Research on High-Frequency PGC-EKF Demodulation Technology Based on EOM for Nonlinear Distortion Suppression

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a study of high frequency PGC-EKF demodulation based on EOM with nonlinear distortion suppression. The results are comprehensive and supportive. Here are minor comments:

1, Please carefully revise the paper regarding the English presentation, including particularly line-14 in the abstract.

2, Please clarify the reason of using the FPGA hardware for implementing the method. Is this the optimized choice?

3, The comparison in Table 2 is too much significant, which make me wonder if the comparison is fair enough? Please clarify.

Author Response

This paper presents a study of high frequency PGC-EKF demodulation based on EOM with nonlinear distortion suppression. The results are comprehensive and supportive. Here are minor comments:

Comment1: Please carefully revise the paper regarding the English presentation, including particularly line-14 in the abstract.

Response 1:

We thank the reviewer for the comment. We have checked and revised the corresponding content in the revised manuscript according to the reviewer's comment.

 

See changes made in Abstract, page 1.

In this study, a phase-generated carrier (PGC) demodulation algorithm combined with the extended Kalman filter (EKF) algorithm based on an electro-optic modulator (EOM) is proposed, which can achieve nonlinear distortion (such as modulation depth drift and carrier phase delay) suppression for high-frequency phase carrier modulation. The improved algorithm is implemented on a field programmable gate array (FPGA) hardware platform. The experimental results by the PGC-EKF method show that total harmonic distortion (THD) decreases from −32.61 dB to −54.51 dB, and SINAD increases from 32.59 dB to 47.86 dB, compared to the traditional PGC-Arctan method. This indicates that the PGC-EKF demodulation algorithm proposed in this paper can be widely used in many important fields such as hydrophone, transformer, and ultrasound signal detection.

 

Comment 2: Please clarify the reason of using the FPGA hardware for implementing the method. Is this the optimized choice?

Response 2:

We thank the reviewer for the suggestion.

The field programmable gate array (FPGA) is a digital signal processing chip, which is widely used to implement various signal processing algorithms and demodulation systems. FPGA is a programmable logic device, and it is more efficient in handling tasks involving a large amount of parallel computation. Compared to digital signal processors (DSPs), FPGA has a faster instruction execution speed. In this paper, FPGA is used for hardware acceleration of the improved algorithm. First, the high-frequency carrier modulation signal is generated by the FPGA and output through a digital-to-analog converter (DAC). Second, the interference signal is digitized by an analog-to-digital converter (ADC) and then demodulated by the FPGA in real-time. Finally, the demodulation result is transmitted to the upper computer. The signal processing speed is increased with the use of FPGA, enabling real-time demodulation of interference signals.

 

See changes made in paragraph 2, section 3, page 5.

… The resolutions of AD9253 and DA9122 are 14 bits and 16 bits, respectively. The FPGA (K7 Xc7k325t, Xilinx) is used as the hardware development platform of the digital PGC demodulation system, which is used for hardware acceleration of the improved algorithm, enabling real-time demodulation of interference signals.

 

Comment 3: The comparison in Table 2 is too much significant, which make me wonder if the comparison is fair enough? Please clarify.

Response 3:

We sincerely appreciate the valuable comment.

In the experiments, the interference signal is affected by the combination of modulation depth (C value) drift and carrier phase delay. The demodulation results of the PGC-Arctan algorithm are sensitive to C value drift and phase delay, and the performance of the demodulated signals is degraded. Compared with the PGC-Arctan algorithm, the proposed PGC-EKF algorithm can eliminate the effects of carrier phase delay and C value drift, which leads to the more outstanding performance of the improved algorithm.

To highlight the advancements of the algorithm in this paper, a comparison of the demodulation results of the traditional algorithm and other algorithms (the PGC-SDD and PGC-SDD-DSM algorithms) is presented. We changed the value of the carrier phase delay by adjusting the initial phase of the reference carriers and completed the experimental part.

 

See changes made in paragraph 4, section 3, page 7.

The time domain waveform of the demodulated signal is shown in Fig. 7. The demodulated signal waveforms of the PGC-Arctan and PGC-SDD algorithms have obvious distortions because of carrier phase delay and C value drift in the system. The PGC-SDD-DSM algorithm integrates the differentiated signal, and the demodulation result introduces nonlinear error in the integration operation. In contrast, the PGC-EKF algorithm is unaffected by C value drift and phase delay, and the demodulated signal waveform is smoother without obvious harmonic distortion, which can effectively recover the phase signal to be measured.

