with each other. - The frequency domain representation of the signal is at most 1 octaveOctaveIn music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
long, in other words, it is bandpass-limitedBandlimitedBandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...
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