I have some difficulties understand how can we generate frequency-dependent beams using unniform linear array. A paper that discusses this is the following: https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9826716.
For a carrier frequency \$f_c\$, I want to generate a beam at frequency \$f_k\$ and at direction \$\theta\$. Here, \$f_k\$ is the frequency of subcarrier \$k=0,\ldots,K-1\$ around \$f_c\$ and is given by: $$f_k=f_c+\frac{B}{K}(k-\frac{K-1}{2}).$$ I am using \$f_c=28\$ GHz and the bandwidth \$B=1\$ GHz and \$K=128\$ subcarriers.
The array response vector of the antenna of size \$N\$ can be written as (after adding a time delay \$\tau_n\$ and a phase shift \$\Phi_n\$):
$$A(f_k,\theta)=\left[1,e^{j\Phi_1}e^{j2\pi(f_k-f_c)\tau_1}e^{-j\pi\frac{f_k}{f_c}\sin(\theta)},\ldots,e^{j\Phi_{N-1}}e^{j2\pi(f_k-f_c)\tau_{N-1}}e^{-j\pi(N-1)\frac{f_k}{f_c}\sin(\theta)}\right]^\top.$$
As I understand, by controlling the values of \$\tau_n\$ and \$\Phi_n\$ we can generate a beam at frequency \$f_k\$ and at direction \$\theta\$. Is this possible?
My first question is: If I want to generate say four beams at \$(f_0,\pi/4)\$, \$(f_0,\pi/2)\$, \$(f_1,\pi/4)\$, and \$(f_1,\pi/2)\$. How can I do this? Should I do a search over the values of \$\tau_n\$ and \$\Phi_n\$ to find the best ones? And the best ones in terms of what? Should we maximize \$|\sum_{n=0}^{N-1}[A(f_k,\theta)]_n|^2\$?
My second question is: Let us say that we have the values of \$\tau_n\$ and \$\Phi_n\$ that give us the fours beams, how can I show in a plot that we have the correct beams? Does a plot of \$|\sum_{n=0}^{N-1}[A(f_k,\theta)]_n|^2\$ when varying \$\theta\$ should have maximum values at \$\pi/4\$ and \$\pi/2\$?
