Examples & Applications of Quadratic Phase Coupling

Question

Quadratic phase coupling is defined as follows:

Consider the signal $$ x(t) = A_{1} \cos(\omega_{1}t+ \phi_{1}) + > A_{2}\cos(\omega_{2}t + \phi_{2}) $$ which is passed through the quadratic nonlinear system $$ h(t) = ax^{2}(t) $$ where $a$ is non-zero constant. On the output of the system, the signal $x(t)$ will include harmonic components: $(2\omega_{1}, 2\phi_{1})$, $(2\omega_{2}, 2\phi_{2})$, $(\omega_{1} + \omega_{2}, \phi_{1} + > \phi_{2})$, $(\omega_{1} - \omega_{2}, \phi_{1} - \phi_{2})$. Such phenomenon, which produces a formation of these phase relations, is called quadratic phase coupling.

My question is if there are real-life examples and/or applications of quadratic phase coupling?

Answer 1

As is called out in your quoted section, any system that contains a quadratic non-linearity will exhibit a quadratic phase coupling phenomenon to some extent. If it is large enough to be measured is another question. In modeling the dynamics of beam properties (displacement, shear/bending stress, etc.), quadratic and higher-order nonlinear terms are commonly used. For small displacements, these terms are usually considered negligible (common assumptions made in Euler-Bernoulli beam theory).

Keep in mind that systems involving any kind of rotation or periodicity are described by trigonometric terms which are rich with non-linearities and can contain quadratic terms, as can be seen by the Taylor-Series of cosine: $$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$$

When measuring the vibrations of rotating machinery, impacts on gears (e.g. from missing gear teeth) cause a periodic impulse which can be modeled as being generated by a non-linear system. From this, studies have been made on the detection of faults in rotating machines using bicoherence, a common measure of quadratic phase coupling, to identify the presence of non-linearities that have been correlated to certain types of damage.

Answer 2

Just about any real-life communications system suffers from passive intermodulation (PIM). One part of PIM distortions is quadratic (though there are other orders possibly involved too).

There're many patents involved in systems to reduce PIM.