The Impact of the Covariance Matrix Sampling on the Angle of Arrival Estimation Accuracy
Round 1
Reviewer 1 Report
The presented paper investigates different values for sampling in order to construct a Projection Matrix to estimate arrival angles. The paper in its current form cannot be considered publishable, it needs heavy revisions both in language and scientific content. Below, i enlist some of my concerns.
The paper needs a thorough revision. The language is not appropriate.
The abstract needs revision. I suggest a total makeover, presenting the paper's motivation and contribution in the best possible way.
How are the spatial movements of the sources are incorporated in equation (13)?
In general, the discussion of related works (as well as the related solutions like those used in the comparison section) is short and in some cases inadequate.
Section 3 needs a better explanation. For example, how does the need for the decomposition of the CM is lifted?
The paper's contribution is narrow. I can only see that the authors play around with the estimation of P and then they simply present the simulations for some values of P..
It is claimed in various parts the proposed solution is of low complexity (mainly because there is no need for decomposing the CM matrix), but there is no complexity analysis to formally support this claim.
For the simulation part, there is no information regarding the parameters used, the programming language(s) and the backing of using these settings according to literature.
When comparing with other approaches, it seems to me that the chosen metrics are quite limited and deliberately in favour of the proposed method.
Fix some figure references within the text (Error! Reference source not found.)
The language used in the manuscript needs a complete revision. I mention just some indicative examples, but overall, I think that almost every paragraph of the text contains a couple of typos and grammar mistakes.
highly computational complexity
The last two mentioned method
It is observed increase the picked sampled
is given the formula below
The PM method in likes MUSIC
can be concluded PM method is still work even
its estimation accuracy increase
Error! Reference source not found.
Author Response
First of all, we would like to thank the reviewer for such valuable comments, which has improved the revised manuscript significantly. We have tried our best to address the required responses to all comments raised. The responses are highlighted below and also in the revised manuscript.
Best Regards
Authors,
Comments and Suggestions for Authors
Reviewer#1, Concern #1: The paper needs a thorough revision. The language is not appropriate.
Authors’ response: Thank you, and we would like to register our apology for that, and therefore the full text has been revised and edited by professional English native in supporting the English language all over the revised manuscript.
Reviewer#1, Concern #2: The abstract needs revision. I suggest a total makeover, presenting the paper's motivation and contribution in the best possible way.
Authors’ response: We do thank the reviewer for this valuable comment, the abstract and introduction have been carefully revised to highlight the work motivation and the new research contributions carried out in this paper.
Reviewer#1, Concern #3: How are the spatial movements of the sources are incorporated in equation (13)?
Authors’ response: thank you for your valuable comment. It should be noted that the covariance matrix needs to be measured frequently to ensure a reliable estimation of the spatial movements. More details have been added after Eq.(13) to address this point.
Reviewer#1, Concern #4: In general, the discussion of related works (as well as the related solutions like those used in the comparison section) is short and in some cases inadequate.
Authors’ response: We understand the reviewer concerns, and therefore, a comprehensive review of the related works has been extended and added into the introduction section of the revised manuscript. The computational complexity has
been carefully revisited in section 4.2 and then the required arithmetic operations calculated and added to sections 5.1 and 5.4 to ensure a fair comparison.
Reviewer#1, Concern #5: Section 3 needs a better explanation. For example, how does the need for the decomposition of the CM is lifted?
Authors’ response: Thank you for your comment; more materials and explanations have been added to section 3 in order to explain this point. We have carefully clarified this point here as well, for example, some AoA methods such as MUSIC, Min Norm, ESPRIT exploit the Eigenvector structure to find the direction of the incoming signals. It should be noted that the number of arrival signals should be known for these methods. For this reason, the decomposition of the CM is needed to determine the number of arrival signals based on the calculated eigenvalues in order to separate the
signal and noise subspaces entirely. With the linear AoA methods such as propagator and projection matrix, the original CM columns are exploited to construct the propagator operator and projection matrix. However, the propagator method still needs to divide the CM into two submatrices based on a number of arrival signals and thus, this number should be estimated accurately. In this work, it is demonstrated the projection matrix can be constructed regardless of the number of arrival signals. It has also been proved any increase in the number of sampled columns in the projection matrix construction stage leads to increase in the Degrees of Freedom (DOFs), which, in turns, will minimize the estimation error and improve the probability of successful detection (PSD) of arrival signals.
Reviewer#1, Concern #6: The paper's contribution is narrow. I can only see that the authors play around with the estimation of P and then they simply present the simulations for some values of P.
