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.



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