Let me define a signal $$f(t)=\sum_{i=1}^N a_i \cos(\omega_it+\theta_i),$$ where $\omega_i = 2\pi f_i$ with a frequency $f_i$.
I want to make a specific $f(t)$ by determining $a_i$'s and $\theta_i$'s on my own will such that $\lvert \max_{t} f(t)\rvert$ and $\lvert \min_{t} f(t)\rvert$ be much different.
For example, when I set $N=2$, $a_1=a_2=1$, $f_1=900\times10^6$, $f_2=1.8\times10^9$, and $\theta_1=\theta_2=0$, i.e.,
$$f(t) = \cos(1.8\pi\times 10^9\,t)+\cos(3.6\pi\times 10^9\,t),$$
$\lvert\max_t f(t) \rvert$ must be $2$, and $\lvert\min_t f(t) \rvert$ is around $1$. The function $f(t)$ is like as follows:
I guess that $\lvert\max_t f(t) \rvert$ and $\lvert\min_t f(t) \rvert$ could be different only when there exist $i$ and $j$ such that $f_i=nf_j$ for a natrual number $n$. But I cannot prove this.
My final goal is to develop a method to determine $a_i$'s and $\theta_i$'s such that $\lvert\max_t f(t) \rvert =c_1$ and $\lvert\min_t f(t) \rvert=c_2$, where $c_1$ and $c_2$ are given value, assuming that $f_i$'s are well given and fixed.
Is there some references about this? Thank you for reading my question.
ps) I am now basically trying to find a method of determining $a_i$'s and $\theta_i$'s when $N=2$ and $f_2=2f_1$. It is also good about the sum of simple two sine waves with frequencies $f$ and $2\times f$.