# Doubts on implementation of a beamforming algorithm to produce narrower beams

I am trying to understand and implement in MATLAB, the algorithm described in the paper A Beamforming Method Based on Polarization Matching but my math background is not strong nor am I used to reading academic papers. Can anyone kindly explain the following questions I have?

1. Since there is no restriction on $$x(t)$$, can we not assume $$x(t)$$ to be a unit DC signal so that the calculations become simpler? If not, why would the radiation pattern depend on the input signal? I thought it only depends on array geometry, frequency, beam direction, and amplitude window.

2. Doesn't a dual polarized antenna mean having 2 polarization states? In other words, shouldn't Q be equal to 2?

3. From equations 3 and 4, $$v(\theta,\phi)*p(\gamma,\eta)$$ is a 3x1 matrix. In equation 14, is the division operation done on element-by-element basis (a Hadamard division)? If not, we can't divide two 3x1 matrices, can we?

4. $$a$$ is Nx1 matrix, $$B = v(\theta,\phi)*p(\gamma,\eta)$$ is 3x1 matrix. So, $$A =$$ kron($$a,B$$) is a 3Nx1 matrix which makes $$y_{qi}, y_{qj}$$ 3NxT matrices, where T = number of time samples. From equation 14, $$Ax_q$$ is a 3x1 matrix. Then, how are $$y_q$$ and $$Ax_q$$ multiplied in equation 17?

5. Equation 17 gives $$Rx$$ but equation 16 uses $$Rx^{-1}$$. Is it $$(Rx)^{-1}$$? If not, how can we compute $$Rx^{-1}$$?

Many thanks.