Given Fourier series coefficients $a_1, a_2, \dots, b_1,b_2, \dots$, what is the maximum value the constructed signal can have?
I am interested in normalizing this signal for which knowing this maximum value makes it possible for me. Another solution for me could also be enforcing a relation between the coefficients so that the resulting signal has a maximum value of one, but I don't know what relation.
A heuristic way for me is of course also to construct the signal on a grid and pick the maximum.
In the end I may only have 10 to 20 coefficients in total, not more. I am not sure though what relation I can enforce between the coefficients. The coefficients are bonded between [0,1]
Best Answer
You could imagine the two terms (cos and sin) being in the same mode to describing describe an ellipses in 2D. Whereby $a_n$ and $b_n$ express the size of the axis. Now you could heuristically limit the normalizing constant to be between $\sum_{i=1}^n{\max(a_n,b_n)}$ (when all of the coefficients are very different in magnitude) and $\sqrt{2} \sum_{i=1}^n{\max(a_n,b_n)}$ (when all of the coefficients are the same in magnitude)