Multi-spectral phase coherence (MSPC) is a generalized cross-frequency

coupling A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mo ...

metric introduced by Yang and colleagues in 2016. MSPC can be used to quantify

nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...

phase coupling between a set of base frequencies and their

harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...

intermodulation Intermodulation (IM) or intermodulation distortion (IMD) is the amplitude modulation of Signal (electrical engineering), signals containing two or more different frequencies, caused by non-linear, nonlinearities or time variance in a system. ...

frequencies. MSPC is a model-free method, which can provide a system description, including (i) the order of the nonlinearity, (ii) the direction of interaction, (iii) the time delay in the system, and both (iv) harmonic and (v) intermodulation coupling. The MSPC is defined as: :

\Psi(f_i,a_i) = \left\langle \exp \left( j\left(\sum_i a_i\varphi(f_i) -\varphi \left(f_\text\right)\right) \right) \right\rangle

where

\varphi(f_i)

is the phase at frequency

f_i

a_i

is the weight of

f_i

to a harmonic/intermodulation frequency

f_\text = \sum_i a_if_i

), and

\langle \cdot \rangle

represents the average over realizations. Bi-phase locking value,{{cite journal, last2=Ojemann, first2=JG, last3=Sorensen, first3=LB, date=15 May 2009, title=Bi-phase locking – a tool for probing non-linear interaction in the human brain., journal=NeuroImage, volume=46, issue=1, pages=123–32, pmid=19457390, last1=Darvas, first1=F, doi=10.1016/j.neuroimage.2009.01.034, pmc=2778057 also called bi-phase coherence in the literature, is a special case of MSPC when

a_1=a_2=1

i = 1, 2.

The time-delay can be estimated from the

phase lag In the aerodynamics of rotorcraft like helicopters, phase lag refers to the angular difference between the point at which a control input to a rotor blade occurs and the point of maximum displacement of the blade in response to that control input ...

when MSPC is computed between signals.

References

Nonlinear functional analysis Spectral theory