I've been given a list of 5 composite signals, where each is composed of 10 sinusoids of different frequencies. For instance, the first composite signal $S_1$ is given by
$$ S_1 = \sum_{i=1}^{10} A_i \sin (2 \pi f_i t- \phi_i) $$ where $A_i, f_i$ and $\phi_i$ are known values. The other 4 composite signals are similar to $S_1$. In total, I have 50 different values for $A_i, f_i$ and $\phi_i$ (10 for each $S_i$).
My goal is to find the orthogonal basis functions for $\{S_1, S_2, \cdots, S_5\}$. I've decomposed them into $\sin$ and $\cos$: $$ \sum_{i=1}^{10} A_i \sin (2 \pi f_i t- \phi_i) = \sum_{i=1}^{10} A_i [\sin(2\pi f_i t)\cos(\phi_i) - \cos(2\pi f_i t)\sin(\phi_i)] $$ and from here I can see that $A_i, \cos(\phi_i), \sin(\phi_i)$ are constants, and a spanning set would be
$$ \bigcup_{k=1}^{50} \{\sin(2\pi f_k t), \cos(2\pi f_k t) \} $$ so I end up with a 100-element set. I presume these are my orthogonal basis functions, since each $f_k$ is unique. Can I check if my train of thought is right?