Evaluation of Correlation-Based Methods for Time Period Estimation in Vehicle Speed Measurement Using Pyroelectric Infrared Sensors

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

1. In row 131 (6÷10), row 165 (5÷10), and Table 2, the use of the ÷ symbol seems to be wrong
2. Some key data should be given in the results and discussion section and the conclusion section
3. The results should reflect the comparison of different methods to verify the effectiveness of the proposed method
4. It is recommended to add some flowcharts and schematic diagrams of key methods to better show the logic of the research
5. The real-time application scenario and limitations of the proposed method need to be explained

Author Response

Comment 1. In row 131 (6÷10), row 165 (5÷10), and Table 2, the use of the ÷ symbol seems to be wrong.

Response: Thank you for pointing out this formatting issue. We have corrected the improper use of the ÷ symbol and replaced it with proper mathematical expressions (e.g., (6)-(10)) in the revised manuscript. We have also thoroughly reviewed the manuscript to correct similar typographical inconsistencies.

Comment 2. Some key data should be given in the results and discussion section and the conclusion section

Response: We appreciate this suggestion. In the revised manuscript, we have added specific numerical outcomes in both the results and conclusion sections to clearly highlight the performance differences between the methods evaluated.

- In Section 3.2 Experimental Validation, we explicitly state that the maximum estimation error did not exceed 5% for the conventional CCF method and 3.5% for the CCFHT method across tested speeds from 20 km/h to 100 km/h.

- In the Conclusion (Section 5), we have reinforced these findings by summarizing the key quantitative performance indicators, emphasizing the lower variance and improved robustness of the CCFHT method in practical conditions.

These additions better communicate the significance of the results and the effectiveness of the proposed evaluation framework

Comment 3.          The results should reflect the comparison of different methods to verify the effectiveness of the proposed method?

Response: Thank you for this helpful comment. Our study focuses on the analytical evaluation of bias and variance in time period estimation (TPE) using two specific correlation-based methods: the Conventional Cross-Correlation Function (CCF) and the Hilbert-Enhanced Cross-Correlation (CCFHT).

In Sections 2.3 and 2.4, we provide mathematical derivations to compare how each method behaves under sensor mismatch, noise, and bandwidth changes.

In Section 3, we provide both numerical simulation results and real-world experimental validation to compare the effectiveness of CCF and CCFHT. The results show that while both methods produce reasonable estimates, CCFHT consistently yields lower bias and variance, especially under noisy or mismatched conditions.

In Section 4.3, we also briefly discuss other TPE methods such as Generalized Cross-Correlation (GCC) and Wavelet-based approaches as potential directions for future research, though our paper maintains its focus on correlation-based estimators

Comment 4. It is recommended to add some flowcharts and schematic diagrams of key methods to better show the logic of the research.

Response: Thank you for this suggestion. We addressed this point in several ways. In Section 2, we added a clear description of the methodological flow, outlining how the mathematical analysis progresses from general theory to specific evaluation of the CCF and CCFHT methods. This structure is summarized in the paragraph at the beginning of Section 2, which now states: “This section presents the analytical and modeling framework developed to evaluate the bias and variance of time period estimation methods applied to PIR sensor signals… The structure of this section follows a logical progression...”.  Additionally, Figure 1 (now Figure 2 after restructuring) includes a schematic diagram of the experimental system with improved resolution and updated labels for clarity. This supports better understanding of the physical setup

Comment 5.  The real-time application scenario and limitations of the proposed method need to be explained.

Response: This is an important observation. In the revised manuscript, we have added Section 4.3: Implications for Practical Deployment to address this. We explain that the proposed system, based on PIR sensors and correlation-based estimation methods, is low-cost, passive, and energy-efficient, making it well-suited for deployment in lightweight ITS applications (e.g., parking management, rural road monitoring). We also discuss current limitations, such as:

- Dependence on proper alignment of sensors,

- Effects of temperature drift and sensor mismatch,

- Limited responsiveness to speed variations over longer distances.

To overcome these challenges, we suggest: Sensor synchronization calibration; Use of other approaches such as machine learning to model waveform patterns; Integration of additional sensor modalities (e.g., IMU or ambient light sensors). These discussions are presented in Section 4.3 and provide a realistic outlook for implementing our method in real-time systems.

