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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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  • \$\begingroup\$ Please ask yourself this question: if the linked document gets moved or deleted, does your question presently contain enough information so that it can be answered. If the answer is no then you should embed the relevant parts of the document into your question. \$\endgroup\$ Commented May 1, 2024 at 7:50
  • \$\begingroup\$ Another question is, does this relate to electrical engineering, or implementing an algorithm in software? \$\endgroup\$ Commented May 1, 2024 at 8:00
  • \$\begingroup\$ @Andyaka, the link is a doi.org link to the academic research paper "A Beamforming Method Based on Polarization Matching". I wasn't sure how to attach/direct to an academic paper other than providing its doi link. Do you have any suggestions? \$\endgroup\$ Commented May 1, 2024 at 8:10
  • \$\begingroup\$ @Justme, I am trying to understand the logic presented in the algorithm. Implementation in a software (MATLAB), can be done if I understand it. My questions are more on the theory/maths presented in the paper than on software program/code. \$\endgroup\$ Commented May 1, 2024 at 8:13
  • \$\begingroup\$ @RajaKrishnappa that's not a DOI link, that's just an easy chair link. This link will definitely die after the conference review is done, pretty soon. Please find a preprint, (maybe on arxiv?) and link to that. \$\endgroup\$ Commented May 1, 2024 at 8:16

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These are actually too many questions. You need to focus on one question per post! However, I'm interpreting your question as "please help me focus what I need to research and ask", and will hence hope to help you by clarifying the things you got right and wrong, very briefly.

  1. no. \$\mathbf{x}(t)\$ is the incident signal. You don't get to choose it. It carries data and depends on the position of the transmitter.
  2. no, see the text after (9); there's two orthogonal dimensions on which you can project the polarizations, but more polarization states.
  3. I'd agree, you can't divide column vectors. I'm not sure what's happening here, the authors might just be extremely sloppy, and just divided (10) and (11) "as if that was a legal operation"; what would actually be mathematically sound would be multiplication: \$\mathbf{\omega}^H y_{qi} \mathbf{v}_i \mathbf{p}_{qj} = \mathbf{\omega}^H y_{qj} \mathbf{v}_j \mathbf{p}_{qi}\$. (This looks like a paper that still needs to be reviewed. Are you sure you should be reading this? It might be wrong!)
  4. Forget the equations based on dividing vectors; trying to read sense into them is like trying to interpret the baying of a hound as English.
  5. Since I don't think \$\mathbf{x}\$ is invertible in general, yes, that interpretation would make sense. The notation here, again, is not too sloppy to be sure what the author really meant, and to be sure they did not make a mistake.
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  • \$\begingroup\$ Thanks for the clarifications. I did think it was too much for one post but thought it might be inconvenient to people if there are multiple posts. Anyways, I will wait until the paper is reviewed before going further. \$\endgroup\$ Commented May 1, 2024 at 8:43
  • \$\begingroup\$ It is also weird that there is no contact information of the authors given in the website. Meanwhile, I will try to contact them for better understanding of their paper. \$\endgroup\$ Commented May 1, 2024 at 8:44
  • \$\begingroup\$ It's less about the convenience aspect – for example, strictly approaching your 3. question in a separate question might have eliminated questions 4. and 5. \$\endgroup\$ Commented May 1, 2024 at 8:45
  • \$\begingroup\$ @RajaKrishnappa contact info is on the paper, as email address. \$\endgroup\$ Commented May 1, 2024 at 8:45
  • \$\begingroup\$ my bad. How did I miss that? :) \$\endgroup\$ Commented May 1, 2024 at 8:46

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