Calibration Methods for Time-to-Digital Converters^†

Round 1

Reviewer 1 Report

In this article, authors have proposed a comparison of two of the most commonly used methods for the calibration of synchronous TDCs which are the bin-by-bin calibration and the average-bin-width calibration. They have further developed “Matrix Calibration” method for the calibration of asynchronous TDCs. Although the authors have tried their best to organize this study, still I feel the literature review must include some important studies in this area. I have a few questions which need proper clarification:

1) Abstract of the manuscript is lengthy and should be short and precise. Details may be moved to relevant sections in the manuscript.

2) Use of the English language in the manuscript should be revised (academic writing style).  In the manuscript, neutral and impersonal language should be used.

3) Major drawback of the manuscript – a literature review. In general, in this manuscript references are used in a very simple way, without a deeper analysis of their content. A proper and detailed literature review must be performed.
  Authors should include  interesting studies like "--"10.1109/TCSII.2023.3234609 "--"10.1049/hve2.12258 "-.

5)  In the introduction of the manuscript aim of the work should be defined very clearly. The aim of the work should be achieved through the analysis. Respectively, what research problem authors are trying to solve?

6) Structure of the manuscript could be improved. The proposed methodology should be explained in a more detailed way.

Author Response

1. Abstract of the manuscript is lengthy and should be short and precise. Details may be moved to relevant sections in the manuscript.
   We thank the reviewer for this concern. As you can note, we rewrote the abstract so that it is shorter and focuses on the important ideas of the paper, the details are discussed more deeply in the paper body.
2. Use of the English language in the manuscript should be revised (academic writing style). In the manuscript, neutral and impersonal language should be used.
   We apologize for our English level; the paper has been read and improved by English-specialized staff in our research Team, hoping that this new version meets the expectations.
3. Major drawback of the manuscript – a literature review. In general, in this manuscript references are used in a very simple way, without a deeper analysis of their content. A proper and detailed literature review must be performed.
4. Authors should include interesting studies like "--"10.1109/TCSII.2023.3234609 "--"10.1049/hve2.12258 "- .                 We thank the reviewer for this comment. Many references have been added to the manuscript with analysis of their content and results in the introduction and the bin-by-bin calibration. Concerning the proposed studies, it seems to be a mistake, because these papers are out of the of TDCs calibration topic. However, other interesting studies have been added and analyzed in the current version.
5. In the introduction of the manuscript aim of the work should be defined very clearly. The aim of the work should be achieved through the analysis. Respectively, what research problem authors are trying to solve?

We thank the reviewer for pointing out the lack of clarity in the objective. The objective is now stated more clearly in the introduction:

“The aim of this paper is twofold: i) to present the methodology and to compare the performance of the aforementioned methods when applied on histograms in the case of synchronous TDCs, ii) to propose an innovative robust calibration method for asynchronous TDCs named the Matrix calibration. This method is performed as a post-processing of raw data for either ASIC or FPGA-based TDCs. For its evaluation, this method is compared to the bin-by-bin one which is usually adopted for the calibration of asynchronous TDCs.”

In addition, to the best of our knowledge, this paper is the first work treating in detail the calibration of asynchronous FPGA TDCs that are used to record histograms representing the measured signals.

The methodology of the proposed method was explained in detail.

This method was compared with the bin-by-bin calibration; the simulation and experimental results showed that the proposed method is 10 times better than the bin-by-bin calibration in terms of the DNL improvement, meaning that the aim of our work is achieved.

6. Structure of the manuscript could be improved. The proposed methodology should be explained in a more detailed way.

We respectfully disagree with this point; giving more details about the methodology would lengthen the manuscript unnecessarily. The paper is already relatively long, and we did our best to make the explanation the clearest possible. We still believe that the proposed methodology is sufficiently detailed in section “3.2.2. Matrix calibration” with step-by-step explanation of the calibration method.

