The problem of reconstruction from zero crossings can be stated as: given the zero crossings of a continuoussignal, 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 transformxt 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 2B.
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.
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as a lower bound for the sample rate for alias-free signal sampling and
as an upper bound for the symbol rate across a bandwidth-limited baseband channel such as a telegraph line or passband channel such as a limited radio frequency band or a frequency division multiplex channel.
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Signal Processing is one of the important research fields that is used widely these days in different aspects of our lives. It is a wide area of research that extends from the simplest form of one-dimensional (1-D) signal processing to the complex form of multi-dimensional (M-D). In signal processing, a filter is used to remove or modify some components or features of a signal. A filter bank is an array of filters that separates the input signal into multiple components, each one carrying a single frequency sub-band of the original signal. Filter banks have a lot of applications these days. In filter banks the process of decomposition performed by the filter bank is called analysis and the reconstruction process is called synthesis. This article provides a short survey of the concepts, principles and applications of Multirate Filter Banks and Multidimensional Directional Filter Banks.
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