Oblique Projection-Based Covariance Matrix Reconstruction and Steering Vector Estimation for Robust Adaptive Beamforming

Round 1

Reviewer 1 Report

This is a very good paper. Well referenced, well structured, very useful for researchers in the field. Difficult to read due to the abundant use of acronyms. A list of acronyms in the introduction would be interesting to improve readability. However, this reviewer has serious doubts when recommending its publication. The theory of this paper is extremely similar to two other papers by the authors, which are adequately referenced, see references [19] and [31], also [35]. The application changes, and obviously the plots shown are different.

Author Response

Thank you very much for your carefully review. I'm also obsessed with acronyms when I read a literature. I've listed the acronyms in the last paragraph of the Introduction Section. For the convenience of discussion, most RAB methods are proposed based on ULA. Therefore, the signal models or backgrounds of different RAB methods have very similar expression forms. And the difference is only the description notation. As a result, the signal model sections of many papers are quite similar. The main goal of all RAB methods is to improve the robustness of the beamformer and get as close to the optimal performance as possible, while also considering the problem of complexity.

Reviewer 2 Report

Please make performance comparison with the other published articles.

Can you propose measured results?

Author Response

Thanks very much for your carefully review. In this article, we cite two previously published articles: Ref.[3] (Duan, Y.; Zhang, S.; Cao, W. Covariance matrix reconstruction with iterative mismatch approximation for robust adaptive beamforming. J Electromagnet Wave 2021, 35, 2468-2479, doi:10.1080/09205071.2021.1952901.), Ref.[6] (Duan, Y.L.; Yu, X.H.; Mei, L.R.; Cao, W.P. Low-Complexity Robust Adaptive Beamforming Based on INCM Reconstruction via Subspace Projection. Sensors-Basel 2021, 21, doi:10.3390/s21237783.). Ref.[6] participated in the performance comparison. Ref.[3] did not participated in the performance comparison. At that time, it was considered that Ref.[3] was similar with Ref.[19] at high SNR and worse than Ref.[19] (Zheng, Z.; Zheng, Y.; Wang, W.Q.; Zhang, H.B. Covariance Matrix Reconstruction With Interference Steering Vector and Power Estimation for Robust Adaptive Beamforming. Ieee Transactions on Vehicular Technology 2018, 67, 8495-8503, doi:10.1109/Tvt.2018.2849646) at medium SNR, the advantage of Ref.[3] is its low complexity. Therefore, we did not add Ref.[3] to the comparison list. In addition, too many lines can lead to degraded figure visibility. I will add Ref.[3] to the comparison list, and the numerical simulation results will be sent to you in tabular form.

Author Response File: Author Response.pdf

Reviewer 3 Report

1.     In this paper, the precise steering vector (SV) associating with desired signal is estimated by employing the minimum norm of subspace projection (MNSP) approach. As depicted in (11) of the manuscript, the algorithm is by solving a quadratically constrained quadratic programming problem.

Firstly, I think that the main optimization of (11) should be “minimize” rather than “maxmize”. Secondly, the algorithm of (11) is quite similar to the idea proposed in the work of [A].

 

2.     The title of section 3.2. “Figures, Tables and Schemes” is not appropriate. It should be modified to reflect the actual content.

3.     In the works of [B, C], Capon spatial spectrum estimator is utilized to estimate the interferers’ SVs. While in this paper, the interferers’ SVs are estimated via the maximum entropy power spectrum, which is depicted in (12). The authors should compare the performance with the Capon spatial spectrum estimator or show the advantages in using (12).

 

4.     In this paper, the interferers’ SVs and powers are obtained by oblique projection method. However, similar ideas have been proposed in the work of [B]. The authors should explain and emphasize the novelty of the method employed in this paper.

 

5.     The authors should analyze or explain in detail why the proposed method outperforms the existing algorithms.

 

6.     The title “Oblique Projection-Based Covariance Matrix Reconstruction and Steering Vector Estimation for Robust Adaptive Beamforming” may be misleading since I can’t find any adaptive or iterative algorithm.

7.     There are quite a few grammatical or syntax errors. The authors need a thorough proofreading.

 

[A] Jia, W.M.; Jin, W.; Zhou, S.H.; Yao, M.L. “Robust adaptive beamforming based on a new steering vector estimation algorithm”, Signal Processing, 2013, 93, 2539-2542.

