Given Fourier series coefficients, what is the maximum value the constructed signal can have?

Question

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]

Answer 1

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)

Answer 2

There is not a closed formula for the maximum, in the general case.
You added that you have between 10 to 20 coefficients (suppose you count the pairs).

With a modern CAS such an amount of coefficients is quite manageable. Take the shortest period, divide it by a suitable factor (say $10$) and compute the sequence of the signal values at $x=k \tau$. Find the absolute maximum and eventually refine the search around it.

Alternatively, speaking of approximated methods, find the power of signal, and determine in which frequencies it concentrates.
If these are a phew, and the remaining frequencies account for a small portion of the power, then you can concentrate the analysis on the more powerful and regard the others as noise (more or less white) for the amplitude of which there are statistical estimates.

Otherwise, just knowing that the coefficients range in $[0,1]$ you can analyze the whole signal statistically.