I know my response is a bit late but I hope it can help someone who got stuck on the bispectrum just as I have in the last few days. Since I've just learnt about it today, you should proof check everything in this comment. Anyway, my goal is mostly to give you an overall and intuitive view of what it is. You should also refer to : Digital Bispectral Analysis and Its Applications to
Nonlinear Wave Interactions by Young C. Kim (1979).
Here's my match plan :
- Introduce moment-generating function (MGF) and cumulant generating function (CGF).
- Introduce the formula for the bispectrum
- Try to interpret what it means
1) The first order MGF $M_X(t)$ can be thought of as the expected value of a function $f(x)$ of a random variable and is written
$$ M_X(t)= E[f(x)] = \int_{-\infty}^\infty e^{tx} f(x) dx $$
where E denote the expected value. An higher order only means we deal with more than 1 function. Knowing probability density function, we do the convolution $*$ of our sequence of function $$ M_{X_1...X_N}(t)= E[f_1(x)...f_N(x)] = \int_{-\infty}^\infty e^{tx} f_1(x)*...*f_N(x) dx $$
The CGF is more complex but as long as we assume stationary wave (which is usually assumed and means $x(t)=\sum_k X_k e^{-i\omega_kt}$ where $X_k$ doesn't depend on $t$, we can write $C_X(t)=M_X(t)$ for second and third order.
2) From Wikipedia : The bispectrum is the third-order cumulant-generating function.
What does it means ? First, we need 3 functions since it is third order : $f_1$, $f_2$ and $f_{1+2}$. The bispectra is defined
$$B(k,l) = F\{C_3(f_1, f_2, f_1+f_2)\} = F\{ E[f_1 \cdot f_2 \cdot f^C(f_1 + f_2)]\} $$ where F is the Fourier transform, $C$ means complex conjugate and we are doing regular multiplication (instead of convolution). To be honest, I'm not really familiar with the cumulant spectra but it seems close to a quantum mechanical wavefunction. Using the convolution theorem, we get
$$ B(f_1,f_2) = \int_{-\infty}^\infty e^{tx} f_1*f_2*(f_1+f_2) dx = F[f_1] \cdot F[f_2] \cdot F^C[f_1 + f_2]$$
3) What is it actually. The bispectrum is the Fourier-transformed counterpart of the three-point correlation function. The three-point correlation function is a function which if given two values at two points, gives us the expected value for a third point. i.e. If we have the density at two points $\rho(r_1), \rho(r_2)$ of a nebulae, it will gives us the correlation at a third point (We assume the value we want to find as a circular distribution which doesn't depend on the angle). A correlation function measure the order of a system as how the spin or the density at different positions are related. So if I understand correctly, and as I said, take this with a grain of salt as I'm no expert, instead of getting the value at two points, we get the value at two frequencies. To continue the example, we would get the density $\rho(f_1), \rho(f_2)$ and determine the correlation with a third frequency.
Normally, we will study correlation function knowing only one position or one frequency and try to determine it at another value. The power of the bispectra is that, since it depends on two values, it can captures higher order effect.
