Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1.    Equation 4. What does the parameter “a” define? Why does it take values from 0 to 1, and what does the specific value depend on?
2.    Please check the equations and assumptions for typos. L 100 why does it gradually decreases? It is equal to 0.
3.    In the current state of the art, there are many improved wavelet threshold methods. There are no comparisons between the methods presented in the literature.  The advantages of the proposed approach over others in the literature have not been demonstrated.
4.    What methodology for wavelet selection was used in the research?
5.    Please show the visualization Soft threshold, hard threshold and improved wavelet threshold function on one figure
6.    Conclusions should be presented in bullet points, presenting the main findings and defined quantitatively
7.    Nomenclature, list of symbols and units are required

Author Response

Comments 1: Equation 4. What does the parameter “a” define? Why does it take values from 0 to 1, and what does the specific value depend on?

Response 1: The parameter ∈[0,1] in Equation (4) controls the local slope of the thresholding curve. As →0 the function converges to the hard threshold, while →1 yields the soft threshold. Based on the analysis of the signal and 500 Monte-Carlo runs at −35 dB input SNR, the output SNR is maximised at , which is therefore adopted throughout the manuscript.

 

Comments 2: Please check the equations and assumptions for typos. L 100 why does it gradually decreases? It is equal to 0.

Response 2: Agree. We apologize for the oversight, the necessary correction has been applied to equation.

 

Comments 3: In the current state of the art, there are many improved wavelet threshold methods. There are no comparisons between the methods presented in the literature.  The advantages of the proposed approach over others in the literature have not been demonstrated.

Response 3: Agree. The improved threshold method has been incorporated into the manuscript, and the corresponding simulation results are presented in Figure 1. The curves demonstrate that the IWTDC threshold mitigates the biases inherent in the soft-threshold rule, the discontinuity of the hard-threshold rule, and the shortcomings of the improved threshold reported in reference [2] (Zhu et al., 2008). Quantitatively, the simulations yield an output SNR 2.3 dB higher than that achieved by the improved threshold function in [2], under identical Monte-Carlo conditions.

 

Comments 4: What methodology for wavelet selection was used in the research?

Response 4: Agree. The wavelet selection has been added to Sections 3.1 and 3.2. Based on signal characteristics, candidate bases (db2–db6, sym4–sym6) were evaluated by 500 Monte-Carlo runs at −35 dB input SNR; db4 achieved the highest output SNR for navigation signals, and db3 for communication satellite signals.

 

Comments 5: Please show the visualization Soft threshold, hard threshold and improved wavelet threshold function on one figure

Response 5: Agree. The simulation figure has been added to Section 2.2 and is shown in Figure 1.

 

Comments 6: Conclusions should be presented in bullet points, presenting the main findings and defined quantitatively

Response 6: Agree. Your point is well-taken. We have revised Section 5 as follows:

In this paper, We propose the IWTDC method to address the limitations of the wavelet threshold denoising algorithm by constructing a new threshold function and adopting a new threshold determination method.

At −35 dB input SNR, navigation and communication satellite signals were simulated to analyze the denoising performance of the IWTDC method. Simulations show that the IWTDC method significantly outperforms other algorithms at extremely low SNR, suppressing noise while preserving correlation peaks of satellite signals. The SNR is improved by about 2-6 dB.

Moreover, real-time testing further confirms its robustness, yielding an SNR enhancement of 7–8 dB with clearly discernible correlation peaks.

In summary, the IWTDC method exhibits strong performance in both theoretical simulations and practical implementations. When spreading gain is unavailable, IWTDC delivers high-SNR data for passive PNT systems, thereby establishing a reliable foundation for nanosecond-level time-difference estimation. Validation on a zero-baseline platform is currently underway, with subsequent campaigns planned across baselines ranging from 50 to 500 km to further optimize the method.

 

Comments 7: Nomenclature, list of symbols and units are required

Response 7: A Nomenclature section has been added after the Conclusions. All symbols are listed in alphabetical order (Greek → Latin → abbreviations) together with their definitions and SI units where applicable.

Reviewer 2 Report

Comments and Suggestions for Authors

Comments for the authors

General comments

The article proposes an improved de-noising method that utilizes wavelet decomposition to analyze received satellite signals that are characterized by low signal-to-noise ratio (SNR). The method seems to improve the SNR by about 2─5 dB.

The content of the article is within the scope of the Journal.

The main contribution of the article is the proposed de-noising method.

The article provides a sufficient literature review, mainly included in section 1 (Introduction). Section 2 describes the method and provides simulation results while section 3 provides results based on actual data. Section 4 provides a short discussion and concludes the article.

A merit of the paper is that, apart from simulation, it also presents results on actual data.

Specific comments

I would recommend section 2 only include subsection 2.1. Sub-section 2.2 should be a separate section 3 with title “Simulation results” while section 3 should be entitled “4. Results from actual data”. Finally, the present section 4 should be entitled “5. Conclusions”.

Figs. 2b and 3b should be enlarged and use different colors (than those used) to improve readability.

I suppose that the values of the amplitude and the noise, in figs. 2a and 3a are in normalized units. If so, this should be mentioned in the respective legends.