Generally, nonlinear errors can be evaluated by total harmonic distortion (THD), signal-to-noise and distortion ratio (SINAD). SINAD is defined as the ratio of the fundamental power to the sum of all noise and harmonic power, which can be measured the degree of nonlinear distortion and accurately reflect the anti-noise performance. THD is defined as the ratio of the equivalent root mean square (RMS) amplitude of all harmonics to the fundamental frequency amplitude. Nonlinear distortion mainly includes modulation depth drift, carrier phase delay, and the nonlinear effects of low-pass filters. The comparison of the evaluation metrics of the four algorithms for signal demodulation is shown in Table 3. The PGC-EKF scheme achieves a SINAD as high as 47.86 dB and a THD of -54.51 dB, which ensures the accuracy of signal demodulation. By contrast, the performance of the other Three schemes is degraded due to the presence of the C value drift and q. The SINAD of the improved PGC demodulation algorithm achieves gains of 15.27 dB, 18.57 dB, and 14.55 dB, compared with PGC-Arctan, PGC-SDD, and PGC-SDD-DSM, respectively. The THD achieves gains of -21.9 dB, -22.14 dB, and -8.17 dB. Based on the achieved results, PGC-EKF confirms its superiority over the basic PGC-Arctan algorithm and the other demodulation algorithms.

 

Table 2. Comparison of performance between PGC algorithms.

Algorithm	THD/(dB)	SINAD/(dB)	Reference
PGC-Arctan	−32.61	32.59	[2]
PGC-SDD	−32.37	29.29	[20]
PGC-SDD-DSM	−46.34	33.31	[32]
PGC-EKF	−54.51	47.86	This work

 

 

 

 

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

This review report is for the manuscript 'Research on high frequency PGC-EKF demodulation technology based on EOM for nonlinear distortion suppression' by Peng Wu, et al.

 

The manuscript proposed a PGC-EKF approach to improve the signal performance on an FPGA. Based on the results, the work demonstrated significant improvement in SINAD and THD, and suppressed AC interference and distortion over literature. 

 

The following are comments for the authors to address in the manuscript:

 

1) Introduction section. The manuscript would benefit from a more comprehensive discussion of EKF's background in other literature and fields. This would better introduce readers to the technique's potential and broader applicability.

 

2) Experimental System section. The manuscript could also be strengthened by discussing the rationale behind the selection of specific components for the experiment, and how the alternative specifications could improve or degrade the system performance. Will reducing the distortion at the hardware level be an alternate solution? 

 

3) Results and Analysis section, and Table 2. The introduction section covers many relevant literature, to strengthen the analysis, the manuscript could consider incorporating a comparative discussion that highlights the findings of this study in relation to the previously cited reports. Demonstrating quantifiable improvements over previous work through a comparative analysis would significantly strengthen the overall competency of this research

 

4) Line 168, Please define PZT and check for other typos. A thorough proofreading would further enhance the clarity of the manuscript. 

 

5) The references list could be improved. If some references are not readily available online, consider including alternative sources from high-impact journals that support the presented research. For example, ref[25].

 

I hope the comments will assist the authors in improving the manuscript. 

Author Response

This review report is for the manuscript 'Research on high frequency PGC-EKF demodulation technology based on EOM for nonlinear distortion suppression' by Peng Wu, et al.

The manuscript proposed a PGC-EKF approach to improve the signal performance on an FPGA. Based on the results, the work demonstrated significant improvement in SINAD and THD, and suppressed AC interference and distortion over literature.

The following are comments for the authors to address in the manuscript:

 

Comment 1: Introduction section. The manuscript would benefit from a more comprehensive discussion of EKF's background in other literature and fields. This would better introduce readers to the technique's potential and broader applicability.

Response 1:

Thanks to the reviewers for their valuable questions.

We have checked and revised the corresponding content in the revised manuscript according to the reviewer's comments.

 

See changes made in paragraph 1, section 1, page 2.

… These methods are not sensitive to changes of modulation depth. But if there is carrier phase delay in the system, the demodulation of the signal still exhibits nonlinear distortion. The Ellipse Fitting Algorithm (EFA) based on least squares can suppress nonlinear distortion caused by phase delay and C value changes [23, 24], and improved ellipse fitting algorithms have also been proposed [25, 26]. However, overly complex models can increase computation time and hardware resource consumption [27]. As the extended form of the Kalman filter (KF), the extended Kalman filter (EKF) is used for state estimation of nonlinear systems. By using a Taylor series expansion, the nonlinear system is linearized, thus transforming the nonlinear problem into a linear one. EKF has been widely used in state-of-charge (SOC) estimation of battery management systems (BMS) [28], abnormal data inspection of photovoltaic power generation [29], and real-time orbit determination of satellite navigation systems [30].