Authors’ response: Thank you for your comment, as far as the authors’ knowledge, the effect of the sampled columns within covariance matrix in the projection matrix construction has not been addressed in the literature. To this end, this manuscript investigates this effect by constructing the projection matrix based on a various number of sampled columns to estimate the arrival angles. A theoretical analysis has been accomplished to illustrate the relationship between the number of the sampled columns and the degrees of freedom (DOFs). The analysis shows that with the same aperture size, the DOFs can be increased by increasing only the number of sampled columns in the projection matrix calculation step. It should be noted increasing the number of sampled columns will not increase the computational burden in the grid searching stage (i.e., the spatial spectrum construction). The PM method, based on the proposed selected sampling methodology gives better estimation accuracy compared to the well-known AoA’s techniques such as Capon and MUSIC with less computational complexity. The work motivation and the new research contributions carried out in this paper are emphasized in the abstract and the penultimate paragraph in the introduction section.
Reviewer#1, Concern #7: It is claimed in various parts the proposed solution is of low complexity (mainly because there is no need for decomposing the CM matrix), but there is no complexity analysis to formally support this claim.
Authors’ response: Thank you for your comment. In fact, the complexity of computations is stated in section 4.2. However, we agree with the reviewer comment that this point is not clearly explicated and therefore this section has been revised and a new table has also been added in section 4.2 to compare the computational complexity with several AoA methods.
Reviewer#1, Concern #8: For the simulation part, there is no information regarding the parameters used, the programming language(s) and the backing of using these settings according to literature.
Authors’ response: Thank you for your comment, we have used MATLAB software to simulate and evaluate the performance of the PM method including the comparisons with other AoA algorithms. All the simulations parameters have been carefully revisited and highlighted into the revised manuscript. The settings of these compared methods and its final formals have been added to section 3.
Reviewer#1, Concern #9: When comparing with other approaches, it seems to me that the chosen metrics are quite limited and deliberately in favour of the proposed method.
Authors’ response: We thank the reviewer for this comment. In fact, in the first and second scenarios, the impact of the sampled columns on the performance estimation of the projection matrix has been investigated intensively. According to this analysis, we found the modified PM method works well with P = 20, 24, 27, as presented in section 5.1. And thus based on these results, we have selected P = 20 to compare the present work with other methods; and in addition to ensuring a fair comparison, the computational load at this scenario has been added to section 5.4. It should be noted
that each compared method has a different methodology and working principle to find the directions of the received signals.
Reviewer#1, Concern # 10: Fix some figure references within the text (Error! Reference source not found.)
Authors’ response: Thank you for your comment; it has been corrected accordingly.
Reviewer#1, Concern #11: The language used in the manuscript needs a complete revision. I mention just some indicative examples, but overall, I think that almost every paragraph of the text contains a couple of typos and grammar mistakes.
highly computational complexity
The last two mentioned method
It is observed increase the picked sampled
is given the formula below
The PM method in likes MUSIC
can be concluded PM method is still work even
its estimation accuracy increase
Error! Reference source not found.
Authors’ response: We do thank the reviewer for letting us know about these grammatical mistakes and references errors, the English have been comprehensively revised by a professional native speaker. Please revisit the revised manuscript for such required changes.
Reviewer 2 Report
The authors presented a method to reduce the matrix size of the projection matrix method for DOA estimation, so as to reduce the computational cost.
Firstly, the reviewer would like to comment on the quality of the text, which is unfortunately, not very acceptable. There are too many grammar errors and typos for the reviewer to believe that the manuscript has gone through a single round of proof-reading. The "Error! Reference source not found" in page 11 further shows that the paper is poorly prepared.
In the result section, the authors demonstrate that the proposed method outperforms all other methods being compared. However, the setting seems somewhat unfair: the proposed method utilizes 20 vector columns, which would be higher than what would be used in the original PM method. This contradicts with the authors' claim of having reduced complexity. The authors may need to consider either modify their claims in the abstract, or re-run the simulation using different parameters to back up the claims.
Finally, please consider adding references to some important equations, such as eq.16 and eq.17.
In the case of eq.16, if Q_ss is a M-by-L matrix, as is suggested in the manuscript, then the product Q_ss^H (Q_ss Q_ss^H)^-1 Q_ss would seem to result in a L-by-L matrix, which does not agree with the dimensions of I_m.
Author Response
First of all, we would like to thank the reviewer for such valuable comments, which has improved the revised manuscript significantly. We have tried our best to address the required responses to all comments raised. The responses are highlighted below and also in the revised manuscript.
Best Regards
Authors,
Reviewer#2, Concern #1: Firstly, the reviewer would like to comment on the quality of the text, which is, unfortunately, not very acceptable. There are too many grammar errors and typos for the reviewer to believe that the manuscript has gone through a single round of proof-reading. The "Error! Reference source not found" in page 11 further shows that the paper is poorly prepared.
Authors’ response: Thank you, and we would like to register our apology for that, and therefore the full text has been revised and edited by professional English native in supporting the English language all over the paper. Please revisit the revised manuscript for such required changes.