Regarding fig. 3a (LMS graph and IWTDC graph) how did the 0.4 and 0.3 values for the noise occur? It seems to me that both are lower.

The legend of fig. 3 should be rephrased and adapted to that of fig. 2.

In the legend of figs. 4 and 5, there should be a reference to the specific navigation signals the graphs refer to (lines 241-242 and 255-256, respectively).

The “Conclusions” section should be enlarged and include a short reference to table 1 and figs. 4 and 5.

Typo: In line 150, “equation (9)” should read “equation (10)”

Use of English

The article is generally well written so, regarding the use of English, a minor editing would be sufficient.

Review decision

The article presents a novel de-noising method for satellite received signals that seems to outperform existing de-noising methods. Given that, I recommend that the article should be published subject to the revisions mentioned above and a minor editing regarding the use of English.

Comments on the Quality of English Language

The article is generally well written so, regarding the use of English, a minor editing would be sufficient.

Author Response

Comments 1: I would recommend section 2 only include subsection 2.1. Sub-section 2.2 should be a separate section 3 with title “Simulation results” while section 3 should be entitled “4. Results from actual data”. Finally, the present section 4 should be entitled “5. Conclusions”.
Response 1: Agree. We have revised the structure of the manuscript. Section 2 is now titled “IWTDC Method”; the former Subsection 2.2 has been renumbered as Section 3 and titled “Simulation Results”; the former Section 3 has been renumbered as Section 4 and titled “Results from Actual Data”; the former Section 4 has been renumbered as Section 5 and titled “Conclusion”.

Comments 2: Figs. 2b and 3b should be enlarged and use different colors (than those used) to improve readability.
Response 2: Agree. Figures 2b and 3b have been enlarged and redrawn with distinct colors for improved readability; the line width was also increased to 1.2 pt.

Comments 3: I suppose that the values of the amplitude and the noise, in figs. 2a and 3a are in normalized units. If so, this should be mentioned in the respective legends.
Response 3: Agree. Supplementary explanatory notes have been included in Figures 2 and Figures 3. The amplitude and noise values are dimensionless; all datasets have been normalized with respect to the maximum correlation peak.

Comments 4: Regarding fig. 3a (LMS graph and IWTDC graph) how did the 0.4 and 0.3 values for the noise occur? It seems to me that both are lower.
Response 4:  The noise baseline depicted in Figure 3(a) is the normalized outcome of 500 Monte Carlo simulations, while the noise levels assigned to the LMS and IWTDC curves—0.4 and 0.3, respectively—correspond to the mean values of the resulting correlation coefficients. The amplitude and noise values are dimensionless.

Comments 5: The legend of fig. 3 should be rephrased and adapted to that of fig. 2. The caption of Figure 3 has been revised as follows:
Response 5: Agree. The legend of Figure 3 has been revised as follows:  (a) Results of each algorithm. (b) Enlarged results of each algorithm. The amplitude and noise values are dimensionless; all datasets have been normalized with respect to the maximum correlation peak.

Comments 6: In the legend of figs. 4 and 5, there should be a reference to the specific navigation signals the graphs refer to (lines 241-242 and 255-256, respectively).
Response 6: Agree. The legends of Figures 4 and 5 now explicitly state the signal types:
Figure 4: Navigation signal, 3826 MHz carrier, 20.46 MHz bandwidth (CDMA pseudo-code).
Figure 5: Communication satellite signal, 3780 MHz carrier, 27 MHz bandwidth (MCPC, Apstar-7 C-band).

Comments 7: The “Conclusions” section should be enlarged and include a short reference to table 1 and figs. 4 and 5.
Response 7: Agree. Section 5 has been reformulated as bullet points and now explicitly refers to Table 1 and Figures 4–5:
In this paper, We propose the IWTDC method to address the limitations of the wavelet threshold denoising algorithm by constructing a new threshold function and adopting a new threshold determination method.
At −35 dB input SNR, navigation and communication satellite signals were simulated to analyze the denoising performance of the IWTDC method. Simulations show that the IWTDC method significantly outperforms other algorithms at extremely low SNR, suppressing noise while preserving correlation peaks of satellite signals. The SNR is improved by about 2-6 dB. 
Moreover, real-time testing further confirms its robustness, yielding an SNR enhancement of 7–8 dB with clearly discernible correlation peaks.
In summary, the IWTDC method exhibits strong performance in both theoretical simulations and practical implementations. When spreading gain is unavailable, IWTDC delivers high-SNR data for passive PNT systems, thereby establishing a reliable foundation for nanosecond-level time-difference estimation. Validation on a zero-baseline platform is currently underway, with subsequent campaigns planned across baselines ranging from 50 to 500 km to further optimize the method.

Comments 8: Typo: In line 150, “equation (9)” should read “equation (10)”
Response 8: Agree. We apologize for the oversight, equation (9) has been corrected to equation (10).

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The Authors have taken into account the suggestions appropriately. The manuscript may be considered for publication.

Reviewer 2 Report

Comments and Suggestions for Authors

I am satisfied with the authors' response to my comments. 

The article can be published practically as it is with only a minor editing regarding the use of English.

Comments on the Quality of English Language

The article is generally well written so, regarding the use of English, a minor editing would be sufficient.