Homework from electronics class, but since it looks like a math problem to me i decided to look for help here.
I am given this composite periodic signal that has 6 harmonics:
$$V(t)=V_0+\sum_{k=1}^6V_k\;cos(k\ \omega t+\phi_k)$$
I also have a table of 13 values for $V(t)$ that correspond to 13 moments in time ($t$) during one period of the signal. From the table i am easily able to find the value of $\ \omega$.
But i also need to determine amplitude($V_k$) and phase($\phi_k$) of each individual harmonic(that's 12 variables) as well as the offset ($V_0$).
So from the table i am able to form 13 equations with 13 variables(unknowns) that look like, for example:
$$19,714=V_0+\sum_{k=1}^6V_k\;cos(\phi_k)\\ -13,15=V_0+\sum_{k=1}^6V_k\;cos(k \frac {2\pi}{13}+\phi_k)\\ 0,1716=V_0+\sum_{k=1}^6V_k\;cos(k \frac {4\pi}{13}+\phi_k)\\ ...\\ -4,851=V_0+\sum_{k=1}^6V_k\;cos(k \frac {24\pi}{13}+\phi_k) $$
Here is the full list of equations for those that are interested:
I realize that i probably should expand the sums and start eliminating variables somehow but the problem is i don't even know where to start.What is the easiest/fastest way to solve this?