 

[28] S. L. Zuo, W. C, Z. B. Fu, et al. Abnormal data inspection technology of photovoltaicpower generation based on EKF algorithm [J]. Electric Power Engineering Technology, 2020, 39(5): 120-125.

[29] C. Y. Tang, H. C. Han, M. M. S, et al. SOC estimation method of battery energy storage system for BMS test platform [J]. Electric Power Engineering Technology, 2021, 40(3): 7-14.

[30] L. Chen, B. W. Jiang, Y. Q. Liu, et al. Application of adaptive EKF in real-time orbit determination. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2021, 43(4): 187.

 

Comment 2: Experimental System section. The manuscript could also be strengthened by discussing the rationale behind the selection of specific components for the experiment, and how the alternative specifications could improve or degrade the system performance. Will reducing the distortion at the hardware level be an alternate solution?

Response 2:

Thanks to the reviewers for the valuable questions.

In the experimental system, a narrow linewidth laser is chosen as the light source to reduce the phase noise of the system and the drift of the interference pattern caused by the uncertainty of the light source frequency. In addition, the narrow linewidth laser provides high-contrast interference fringes, which is favorable to the stability and accuracy of the measurement [R1]. In this paper, a narrow linewidth laser module produced by RIO Company is chosen as the light source, which has a center wavelength of 1550.11 nm and a linewidth of less than 1 kHz.

For the selection of electro-optical modulators (EOM), EOM with lower half-wave voltage (V_π) should be chosen to obtain higher modulation efficiency [R2]. In this paper, a LiNbO3 EOM is used to modulate the optical signal. The operating wavelength is 1550 ± 20 nm, an insertion loss is 3.5 dB, the V_π is 3.5 V, and the bandwidth is 10 GHz. The EOM has the advantages of low half-wave voltage, low insertion loss, and low drive voltage.

Interferometric fiber optic sensor based on Michelson interferometric structures is used to measure of external signals, theoretically eliminating polarization fading completely and requiring no additional devices.

[R1] Z. C. Zhang. Research on Signal Demodulation Technology of Optical Fiber Sensor Vibration System [D]. Beijing: North China Electric Power University, 2019.

[R2] J. T. Lin, R. H. Gao, J. L. Guan, et al. Advances in Low-Loss Thin-Film Lithium Niobate Photonic Integrated Devices [J]. Journal of Syntheticcrystals, 2024, 53(03): 372-394.

 

See changes made in paragraph 2, section 3, page 5.

In the demodulation system, a narrow linewidth laser is chosen as the light source to reduce the phase noise of the system and the drift of the interference pattern caused by the uncertainty of the light source frequency [32]. A narrow linewidth laser module produced by RIO company is chosen as the light source, which has a center wavelength of 1550.11 nm and a linewidth of less than 1 kHz. The operating wavelength of EOM is 1550 ± 20 nm, an insertion loss is 3.5 dB, the V_π is 3.5 V, and the bandwidth is 10 GHz. The EOM has the advantages of low half-wave voltage, low insertion loss, and low drive voltage. The resolutions of AD9253 and DA9122 are 14 bits and 16 bits, respectively. The FPGA (K7 Xc7k325t, Xilinx) is used as the hardware development platform of the digital PGC demodulation system, which is used for hardware acceleration of the improved algorithm.

 

Comment 3: Results and Analysis section, and Table 2. The introduction section covers many relevant literature, to strengthen the analysis, the manuscript could consider incorporating a comparative discussion that highlights the findings of this study in relation to the previously cited reports. Demonstrating quantifiable improvements over previous work through a comparative analysis would significantly strengthen the overall competency of this research

Response 3:

The reviewer's suggestion is thoughtful.

In the experiments, the interference signal is affected by the combination of modulation depth (C value) drift and carrier phase delay. The demodulation results of the PGC-Arctan algorithm are sensitive to C value drift and phase delay, and the performance of the demodulated signals is degraded. Compared with the PGC-Arctan algorithm, the proposed PGC-EKF algorithm can eliminate the effects of carrier phase delay and C value drift, which leads to the more outstanding performance of the improved algorithm.

