# Finding set of orthogonal basis functions for composite signals

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?

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 $$\left=0 \iff a\ne b$$.
\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}
• Could you include a more detailed and less hand-wavy proof of your claim that the 5 signals are indeed orthogonal? Let's simplify the question even more with each $S_i$ being just one sinusoid instead of the sum of 5 different sinusoids. Why are $\sin(2\pi 1000t)$ and $\sin(2\pi (1000\sqrt 2)t)$ orthogonal signals? And over what time interval are they orthogonal? – Dilip Sarwate Mar 5 at 20:30
• @DilipSarwate in undergrad courses, harmonic oscillations are often axiomatically introduced as base functions. However, showing the orthogonality of your example is relatively straightforward through properties of any inner product: Iff $x,y$ orthogonal, then $<x+y,x+y>=<x,x>+<y,y>$. The classical power signal space $\tilde L_2(\mathbb R)$-inner product $<x,y>=\lim_{T\to\infty}\frac1{2T}\int_{-T}^Tx(t)y^*(t)\,\mathrm dt$ fulfills exactly that. We can also take the cross-spectrum detour, showing their Fourier-domain product is a constant 0, but I'm sure you'll appreciate the direct 0-integral. – Marcus Müller Mar 5 at 21:01