Power spectra, coherence spectra, and linear transfer functions are ubiquitous tools of experimental physics. However, our instruments often retain small nonlinear effects which can contaminate measurements. It appears that higher order spectra, in particular the bispectrum, would be ideal tools to investigate nonlinear interactions. Nonetheless, I've never actually seen them put to use in experimental physics.
For example, consider the (frequency-domain) coherence:
$C_{xy}(f) = \frac{\langle X(f)Y(f)^*\rangle}{\sqrt{\langle X(f)X(f)^*\rangle\langle Y(f)Y(f)^*\rangle}}$
The bicoherence considers not two but three signals, and looks for correlations between oscillations at frequencies $f_1$ and $f_2$ combining nonlinearly to produce a signal at $f_1+f_2$:
$C_{xyz}(f_1,f_2) = \frac{\langle X(f_1) Y(f_2) Z^*(f_1 + f_2)\rangle}{ \sqrt{\langle X(f_1)X^*(f_1)\rangle\langle Y(f_2)Y^*(f_2)\rangle\langle Z(f_1+f_2)Z^*(f_1+f_2)\rangle} } $
…which seems like a useful thing to do.
Why are the bispectrum and bicoherence not used more frequently in experimental physics?
I am specifically thinking about time domain, multi-input/multi-output systems where one is looking for nonlinear couplings between various signals.
One of the top Google hits on the subject is for the Matlab Higher Order Spectral Analysis (HOSA) Toolbox, which seems like a nice resource (though it appears to be no longer maintained and now suffering from bit-rot).
Best Answer
It's used a lot in cosmology. Often, to a decent approximation, the quantities we try to measure in cosmology (e.g., CMB temperature and polarization maps, galaxy distributions) are realizations of Gaussian random processes to a decent approximation, but have (or are predicted to have) interesting non-Gaussian features at some low level. People estimate the bispectra of these things all the time to get a handle on various non-Gaussian effects.