To highlight the advancements of the algorithm in this paper, a comparison of the demodulation results of the traditional algorithm and other algorithms (the PGC-SDD and PGC-SDD-DSM algorithms) is presented. We changed the value of the carrier phase delay by adjusting the initial phase of the reference carriers and completed the experimental part.

 

See changes made in paragraph 4, section 3, page 7.

The time domain waveform of the demodulated signal is shown in Fig. 7. The demodulated signal waveforms of the PGC-Arctan and PGC-SDD algorithms have obvious distortions because of carrier phase delay and C value drift in the system. The PGC-SDD-DSM algorithm integrates the differentiated signal, and the demodulation result introduces nonlinear error in the integration operation. In contrast, the PGC-EKF algorithm is unaffected by C value drift and phase delay, and the demodulated signal waveform is smoother without obvious harmonic distortion, which can effectively recover the phase signal to be measured.

Generally, nonlinear errors can be evaluated by total harmonic distortion (THD), signal-to-noise and distortion ratio (SINAD). SINAD is defined as the ratio of the fundamental power to the sum of all noise and harmonic power, which can be measured the degree of nonlinear distortion and accurately reflect the anti-noise performance. THD is defined as the ratio of the equivalent root mean square (RMS) amplitude of all harmonics to the fundamental frequency amplitude. Nonlinear distortion mainly includes modulation depth drift, carrier phase delay, and the nonlinear effects of low-pass filters. The comparison of the evaluation metrics of the four algorithms for signal demodulation is shown in Table 3. The PGC-EKF scheme achieves a SINAD as high as 47.86 dB and a THD of -54.51 dB, which ensures the accuracy of signal demodulation. By contrast, the performance of the other Three schemes is degraded due to the presence of the C value drift and q. The SINAD of the improved PGC demodulation algorithm achieves gains of 15.27 dB, 18.57 dB, and 14.55 dB, compared with PGC-Arctan, PGC-SDD, and PGC-SDD-DSM, respectively. The THD achieves gains of -21.9 dB, -22.14 dB, and -8.17 dB. Based on the achieved results, PGC-EKF confirms its superiority over the basic PGC-Arctan algorithm and the other demodulation algorithms.

 

Comment 4: Line 168, Please define PZT and check for other typos. A thorough proofreading would further enhance the clarity of the manuscript.

Response 4:

Thanks for your careful checks. The reviewer's comment is correct. We have checked and revised the corresponding content in the revised manuscript according to the reviewer's comment.

 

See changes made in paragraph 1, section 3, page 5.

The high-frequency PGC demodulation system based on EOM is composed of the RIO narrow linewidth laser, lithium niobate EOM, Michelson interferometer (MI), radio frequency (RF) voltage amplifier, signal generator, and signal processing board integrated with PD, ADC, and digital-to-analog converter (DAC), as shown in Fig. 4. The light wave generated by laser is modulated by a phase modulator, and enters the Michelson interferometer. The interference light signal from the Michelson interferometer reaches PD, and then the photoelectrically converted signal is quantized into a digital signal by the ADC. Due to the small modulation carrier voltage output by DAC, the phase modulator must be amplified through a high voltage amplifier (HVA) with a magnification of 20 times, then it is driven. To verify the effectiveness of the PGC-EKF algorithm, a piezoelectric transducer (PZT) is wrapped around the sensing arm of the Michelson interferometer. The signal generator (SG) applied a sine signal to the PZT, which is the phase signal to be measured.

 

Comment 5: The references list could be improved. If some references are not readily available online, consider including alternative sources from high-impact journals that support the presented research. For example, ref [25].

Response 5:

We thank the reviewer for the suggestion.

The reviewer's comment is correct. We have checked and revised the corresponding content according to the formatting of the references in the Photonics template.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

This is the second review report for the manuscript titled 'Research on high frequency PGC-EKF demodulation technology based on EOM for nonlinear distortion suppression' by Peng Wu et al.

 

A minor change to the reference list is suggested:

-Cite related and high-impact literature

-Not cite a thesis or documents that are not publicly available globally.

-To indicate in the reference that the cited report's original language (non-English references)

 

The authors have addressed all the other comments.

Author Response

Comment: A minor change to the reference list is suggested: Cite related and high-impact literature. Not cite a thesis or documents that are not publicly available globally. To indicate in the reference that the cited report's original language (non-English references)

Response: We thank the reviewer for the comment. We have checked and updated the high-quality English reference.

Author Response File: Author Response.docx