We are grateful to the reviewers for their insightful feedback, which has directly led to an improved and more coherent manuscript. We believe the revised version offers a more complete and impactful contribution to the field of low-cost vehicle speed measurement systems using passive sensors.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The foundational explanations in this paper are highly detailed, and the overall structure is well-organized. Below are my specific comments:

1. The authors devote significant space to introducing fundamental mathematical formulas and theories, including references to prior works (e.g., Gaussian noises in Refs. [21]-[23]). Please explicitly clarify:
• Which formulas are derived by the authors?
• Which are directly cited from references?
• Also, please specify what improvements have been made to the derived formulas and what conclusions have been drawn.
2. Throughout the paper, the authors use theoretical derivations and numerical solutions to present conclusions and analyses. However, there is a lack of simulation or experimental validation to support the correctness of these conclusions. Please add simulation or experimental verification.
3. Additionally, if the vehicle speed between two sensors is variable and non-uniform, how will the proposed model handle this? Will it calculate an average speed or treat speed variation as a sensor delay issue? Could this lead to system instability?

Author Response

We would like to present out responses to the reviewer’s comments point-to-point.
Comment 1. 1.     The authors devote significant space to introducing fundamental mathematical formulas and theories, including references to prior works (e.g., Gaussian noises in Refs. [21]-[23]). Please explicitly clarify:

• Which formulas are derived by the authors?
• Which are directly cited from references?
• Also, please specify what improvements have been made to the derived formulas and what conclusions have been drawn.

Response: We thank the reviewer for highlighting this important issue. In the revised manuscript, we have made explicit distinctions between the formulas cited from previous work and those derived in our study. The mathematical expressions related to the statistical properties of Gaussian noise (e.g., autocorrelation, power spectral density) are cited from previous studies—originally listed as [21–23] in the old manuscript, now revised and renumbered as [14], [32], and [33]. These expressions serve as foundational elements for our derivations. Our main contribution lies in deriving closed-form expressions for bias and variance of time period estimators based on correlation methods (CCF and CCFHT). This is achieved by applying Mean Value Theory and first-order Taylor expansion around the extremum point of the cross-correlation function. The derived formulations allow us to analytically assess the impact of sensor mismatch, signal bandwidth, and SNR on the performance of CCF and CCFHT, as shown in Sections 2.3 and 2.4. The conclusions drawn from these derivations form the basis of our performance evaluations in Section 3 (Numerical Results and Experimental Validation).

Comment 2. Throughout the paper, the authors use theoretical derivations and numerical solutions to present conclusions and analyses. However, there is a lack of simulation or experimental validation to support the correctness of these conclusions. Please add simulation or experimental verification.

Response: We deeply appreciate the reviewer’s recommendation, and we have fully addressed this concern in the revised manuscript.

In Section 3.1, we present comprehensive numerical simulations to validate the analytical predictions regarding the impact of sensor mismatch, noise, and bandwidth on estimation bias and variance for both CCF and CCFHT methods.

 

In Section 3.2, we now include an experimental validation conducted using our custom-built dual-PIR sensor system. Real vehicles traveling at speeds ranging from 20 km/h to 100 km/h were measured and compared against reference speeds obtained from high-speed video footage. As shown in Figure 7(a), both methods provide consistent estimates of speed, but Figure 7(b) and the accompanying discussion highlight that CCFHT yields lower estimation error, especially under mismatch conditions.

Comment 3.          Additionally, if the vehicle speed between two sensors is variable and non-uniform, how will the proposed model handle this? Will it calculate an average speed or treat speed variation as a sensor delay issue? Could this lead to system instability?

Response: This is an excellent and insightful question, and we are grateful to the reviewer for raising it. In our work, the primary aim is to evaluate the bias and variance of time period estimators using analytical methods under idealized but practically relevant conditions. For this reason, we assume that the differential infrared flux from the vehicle can be approximated by a single-frequency sinusoidal signal, as described in Section 2.2.

In practice, we acknowledge that non-uniform vehicle speeds can occur. To mitigate the resulting error: The PIR sensors in our system are placed only 1 meter apart, minimizing the temporal window in which speed variations may occur. The deterministic component of the PIR signal dominates in our setup, and this component is modeled as sinusoidal based on the Field of View (FOV) and vehicle speed. While this assumption simplifies derivations, it captures the essence of the signal behavior for analytical tractability.

