Finding set of orthogonal basis functions for composite signals

Question

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?

Answer 1

Your $S_1,\ldots, S_5$ already are your orthogonal bases, exactly the way your first equation writes them down.

To check that: write down the definition of your inner product, and insert, for example, $S_a$ and $S_b$. You'll see that $\left<S_a,S_b\right>=0 \iff a\ne b$.

This follows quite directly from the linearity that a (real) inner product needs to have:

\begin{align} \Big<\sum_i A_{a,i}\sin(2\pi f_{a,i}t -\phi_{a,i}), \sum_k A_{b,k}\sin(2\pi f_{b,k}t -\phi_{b,k}) \Big>&\\ = \sum_i \Big< A_{a,i}\sin(2\pi f_{a,i}t -\phi_{a,i}), \sum_k A_{b,k}\sin(2\pi f_{b,k}t -\phi_{b,k}) \Big>&\\ =\sum_i\sum_k \Big< A_{a,i}\sin(2\pi f_{a,i}t -\phi_{a,i}), A_{b,k}\sin(2\pi f_{b,k}t -\phi_{b,k}) \Big> \end{align}

and sines of different frequencies are orthogonal.