Reviewer#2, Concern #2: In the result section, the authors demonstrate that the proposed method outperforms all other methods being compared. However, the setting seems somewhat unfair: the proposed method utilizes 20 vector columns, which would be higher than what would be used in the original PM method. This contradicts with the authors' claim of having reduced complexity. The authors may need to consider either modify their claims in the abstract, or re-run the simulation using different parameters to back up the claims.
Authors’ response: Thank you for this comment, of course, we agree with the reviewer, and therefore the abstract has been modified accordingly to explain this claim carefully. The computational complexity has been analyzed and compared in section 4.2, while the number of the required number of arithmetic operations for each applied method in the comparison scenario (i.e., section 5.4) has been provided to ensure a fair comparison.
Reviewer#2, Concern #3: Finally, please consider adding references to some important equations, such as eq.16 and eq.17.
Authors’ response: thank you, we do agree with the comment and thus references have been added accordingly as requested.
Reviewer#2, Concern #4: In the case of eq.16, if Q_ss is a M-by-L matrix, as is suggested in the manuscript, then the product Q_ss^H (Q_ss Q_ss^H)^-1 Q_ss would seem to result in a L-by-L matrix, which does not agree with the dimensions of I_m.
Authors’ response: Thank you very much for your concentrated reading. The math
manipulation of matrix dimensions has been corrected accordingly.
Round 2
Reviewer 1 Report
First, I thank the reviewers for taking into consideration my previous comments. I appreciate their effort in presenting a revised version of their work. I believe that the current shape of the paper constitutes an improvement compared to the initial submission. Nevertheless, I still have some concerns. The language should be revised, there are some remaining mistakes even in the first sentence of the abstract! Regarding the discussion of related works, it is useful to have all these new references. Still, I would appreciate if the authors discussed in detail which of them are the most closest to the proposed one.
The inclusion of some quantitative results regarding the solution's approach is appreciated and in my opinion make the proposed method more solid. have in mind, though, that the big O notation shouldn't be followed by a space (that is, O(M) instead of O (M), like we write a typical function).
Overall, I fell that scientifically, the new version is a lot better and could be considered for publication. The language, though, still requires improvement. Maybe the authors should be given a wider time window to correct their manuscript.
Author Response
We would like to thank the reviewer for the valuable comments and positive feedbacks, which improved our manuscript significantly and helped us to emphasize the contribution and motivation of this work. The responses are highlighted below and also in the revised manuscript.
Best Regards
Authors,
Comments and Suggestions for Authors
Reviewer#1, Concern #1: First, I thank the reviewers for taking into consideration my previous comments. I appreciate their effort in presenting a revised version of their work. I believe that the current shape of the paper constitutes an improvement compared to the initial submission. Nevertheless, I still have some concerns. The language should be revised, there are some remaining mistakes even in the first sentence of the abstract! Regarding the
discussion of related works, it is useful to have all these new references. Still, I would appreciate if the authors discussed in detail which of them are the most closest to the proposed one.
Authors’ response: We understand the reviewer concerns, and therefore the English again have been revised carefully by a professional native speaker. Please revisit the revised manuscript for such required changes. The projection matrices methods which defined in (24) and (28) in addition to the propagator and MUSIC methods can be considered as the closet methods to the proposed method. The idea, working principles and computational complexity of these methods have been already presented in section 3 and compared in section 5.
Reviewer#1, Concern #2: The inclusion of some quantitative results regarding the solution's approach is appreciated and in my opinion make the proposed method more solid. have in mind, though, that the big O notation shouldn't be followed by a space (that is, O(M) instead of O (M), like we write a typical function).
Authors’ response: We do thank the reviewer for this comment, the original stamped matrices have been linked with their eigenvalues and eigenvectors to illustrate the relationship between them. It is known the estimation accuracy of AoA method is directly propositional with the power of the Eigenvalues. To this end, an analysis and quantitative results regarding to the impact of the proposed sampling methodology have been added to section 4. It has been demonstrated the power of the eigenvalues / singular values increases significantly with the increasing number of sampled columns. Also, a table has been added to summarize the procedures of the proposed method. The required space after O symbol has been added.
Overall, I fell that scientifically, the new version is a lot better and could be considered for publication. The language, though, still requires improvement. Maybe the authors should be given a wider time window to correct their manuscript.
Authors’ response: Thank you for your support; the English style have been carefully checked, the changes have been highlighted. Please revisit the revised manuscript for such required changes.
Reviewer 2 Report
This version has improved greatly in overall quality compared to the previous draft. The new claims in the abstract and conclusion are appropriately supported by the simulations, and the math is also better presented.
Author Response
We do thank the reviewer for the valuable comments, which have improved our manuscript significantly.
Best regards
Authors