Additionally, we note in Section 4.3 (Implications for Practical Deployment) that non-uniform motion can be a source of uncertainty, and suggest future enhancements, including: Sensor synchronization; Incorporation of complementary sensors (e.g., IMUs), and other TPE methods to better handle speed variability.

From our experimental validation in Section 3.2, we observe that the system performs acceptably even when vehicle speeds vary slightly during measurement, suggesting reasonable robustness of the current model.

We are grateful to the reviewers for their insightful feedback, which has directly led to an improved and more coherent manuscript. We believe the revised version offers a more complete and impactful contribution to the field of low-cost vehicle speed measurement systems using passive sensors.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Authors analyze time delay estimation in vehicle speed measurement which is performed with Pyroelectric infrared sensors. 

Authors should check and correct:

TECHNICAL ASPECT
1. Figure 1. has the right part with a schema of the PIR sensors integrated in speed measurement system. Obviously, this is raster graphic and during image resize, the integrated text lost precision. Authors should maybe recreate the right part of the figure so the labels next or in schema elements are all precise, visible and compact.

CONTENT
1. The idea - pyroelectric infrared sensors are based on detecting naturally emitted infrared radiation (changes in temperature) and they are used to detect human or animal presence by sensing emitted infrared radiation. (See: https://www.electricity-magnetism.org/pyroelectric-infrared-sensor/). They are not commonly used for speed measurement.
There is a published paper where it could be seen that this sensor is used for human movement detection (https://www.mdpi.com/1424-8220/14/5/8057).

2. Title - 
2.1. it is not precise in the contribution part, but it only presents the problem of delay estimation when measuring vehicle speed with sensors. When you say - "uncertainty analysis" this does not provide any contribution - which method has been proposed for this analysis etc. Is this paper contributing with analysis that uses existing methods...Which methods are used in uncertainty analysis? Having "uncertainly analysis" in the title is too broad and imprecise.
2.2. Why "time delay" estimation in speed measurement? Is this term in fact related to "time period" estimation, having focused the time period between two sensor detections used for speed measurement, or "time delay" is an error in speed measurement system, that appears due to differences in thermal and electrical part of sensors and the whole measurement system?

2. Introduction -It is not a common practice to use equations in the introduction (line 40). It is better to shift it into the materials and methods section, in line 86 (instead of "which is mentioned in the previous section".

3. Background - authors should explain sensors and methods in use for speed measurement, types of errors in measurements, as well as basic methods for estimation of speed measurement accuracy with special emphasis on explaining methods that are integrated in the proposed method in this manuscript.  

4. Related work - not presented with details, but just briefly in the introduction. There should be more detailed related work, preferably in a separate section after introduction. The key aspects of this research are related to types and causes of errors in speed measurements and the method for evaluation of measurement results. Therefore, in the related work there should be presented previous results about these key aspects. 

5. In introduction's related work - Authors evaluate CCF with Hilbert's transform, but they also state that this was already presented in [14-16]. What is the contribution of this manuscript comparing to [14-16]?

6. The manuscript structure should have expected sections - common in research papers: Introduction, Background (basic knowledge of key terms and existing methods), Related work (previous similar published works),  Materials and methods (the proposed method or system), Experiment (research methodology - hypotheses and methods, experimental setup - with details about experimental working environment and sample - here should be placed the photo and schema of the experimental system currently presented at Figure 1, results - results of experiment with sample), Discussion (about contribution compared to previous research, about experimental results), Conclusion.

6. Section "2. Mathematical framework..." starts with subsection 2.1. Signal modeling, but this section does not present the method, framework...but starts with a system construction with the use of PIR (Pyroelectric Infrared Sensor). This section should emphasize proposed metrics on possible types and causes of errors in speed measurements and the method for evaluation of measurement results, based on these metrics.

OVERAL OPPINION: The manuscript deals with a practically oriented issues regarding sensors'-based measurements quality, but should be reorganized in title, structure and presentation level of details.

 

Author Response

we would like to present out responses to the reviewer’s comments point-to-point.

COMMENTS ON TECHNICAL ASPECT

Figure 1. has the right part with a schema of the PIR sensors integrated in speed measurement system. Obviously, this is raster graphic and during image resize, the integrated text lost precision. Authors should maybe recreate the right part of the figure so the labels next or in schema elements are all precise, visible and compact.

