I am trying to model something in wireless channels, but I am stuck with something. I am trying to find the absolute value of a 'complex' complex function. The first complex refers to the number of terms (complexity) in the function, while the second one refers to the usual meaning of complex in math.
So here is the function
$$ e^{j\phi} \frac{1-e^{j\pi N(cos\theta_s-cos\theta_1)}}{1-e^{j\pi (cos\theta_s-cos\theta_1)}}+e^{-j\phi} \frac{1-e^{j\pi N(cos\theta_s-cos\theta_2)}}{1-e^{j\pi (cos\theta_s-cos\theta_2)}} $$
or in a more compact form $$ f(N,\Delta_1, \Delta_2)=e^{j\phi} \frac{1-e^{jN\Delta_1}}{1-e^{j\Delta_1}}+e^{-j\phi} \frac{1-e^{jN\Delta_2}}{1-e^{j\Delta_2}} $$
where $\Delta_m = \pi(cos\theta_s - cos\theta_m)$, $\theta_s, \theta_1, \theta_2\in[\frac{-\pi}{2},\frac{\pi}{2}]$.
I want to find the $\theta_s$ points that satisfy $|f(N, \Delta_1, \Delta_2)|<1$, when $N$, $\theta_1$ and $\theta_2$ are given.
There are a few things that I tried:
There are 2 ways for finding the abs of a complex number: i) $|g|=\sqrt{g g^*}$, where $*$ denotes the conjugate operator, and ii) $|g|=\sqrt{Real\{g\}^2+Imag\{g\}^2}$. So, whatever I do to simplify the given function, it should help me to change its form such that computing its abs (with one of the methods) is possible.
This term $ \frac{1-e^{j\pi N\Delta_1}}{1-e^{j\pi \Delta_1}} $ can be further simplified as $ e^{j\pi \frac{N-1}{2}\Delta_1} \frac{sin(\pi \frac{N}{2}\Delta_1)}{sin(\pi \frac{1}{2}\Delta_1)}$. However, it doesn't yield anywhere near which I want to reach.
Writing everything in polar form, so $cis(\pi \Delta_1)$, etc. didn't help me to simplify the function.
I wonder if there is a specific tool in mathematics that can help me solve this problem. Any ideas are welcomed!
Cheers.