Bispectrum

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?

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)!*

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.
> 
> 
> 
> 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?

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