Response: We appreciate this helpful technical comment. As suggested, we have revised the schematic diagram in the below part of the original Figure 1 to improve label clarity and resolution. All annotations have been adjusted for compactness and visibility. Additionally, due to the restructuring of the manuscript, this figure has now been repositioned and renumbered as Figure 2, which better aligns with the flow of the new Section 2 (Materials and Methods). This ensures both technical readability and structural consistency.

 

COMMENTS ON CONTENT

Comment 1. The idea - pyroelectric infrared sensors are based on detecting naturally emitted infrared radiation (changes in temperature) and they are used to detect human or animal presence by sensing emitted infrared radiation. (See: https://www.electricity-magnetism.org/pyroelectric-infrared-sensor/). They are not commonly used for speed measurement.
There is a published paper where it could be seen that this sensor is used for human movement detection (https://www.mdpi.com/1424-8220/14/5/8057).

Response: Thank you for this thoughtful remark. We sincerely appreciate the reviewer’s attention to the fundamental principle of PIR sensors. Indeed, PIR sensors are more commonly applied in motion detection and occupancy sensing. We fully agree that using PIR sensors for speed measurement is not widely adopted. However, this challenge is precisely what motivated our study. While rare, PIR-based speed detection has been explored in a few early studies, including:

- [11] Tarik, M. H.; Ahsen, M.B.; Tarek, N. S.; Samir A. A. Infrared Pyroelectric Sensor for Detection of Vehicular Traffic Using Digital Signal Processing Techniques. IEEE Trans. Veh. Technol., 1995, 44 (3), pp. 683 – 689.

- [7] E. Y. Luz; A. Mimbela. Summary of vehicle detection and surveillance technologies used in intelligent transportation systems. Technical Report, The Vehicle Detector Clearinghouse, Southwest Technology Development Institute (SWTDI) at New Mexico State University, 2007.

Moreover, our prior work developed a dual-PIR prototype for speed estimation: [25] Vu Van Quang, Vu Toan Thang. A novel system for measuring vehicle speed via analog signals of pyroelectric infrared sensors. Int. J. Mod. Phys. B, 2021, 35 (14), p. 2140028.

These studies demonstrate feasibility but lack in-depth analysis of uncertainty. Our current paper builds on this foundation by introducing a full analytical framework to assess bias and variance in time period estimation using correlation-based methods under realistic sensor imperfections.

Comment 2. On Title

Comment 2.1. it is not precise in the contribution part, but it only presents the problem of delay estimation when measuring vehicle speed with sensors. When you say - "uncertainty analysis" this does not provide any contribution - which method has been proposed for this analysis etc. Is this paper contributing with analysis that uses existing methods...Which methods are used in uncertainty analysis? Having "uncertainly analysis" in the title is too broad and imprecise.

Response: We thank the reviewer for this important observation. In response, we have changed the title of the manuscript to more accurately reflect the scope and methodology: “Evaluation of Correlation-based Methods for Time Period Estimation in Vehicle Speed Measurement Using Pyroelectric Infrared Sensors”.

This revised title explicitly mentions the methods under evaluation (CCF and CCFHT), the task (time period estimation), the context (speed measurement), and the sensor type (PIR). We believe this better represents the actual contributions of the work.

Comment 2.2. Why "time delay" estimation in speed measurement? Is this term in fact related to "time period" estimation, having focused the time period between two sensor detections used for speed measurement, or "time delay" is an error in speed measurement system, that appears due to differences in thermal and electrical part of sensors and the whole measurement system?

Response: We greatly appreciate this nuanced question. Initially, we used “time delay” in the same spirit as prior literature ([2–4]), where it refers to the time interval between two sensor signals. However, to improve clarity and avoid misinterpretation, we have now consistently used “time period estimation (TPE)” throughout the paper and stated in Section 1.1: “Among the commonly used methods for speed measurement, time period (or time delay, time shift, time interval in other literatures [2-4]) estimation (TPE) between two spatially separated sensors remains one of the most intuitive and robust strategies, especially for point-based systems”. This change ensures consistency and clarity for readers unfamiliar with multiple terminologies.

Comment 3. Introduction -It is not a common practice to use equations in the introduction (line 40). It is better to shift it into the materials and methods section, in line 86 (instead of "which is mentioned in the previous section").

Response: We fully agree with this recommendation. The equation originally placed in the Introduction (Eq. for speed from time delay and distance) has been moved to Subsection 2.2, where signal modeling and system setup are discussed. The sentence in the Introduction has also been rephrased to refer descriptively to the speed computation principle without using a formula.