[B] XIAOYU AI and LU GAN, “Robust Adaptive Beamforming With Subspace Projection and Covariance Matrix Reconstruction,” IEEE Access, vol. 7, pp. 102149-102159, 2019.

[C] P. Stoica, J. Li, and X. Tan, “On spatial power spectrum and signal estimation using the Pisarenko framework,” IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, pp. 5109-5119, OCTOBER 2008.

Author Response

1, I‘am sorry for this mistake, the objective function of (11) should be "min" and has been modified. The optimization problem in [A], [19] and others. It’s all based on the principle of estimating the signal steering vectors via maximizing the beamformer output power. The MNSP method estimates the desired signal SV by minimizing the projection norm of the SV on the noise subspace. It is implemented based on the orthogonality between the signal SV and the noise subspace. Therefore, they are not the same.

2, I`m sorry for this mistake, this mistake occurred when I applied the ‘Electronics’ template, and it has been modified.

3, Thanks very much for this tip. The main reason we use maximum entropy spectrum to estimate nominal SV in this paper is that it has a lower complexity. Compared to Capon, its estimation accuracy is very high. It has been proved and compared in [26]. Therefore, considering the problem of space, we do not carry out the proof and comparison in this manuscript, only a brief description is given in the introduction.

4, Oblique projection Oblique projection is a common mathematical method that was not first proposed by Ref[B]. Oblique projection is an extension of orthogonal projection.  is an oblique projection operator which project a vector on subspace Rang(A) along the direction parallel to the subspace Rang(B). Oblique projection operator is used to correct signal SV in this paper and Ref.[B]. Compares with Ref.[B], we assign different Rang(A) and Rang(B) to construct .  and  come from the previous calculation result. In our manuscript, we construct a new optimization problem to estimate desired signal SV, and construct an oblique projection operator via new Rang(A) and Rang(B) to correct interference SV. From the simulation result, our method is better than the SPCMR in Ref.[B].

5, The effectiveness of our method is described in 3.1 and 3.2. Firstly, converting SCM  to Hermite’s Toeplitz matrix  can effectively combat the possible coherent scattering mismatch problem, which is proved in Ref.[42]. Secondly, we reconstruct the orthogonality between desired signal SV and noise subspace which is destroyed by various types of mismatches. Three constraints ensure that the estimated mismatch vector meets the design requirements (the norm of mismatch is small, orthogonal to nominal SV and guarantees the validity of desired signal SV estimation at low SNR). The effectiveness of this optimization problem is proved in Fig-1. From Ref.[B] and Ref.[44], it can be seen that the oblique projection can be used to correct interference SV. Rang(B) in this manuscript is spanned by desired signal SV. Fig-2~5 proved that the performance of our method in this manuscript is better Ref.[B]. Besides, it can be seen that the complexity of calculating  is lower than SPCMR in Ref.[B].

6, Thanks very much for this tip. In fact, most robust beamforming methods based on LCMV or MVDR are called RAB methods. They are not static beamforming methods and are often used in conjunction with adaptive methods. Therefore, the title of this manuscript is called “Oblique Projection-Based Covariance Matrix Reconstruction and Steering Vector Estimation for Robust Adaptive Beamforming”.

7, I‘am sorry for this problem. I will check this manuscript carefully and correct grammatical and syntax errors

Reviewer 4 Report

The MNSP approach and established oblique projection matrix are used in this paper to estimate desired signal SV and reconstruct INCM. To estimate the desired signal SV, the MNSP approach minimizes the norm of the projection of SV on the noise subspace. The oblique projection matrix is used to correct interference SVs and alleviate the leakage problem of spatial power spectrum estimation. The proposed method can handle various types of SV mismatches while requiring only a basic understanding of the array configuration and angular region in which the desired signal is located. Multiple simulations show that the proposed RAB method performs well and can effectively combat various mismatches.

Author Response

Thanks very much for your carefully review. And I will make further revisions to my manuscript.

Round 2

Reviewer 1 Report

Authors apply a variant of a technique used in previous applications, for that reason the theoretical background seems very similar to past publications of the same authors.