Fourier transform, same frequencies, different amplitudes

Question

I understand that the Fourier transform is changing the domain (time/space) to frequency domain and provides the sin waves. I have seen the visualizations of Fourier transform and they are all showing the transform results as the list of frequencies and their amplitude. My question is, what if the result has same sin waves with similar frequencies, but different amplitudes; are they going to be summed up (added/subtracted) in the final result?

Answer 1

Yes. The Fourier Transform is a linear operator and possesses the properties:

$$\alpha \mathscr{F}\left\{f(x)\right\} = \mathscr{F}\left\{\alpha f(x)\right\}$$ $$\mathscr{F}\left\{f(x)\right\} + \mathscr{F}\left\{g(x)\right\} = \mathscr{F}\left\{ f(x) +g(x)\right\}$$

For your specific case:

$$\mathscr{F}\left\{a \sin(2\pi x - \theta_a)\right\} + \mathscr{F}\left\{b \sin(2\pi x -\theta_b)\right\} = \mathscr{F}\left\{a \sin(2\pi x - \theta_a) +b \sin(2\pi x - \theta_b)\right\}$$