Comment 4. Background - authors should explain sensors and methods in use for speed measurement, types of errors in measurements, as well as basic methods for estimation of speed measurement accuracy with special emphasis on explaining methods that are integrated in the proposed method in this manuscript.  

Response: We are very grateful for the Reviewer's comment, and indeed we find it will make our paper more coherent. After restructuring the paper, we have dedicated subsection 1.1 Background in Section 1. Introduction. Particularly, in subsection 1.1 – Background, we detail: Common sensor types for vehicle speed detection (LiDAR, radar, magnetic loops, cameras); The advantages and limitations of PIR sensors in comparison; Common sources of uncertainty in TPE: sensor response mismatch, temporal resolution, environmental noise.

Comment 5. Related work - not presented with details, but just briefly in the introduction. There should be more detailed related work, preferably in a separate section after introduction. The key aspects of this research are related to types and causes of errors in speed measurements and the method for evaluation of measurement results. Therefore, in the related work there should be presented previous results about these key aspects. 

Response: Thank you for pointing this out. We have restructured and expanded Section 1.2 – Related Work, where we now present:

-Previous applications of PIR sensors for detection or basic speed estimation,

- Work on cross-correlation methods in general signal processing (e.g., [18–20]),

- A clear identification of the gap: no prior analytical evaluation of correlation-based methods (CCF, CCFHT) under sensor mismatch and noise in PIR-based systems.

This is now stated clearly: “To the best of our knowledge, no prior study has systematically evaluated the application of correlation-based estimators—such as CCF and CCFHT—to PIR-based speed measurement systems under conditions involving sensor response mismatch and environmental noise.”

Comment 6. In introduction's related work - Authors evaluate CCF with Hilbert's transform, but they also state that this was already presented in [14-16]. What is the contribution of this manuscript comparing to [14-16]?

Response: Thank you for this excellent point. The methods in [14–16] (now [21–23] in the revised manuscript) applied CCFHT for white-noise-like signals or stationary sources. Our contribution is distinct in several ways: PIR signals are non-white, with deterministic waveform components shaped by vehicle speed and sensor FOV. Our work addresses sensor mismatch, especially time constant asymmetry, which prior works do not. We derive bias and variance expressions specific to PIR-like signal shapes and validate them through simulation and real-traffic data. These distinctions are now explained in Section 1.3 (Contributions).

Comment 7. The manuscript structure should have expected sections - common in research papers: Introduction, Background (basic knowledge of key terms and existing methods), Related work (previous similar published works),  Materials and methods (the proposed method or system), Experiment (research methodology - hypotheses and methods, experimental setup - with details about experimental working environment and sample - here should be placed the photo and schema of the experimental system currently presented at Figure 1, results - results of experiment with sample), Discussion (about contribution compared to previous research, about experimental results), Conclusion.

Response: We are really grateful for this comment. In fact We have revised the structure as follows:

- Section 1: Introduction (with Background and Related Work subsections),

- Section 2: Materials and Methods (framework, signal modeling, CCF and CCFHT),

- Section 3: Numerical Results and Experimental Validation,

- Section 4: Discussion (including robustness analysis),

- Section 5: Conclusion.

Figures and experimental setup descriptions have also been relocated accordingly (e.g., Figure 1 is now in Section 2.2).

Comment 8. Section "2. Mathematical framework..." starts with subsection 2.1. Signal modeling, but this section does not present the method, framework...but starts with a system construction with the use of PIR (Pyroelectric Infrared Sensor). This section should emphasize proposed metrics on possible types and causes of errors in speed measurements and the method for evaluation of measurement results, based on these metrics.

Response: Thanks for the reviewer's comments. We have taken them up by moving subsection 2.1 to the top to discuss the approach of using Mean value theory or Taylor expansion to determine the bias and variance of an estimator for the zero-point of a target function. To that end, in the remaining sections, we attempt to derive the target function formulation for each correlation-based method, starting from defining them in the frequency domain.

We are grateful to the reviewers for their insightful feedback, which has directly led to an improved and more coherent manuscript. We believe the revised version offers a more complete and impactful contribution to the field of low-cost vehicle speed measurement systems using passive sensors.

Sincerely,
The Authors

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised manuscript has dispelled my doubts and I have no further comments