# Examples & Applications of Quadratic Phase Coupling

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?

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: $$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+...$$