Reviewer 2 Report

1. Cite the figure numbers like figure 4, figure 5 etc., after the respective figures. Let the figure comes first and let the citation of those figures come later.
2. Make all the captions of the figures and tables to the center.
3. Add a separate section named “Related work” with references cited closely related to your work in addition to introduction section. Let the introduction section has the common cited references.
4. Make sure that the Figures and tables like figure 8, 9, 13, 16, table 2,table 3 fit in the textwidth.
5. Suggest some future work for the budding researchers in the Conclusion section.
6. Check grammatical errors all over the paper.
7. Enhance the quality of the figures.
8. Check in line number 304, average delay of 2,5 ns or 2.5 ns. kindly check.
9. You can list out the simulation parameters as a table for each simulations of synchronous and asynchronous for the easy understanding of the readers.
10.  

Author Response

1. Cite the figure numbers like figure 4, figure 5 etc., after the respective figures. Let the figure comes first and let the citation of those figures come later.
   According to the template provided by the journal, the citation of figures and tables comes before these figures and Hence, unfortunately, we cannot fulfill this request.
2. Make all the captions of the figures and tables to the center.
   Unfortunately, we can’t fulfill this request. Indeed, all the figures and tables are formatted according to the journal template imposing the captions to be formatted to the left.
3. Add a separate section named “Related work” with references cited closely related to your work in addition to introduction section. Let the introduction section has the common cited references.
   We thank the reviewer for this suggestion. We discussed additional related works and cited more closely related references. However, to avoid lengthening the paper, which is already relatively long, the discussion of related works has been integrated in the introduction rather than adding a separate section for it.
4. Make sure that the Figures and tables like figure 8, 9, 13, 16, table 2,table 3 fit in the textwidth.
   All the tables and figures in this paper respect the journal template. In the template, large tables and figures with multiple panels are larger than the text width.
5. Suggest some future work for the budding researchers in the Conclusion section.
   We thank the reviewer for this suggestion. Future works are suggested in the conclusion. “The proposed method has been effectively applied in a Time-Correlated Single Pho-ton Counting (TCSPC) system including asynchronous TDCs. Another paper about this system will published in the future. Furthermore, this method can be extended to be applied for the calibration of other systems dealing with multidimensional histograms involving more than three dimensions.”.
6. Check grammatical errors all over the paper.
   We thank the reviewer for raising these typos. Some grammatical errors have been corrected during the revision of the manuscript.
7. Enhance the quality of the figures.
   We thank the reviewer for highlighting this point. Indeed, quality of the figures has been enhanced by using the “enhanced metafile” option in the “paste special”.
8. Check in line number 304, average delay of 2,5 ns or 2.5 ns. kindly check.
   Again, we thank the reviewer for raising these typos. Two typos have been found and corrected.
9. You can list out the simulation parameters as a table for each simulations of synchronous and asynchronous for the easy understanding of the readers.
   Again, to avoid lengthening the paper, we preferred to mention the simulation parameter in sentences in the context of the simulation explanation rather than in tables. However, did not find relevant parameters to be listed in tables.

Reviewer 3 Report

Attached please kindly find the detailed comments.

Comments for author File: Comments.pdf

Author Response

First of all, we thank the reviewer for pointing out that this work is relevant and positive.

1. The level of error risks when making decisions based on TDC estimates depending on the signal class and type of generating system without calibration and calibration is not justified
   We agree with the reviewer that the error risks when making decisions based on TDC depends on the signal class, but it is very difficult to provide extensive studies for all signal classes. However, we added some sentences mentioning that: the proposed calibration method does not improve the TDC precision in the case of single-shot measurements. In addition, we emphasized that this method mainly addresses the calibration of measurements’ histograms.
2. The choice of intervals in synchronous and asynchronous mode, depending on the signal statistics and the goals of the measurement process, is not justified.
   We thank the reviewer for pointing out this issue, we agree with him that it is not justified. In fact, the intervals have been arbitrary chosen. We have mentioned that in the current version. In addition, we justified the goal of these measurements which it to show the advantage of the proposed method over the bin-by-bin when applied on histograms.
3. The list of references does not contain basic monographs on mathematical bases on temporal - digital transformations of signal parameters.
   Unfortunately, this question is not clear enough for us, we did not understand which parameters the reviewer meant. However, we added some references discussing some important parameters such as the RMS error in addition to the DNL and INL.