The problem of reconstruction from zero crossings can be stated as: given the
zero crossing
A zero-crossing is a point where the sign of a mathematical function changes (e.g. from positive to negative), represented by an intercept of the axis (zero value) in the graph of the function. It is a commonly used term in electronics, mathemat ...
s of a
continuous signal
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
, 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
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mult ...
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 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
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...
''x''
''t'' have no zeros in common with each other.
# The frequency-domain representation of the signal is at most 1
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
long, in other words, it is
bandpass
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two- ...
-
limited
Limited may refer to:
Arts and media
*''Limited Inc'', a 1988 book by Jacques Derrida
*Limited series (comics), a comic book series with predetermined length
Businesses
*Limited Brands, an American company - owners of Victoria's Secret, Bath & Bo ...
between some frequencies ''B'' and 2''B''.
Further reading
*
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
Signal processing