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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for the Matlab Higher Order Spectral Analysis (HOSA) Toolbox, which seems like a nice resource (though it appears to be no longer maintained and now suffering from bit-rot).

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3 Answers

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Such objects are used all the time. The mathematics is done in terms of quantum fields, which to some extent conceals what's going on.

For example, your "(frequency-domain) coherence" is a correlation coefficient, which is normalized, whereas Physicists typically work in terms of correlation functions, which typically are not, but they largely amount to the same thing. Your observables $X(f_1)$, etc., are constructed as functions of frequency, however this is a singular object in quantum field theory. In quantum field theory, we instead construct observables $\phi(F_1)$, etc., as functionals of test functions $F_1(x)$, etc. One singular choice would be $F_1(x)=\exp(if_1\cdot x)$, which makes $\phi(F_1)$ essentially the same object as your $X(f_1)$; it's singular, however, because $F_1(x)$ is not square integrable.

Another choice of singular test function is, of course, $\delta(x-y)$, which gives the value of the field at a point, which we might write in your terms as something like $X(y)$. For a quantum field, this is also a rather singular object.

In fact, when you say $X(f_1)$, what you really mean is $\int X(f) {\mathrm d}f$, over some small range of frequencies, and in the mathematical and experimental details this has to be taken into account. Making everything precise requires that we know what the frequency ranges of each of the measurements is, which an experimenter either must characterize or must read off from a manufacturer's data sheets. In even more detail, we will have to construct a weight function, saying that frequencies near $f_1$ are still registered by the measurement device, but not as much as near $f_1$. We may well take the weight function, as a first approximation, to be Gaussian. This corresponds to taking the test function $F_1(x)$ to be that Gaussian. Signal analysis usually calls the test function a window function. Test or Window functions can be difficult to become familiar with, but I believe it's well worth getting there.

In these terms, your $C_{xyz}$ is a particular choice of (normalized) 3-point function. The choice of $f_1+f_2$ for the third frequency is of course not necessary, we can consider 3-point correlations between any three frequencies $f_1,f_2,f_3$ (and their vicinities). In quantum field theory, we would represent the 3-point correlation function, in the vacuum state, as $\left<0\right|\phi(F_1)\phi(F_2)\phi(F_3)\left|0\right>$. Replace the vacuum vector by some other state vector, if you like.

In the particular case when quantum field observables are mutually commutative, it can be understood to generate probability measures that correspond to an equivalent description in terms of probability measures over classical random variables, and hence quite precisely to a stochastic signal analysis. When quantum field observables do not commute, everything gets lots more complicated, but a remnant of the signal processing point of view can be maintained.

There is a mathematics of random fields that is used in cosmology because it is generally not necessary to worry about measurement incompatibility in that context. Mathematicians generally present signal analysis in Hilbert space terms unless they are writing for an engineering audience.

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Thanks for the intriguing response! –  nibot Feb 22 '11 at 23:02
    
The signal analysis perspective on quantum fields is not much discussed, AFAIK, but I find it extremely effective. It is very much a fields instead of particles perspective on QFT, which sets it somewhat at odds with the traditional textbook presentation of QFT in Physics. I'm working on constructing interacting quantum fields in ways that are motivated by this approach (though there are so many approaches out there, and my stuff rarely works nicely enough). Some of my perspective on non-interacting random/quantum fields can be found in my published papers (flawed though they are). –  Peter Morgan Feb 23 '11 at 4:17
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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.

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Thanks for the response. Are you talking about, for instance, a three-point correlation function on the sphere? Do you have any references to papers that show this stuff? –  nibot Feb 18 '11 at 20:20
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Yes, in the context of the microwave background, a 3-point function on the sphere is exactly what's computed. Usually it's done in spherical harmonic space, not in real space, though. See, for instance, prd.aps.org/abstract/PRD/v80/i4/e043510 . I think that people talk about the bispectrum of various functions (density perturbations, e.g.) in 3-D space too. Once again, this is generally done in Fourier space, not in real space. –  Ted Bunn Feb 18 '11 at 20:33
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My guesses are:

  1. Much more data is needed to produce a low-noise bispectrum than power spectrum(?).
  2. The results are confusing to interpret (or at least unfamiliar). Part of this is due to the large amount of redundancy in the output.
  3. The sheer amount of data in the bispectrum can be huge. A bispectrum with the same frequency resolution as a power spectrum requires the square of the amount of memory!
  4. Any good clocks will appear to be bispectrally coherent. Transients also create spurious bicoherence.
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