On Saturday, February 22, 2014 5:36:29 PM UTC-5, gyans...@gmail.com wrote:
> I am trying to understand Bispectrum by doing some simple examples. I have looked at most of the literature but few of them give any simple examples.
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> Suppose I have an FIR filter (NMPhase). Then it's z-transfer function spectrum I can find easily suppose a(z^-1)=1+a1z^-1 and it is driven with zero-mean non-Guassian white noise unit variance.
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> P(z^-1)=a(z^-1)a(z)
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> It's Bispectrum - is it
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> P(z1^-1,z2^-1)=a(z1^-1)a(z2^-1).a(z1+z2)
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> so if a(z^-1)=1+2z^-1 its Bispectrum is
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> (1+2z1^-1)(1+2z2^-1)(1+2(z1+z2))
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> which one multiplies out.
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> Also, to find the Bispectrum from data I read that you can get it from the Cumulant (no 3). I assume you then need the 2D FFT to get the BiSpectrum?
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> But I saw this
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> B(w1,w2)=a(w1)a(w2)a*(w1+w2)
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> P(w1,w2) =!X(n1)!!X(n2)! X!(n1+n2)!*
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> (sorry I don't have the pipe symbol on this keyboard).
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> So do you use a 1D FFT and a 2D FFT?
My exposure and use of bispectral analysis comes from non-redundant masking of telescopes in astronomy. If my telescopes are in a straight line, I use 1-D fourier analysis to get my closure phases.
What you are doing is essentially the same, so I'd say you use a 1-D FFT.
Clay
Reply by ●February 22, 20142014-02-22
On Sunday, February 23, 2014 11:36:29 AM UTC+13, gyans...@gmail.com wrote:
> I am trying to understand Bispectrum by doing some simple examples. I have looked at most of the literature but few of them give any simple examples.
>
>
>
> Suppose I have an FIR filter (NMPhase). Then it's z-transfer function spectrum I can find easily suppose a(z^-1)=1+a1z^-1 and it is driven with zero-mean non-Guassian white noise unit variance.
>
>
>
>
>
> P(z^-1)=a(z^-1)a(z)
>
>
>
> It's Bispectrum - is it
>
>
>
> P(z1^-1,z2^-1)=a(z1^-1)a(z2^-1).a(z1+z2)
>
>
>
> so if a(z^-1)=1+2z^-1 its Bispectrum is
>
>
>
>
>
> (1+2z1^-1)(1+2z2^-1)(1+2(z1+z2))
>
>
>
> which one multiplies out.
>
>
>
> Also, to find the Bispectrum from data I read that you can get it from the Cumulant (no 3). I assume you then need the 2D FFT to get the BiSpectrum?
>
>
>
> But I saw this
>
>
>
> B(w1,w2)=a(w1)a(w2)a*(w1+w2)
>
>
>
> P(w1,w2) =!X(n1)!!X(n2)! X!(n1+n2)!*
>
>
>
> (sorry I don't have the pipe symbol on this keyboard).
>
>
>
> So do you use a 1D FFT and a 2D FFT?
actually that FFT calculation should read
P(w1,w2) =!X(n1)!!X(n2)! X!(n1,n2)!*
Reply by ●February 22, 20142014-02-22
I am trying to understand Bispectrum by doing some simple examples. I have looked at most of the literature but few of them give any simple examples.
Suppose I have an FIR filter (NMPhase). Then it's z-transfer function spectrum I can find easily suppose a(z^-1)=1+a1z^-1 and it is driven with zero-mean non-Guassian white noise unit variance.
P(z^-1)=a(z^-1)a(z)
It's Bispectrum - is it
P(z1^-1,z2^-1)=a(z1^-1)a(z2^-1).a(z1+z2)
so if a(z^-1)=1+2z^-1 its Bispectrum is
(1+2z1^-1)(1+2z2^-1)(1+2(z1+z2))
which one multiplies out.
Also, to find the Bispectrum from data I read that you can get it from the Cumulant (no 3). I assume you then need the 2D FFT to get the BiSpectrum?
But I saw this
B(w1,w2)=a(w1)a(w2)a*(w1+w2)
P(w1,w2) =!X(n1)!!X(n2)! X!(n1+n2)!*
(sorry I don't have the pipe symbol on this keyboard).
So do you use a 1D FFT and a 2D FFT?