Inverse of bispectrum?
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From: Anindya G on 21 Jun 2010 16:15 Hello, I am a newbie in higher-order spectra analysis. Please excuse me if anything appears really incoherent in my question. Some of my initial readings suggest that for a Gaussian signal, its bispectrum is zero. In this connection, suppose I have an additive mixture of Gaussian and Non-Gaussian signals and I take the bispectrum of this mixture. Then the bispectrum of Gaussian signal being zero, I should only have the bispectrum of the non-Gaissian signal as the final result. Is it now possible to take the inverse of this bispectrum and recover the non-Gaussian signal? Is there something called "Inverse Bispectrum"? Any insight will be greatly appreciated. If there is some recommended reading in this filtering process, please do suggest. Regards, Anindya G.
From: illywhacker on 22 Jun 2010 09:22
On Jun 21, 10:15 pm, Anindya G <kamaskar1...(a)gmail.com> wrote: > Some of my initial readings suggest that for a Gaussian signal, its > bispectrum is zero. In this connection, suppose I have an additive > mixture of Gaussian and Non-Gaussian signals and I take the bispectrum > of this mixture. Then the bispectrum of Gaussian signal being zero, I > should only have the bispectrum of the non-Gaissian signal as the > final result. Is it now possible to take the inverse of this > bispectrum and recover the non-Gaussian signal? Is there something > called "Inverse Bispectrum"? > > Any insight will be greatly appreciated. If there is some recommended > reading in this filtering process, please do suggest. Be careful not to confuse two different things. Thr first are properties of a probability distribution (e.g. the second- and third- order cumulants). The second are properties of an individual signal (e.g the covariance and the bispectrum). The latter are frequently estimates of the former, but they are not the same. A Gaussian distribution has vanishing higher-order cumulants. Any given signal, however, even if simulated from a Gaussian distribution, will not in general have vanishing bispectrum, although probably it will be 'small' in some sense. So, if you compute the bispectrum of your signal, the contribution of the Gaussian part will probably be small. The bispectrum is thus some kind of an estimate of the third-order cumulant of the probability distribution for the second, non-Gaussian component of your signal. On its own, however, it does not permit to reconstruct this distribution in a strict sense (unless you have a model in which it is the only unknown parameter). You could use the information as part of a maximum entropy procedure, though. This does not seem to be what you are asking, however. You seem to want to reconstruct the signal itself. This is possible. See this wikipedia entry http://en.wikipedia.org/wiki/Triple_correlation for discussion of, and references to, the uniqueness of a function given its bispectrum. illywhacker; |