# How to explain the meaning of the intersection in a Fourier series representation of periodic signals?

I saw a piece of code on github which transforms the planetary movement into the Fourier wave function.

These circles are given by the $$x$$ and $$y$$ ordinates: $$x=\cos (\omega t)$$, $$y=\sin (\omega t)$$, which are periodic. To help us see this question more clearly, we can imagine it as a series of signals with different frequencies. As we can see visually in the figure below, there are four wave functions (four signals) in the plot. They will intersect at specific time. At the intersection of two wave functions,say, $$\dfrac{4\sin (3\theta )}{3\pi}$$ and $$\dfrac{4\sin (5\theta )}{5\pi}$$, it suggests that these two signals will have the same phase and have the same projected displacement (same y value). But how can we use this information in signal processing? In other words, what may happen when we observe two signal intersect, will the intersection cause phase shift?

Any thoughts would be greatly appreciated.

• This clip might help. Also it would help us to know what you mean by "intersect" and "phase shift" (phase of what?). Jul 28 at 11:19

The shown signals, as you can realize, though periodic and sinusoidal, are not related to Fourier in any way. You are analyzing periods of specific sinusoids in time. Hence, when all sinusoids have the same phase shift, as depicted in your image, if a sinusoid intersect another, i.e. if two sinusoids have the same value and the same derivative at zero: $$x_1(t_0)=x_2(t_0)=0\\ x_1'(t_0)=x_2'(t_0)$$ they will have periods in a rational proportion. You can say only the sign of the derivative matters, since every sinusoidal was normalized by the reciprocal of the period (i.e. the frequency) in this example, making all derivatives equal in amplitude. It is evident that for $$sin(3\theta)$$ and $$sin(7\theta)$$, their periods $$2\pi/3$$ and $$2\pi/7$$ share a proportion $$7:3$$ which is rational.
Moreover, you can easily prove, that if that intersection happens for both a positive (+1) and a negative (-1) derivative for some value $$\theta_1$$ and $$\theta_0$$, their periods will share a rational proportion composed by odd numbers, which happens when the argument of the sinusoidals reach $$2\pi$$ and $$\pi$$ respectively.