Efficient Implementation for SBL-Based Coherent Distributed mmWave Radar Imaging

Round 1

Reviewer 1 Report

The authors deal with high precision direction of arrival estimates. To this end the authors use coherently distributed radar systems and introduce a fast and precise method. The topic and the used method are interesting, however the presentation is rather poor. Firstly, English grammar and spelling have to be improved! Almost on each page there are multiple spelling errors, e.g in the first lines of page 3:

- "no any"

- "approaxiamations"

- "converge rate"

- "dispalcement rank"

Moreover, the notation is sometimes confusing, mathematical concepts are not very well explained. Here are only some comments:

- p.3: Check spelling after Formula (1) and (2).

- p.3.: f_0 should probably the start fequency of the chirp.

- p.3.: The use of "o" and "0" may be confusing

- p.4.: I don't think there should appear delta functions as one use a finite bandwidth and a finite aperture size.

- p.5.: Explanation of Formula (9). Shoulnd't "x" contain the reflectivity of the targets. They are not present in Formula (8). How is the Fourier dictionary formed? Do you use an underlying grid?

- p.5: How do CFAR (which is a detection algorithm) and high frequency limit the number of measurements?

- p.6.: The parameter beta is not explained.

- p.6. (Section 3): What is E_0 and what is v? It seems that the use of lowercase for vectors and uppercase letters for matrices (as stated in the introduction) does not apply here.

- p.6.. If x and y are defined as in Equation (9), how can they satisfy Equations (25)-(27)? What is s?

- What is the aim of Section 3.1?

The subsequent thereotical parts are also hard to follow (I do not want to go into detail). Although the results are interesting,  I cannot recommend acceptance in its current form. Major revisions are necessary. Some further comments:

- References: It seems that there is a an algorithm in "Efficient Inversion of Toeplitz-Block Toeplitz Matrix" in 1983. Does it apply a similar scheme?

- Figure 3: Which SBL-algorithm has been applied? What would be the result of a full array?

- Figure 4: Normally, one obtains a sparse scene. How many sparse elements did you use, and how many elements did the grid have?

 

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Reviewer 2 Report

This paper develops a new decomposition method to speed up the algorithm by transposing the inverse of TBT matrix as the sum of the products of the block lower triangular Toeplitz matrix and the block circulant matrix. Through this method, the time cost for SBL algorithm is decreased by several orders of magnitude. At last, simulations and experiments ensure the effectiveness of the proposed algorithm. But the following comments should be improved.

1. The proposed fast SBL method should be compared with other fast SBL methods.

2. The authors should add some other up-to-date angular superresolution methods for comparison.

3. Some additional indicators should be added to evaluate the angular resolution performance of the simulated and measured data.

4. The English writing in this article should be improved, such as:

 “Sparse Bayesian learning framework based on RVM has been widely used to so solve CS reconstruction problem [40-42]”-à “Sparse Bayesian learning framework based on RVM has been widely used to solve CS reconstruction problem [40-42]”.

 

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Round 2

Reviewer 1 Report

I would like to the thank the authors, the quality of the spelling and the presentation has been significantly improved. There are still some remarks. Once they have been clarified, I recommend publication. Remarks concerning the authors responses:

- Concern #5: I do not agree, the authors state in the article (after Formula 1) that the variable t ranges from 0 to T. So the delta has to be convulated with a sinc-function. A similar statement holds true for the azimuth direction. Otherwise, there would be no need to increase the aperture size.

- Concern #9: I dind't see the explanation of E_0 in the text. Moreover, E_{n-1} should also be explained.

- Concern #13: In Figure 4 the authors depict the results for S-SBL and LC-SBL. The results are different, so they should at least explain in Figure 3 which SBL algorithm they used.

Please make sure to be clear concerning the mathematical context:

- Notations section, example: The authors write that the elements of a consist of 2x2 matrices. So is a_1 a 2x2 matrix? In any case the example seems to be wrong. The first line of a^T should be [a_1^T, a_5^T]. If a_1 is a scalar then a_1 = a_1^T.

- Notations section, example (continued): I get an idea of the definition of a', but it seems highly ambiguous as it seems to depend on the block size.  Would the result be different, if a_1 would be a 2x2 matrix? I do not recommend this notation!

- The two previous points becomes also apparent in formula (18), (19) and (27). Do the authors really want to transpose every single matrix inside?

- Moreover, the authors should write sometiomes what are block matrices and what are vectors (resp. their size). Example: The elements of x and y in (18) and (19) seem to be scalars, but then in (41) they seem to matrices. 

To sum up the previous points: The notation for block matrices demand a careful handling.  The transpose of a matrix a^T has a fixed definition and the author's definition  of a' of seems to be ambigious. Please find a clear way to refer to the corresponding matrices. Check also the rest of the text, e.g. (36), ....

- Equation (31): r is introduced but does never appear.

Spelling (among others):

- P.6. "In matrix form, The"

- P.7. "linar system"

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