Efficient Frequency-Domain Block Equalization for Mode-Division Multiplexing Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript “Efficient frequency-domain block equalization for mode-division multiplexing systems” proposes an adaptive frequency-domain block equalizer (FDBE) based on the adaptive moment estimation (Adam) algorithm for mode-division multiplexing (MDM) optical fiber communication systems. To achieve high accuracy and fast convergence, the author designed an adaptive FDBE to compensate for multiple frequency components and simulated and analyzed its properties. The simulation results suggested the FDBE with the Adam adaption algorithm has the fastest adaption time and the best SER performance. This simulation analysis work has good innovation and positive significance for optimizing MDM systems. I think this manuscript can be accepted and published in Photonics.

Here are some suggestions:

         1.            Page 6, section 5.1, Why are these parameters chosen as simulation conditions? The author should supplement some content to demonstrate the rationality of selecting the current simulation parameters and clarify the applicability of the simulation results.

         2.            The simulation results focused on comparing adaptation speed and the SER performance. However, the author mentioned in the introduction section that “However, the traditional FDEs only compensate for a single frequency at each time in the receiver”. What is the main purpose of this work? What are the key parameters for designing FDBE in this manuscript? The author needs to modify or add some content to clarify the research focus.

Author Response

Comment 1.1:

Page 6, section 5.1, Why are these parameters chosen as simulation conditions? The author should supplement some content to demonstrate the rationality of selecting the current simulation parameters and clarify the applicability of the simulation results?

Response 1.1:
  Thank you for the careful review and valuable comments. We appreciate your constructive feedback and are glad to address your question regarding the choice of simulation parameters in Section 5.1.

  Regarding the choice of simulation parameters, we have selected them based on a comprehensive consideration of practical system requirements and the current state-of-the-art in mode-division multiplexing (MDM) systems. Below, we provide detailed justifications for our parameter selections and clarify the applicability of our simulation results:

1. Rationale for Choosing Simulation Parameters

• Number of Modes (D=12): This number of modes is chosen to represent a moderate-scale MDM system, balancing between complexity and performance. It allows us to demonstrate the effectiveness of our proposed adaptive frequency-domain block equalizer (FDBE) with a reasonable computational load. Moreover, 12-mode systems have been extensively studied in the literature and are feasible with current technology [1,2].
• Transmission Distance (2000km): The transmission distance of 2000 km is typical for long-haul optical fiber communication systems. Dividing the link into shorter sections (1km each) enhances mode coupling, which is crucial for managing the large group delay spread from modal dispersion. This approach is supported by our reference [3], which shows that strong mode coupling reduces the effective group delay spread.
• Section Length (1km): Dividing the transmission link into 2000 sections, each of length 1 km, allows us to model the gradual accumulation of MDL and MD over the link. This granularity is sufficient to capture the key characteristics of the channel response while maintaining a reasonable computational complexity for simulations.
• Coupled Group Delay Statistics: The statistics of the coupled group delay, determined by the root mean square (rms) uncoupled modal dispersion of 29ps/km, are based on typical modal dispersion characteristics of few-mode fibers (FMFs) and multi-mode fibers (MMFs) [4]. It represents a realistic level of modal dispersion that needs to be compensated in MDM systems.
• Symbol Rate (32Gbaud): The symbol rate of 32Gbaud is chosen to align with current and emerging high-speed optical transmission technologies. This rate enables us to evaluate the performance of the proposed equalizer in high-throughput communication systems.

2. Applicability of Simulation Results

  The simulation results presented in our manuscript are applicable to MDM optical fiber communication systems employing FMFs and operating at high symbol rates over long distances. The chosen parameters represent a practical setup that is relevant to current research and development efforts in this field. The performance analysis of the proposed FDBE, in terms of adaptation time and symbol error rate (SER), provides valuable insights into the feasibility and benefits of this approach for mitigating the effects of MDL and MD in MDM systems.

 

[1] S. Ö. Arık, D. Askarov, and J. M. Kahn, "MIMO signal processing for mode-division multiplexing: An overview of channel models and signal processing architectures," IEEE Signal Processing Magazine, vol. 31, no. 2, pp. 25-34, March 2014.

[2] K.-P. Ho and J. M. Kahn, "Mode coupling and its impact on spatially multiplexed systems," in Optical Fiber Telecommunications VI, I. P. Kaminow, T. Li, and A. E. Willner, Eds. Amsterdam: Elsevier, 2013.

[3] S. Ö. Arık, D. Askarov, and J. M. Kahn, "Effect of mode coupling on signal processing complexity in mode-division multiplexing," Journal of Lightwave Technology, vol. 31, no. 3, pp. 423-431, February 2013.

[4] N. Bai and G. Li, "Adaptive frequency-domain equalization for mode-division multiplexed transmission," IEEE Photonics Technology Letters, vol. 24, no. 21, pp. 1918-1921, November 2012.

 

Author’s action:

1.section 5.1 and 5.2

“…a long-haul MDM transmission system is studied, which is described by the system model of Section 2…”

“…We select the 12-mode FMFs with low uncoupled GD spread, relying on strong mode coupling [20]...”

“The total frequency block length  is  to ensure fast adaption.”

“…The parameters of different algorithms were initially optimized using training sequences, in order to improve performance of the algorithms…”

2.Add reference 31

31. Ip, Ezra, and Joseph M. Kahn. "Digital equalization of chromatic dispersion and polarization mode dispersion." Journal of Lightwave Technology 25.8 (2007): 2033-2043.

 

Comment 1.2:

The simulation results focused on comparing adaptation speed and the SER performance. However, the author mentioned in the introduction section that “However, the traditional FDEs only compensate for a single frequency at each time in the receiver”. What is the main purpose of this work? What are the key parameters for designing FDBE in this manuscript? The author needs to modify or add some content to clarify the research focus?

Response 1.2:
  Thank you for the comment. We would like to clarify the main purpose of our work and the key parameters for designing the proposed frequency-domain block equalizer (FDBE).

1. The main purpose of this work

  The main purpose of this work is to develop an adaptive FDBE that can simultaneously compensate for multiple frequency components at each block in MDM optical fiber communication systems. This is motivated by the fact that traditional FDEs only process a single frequency component at a time, which can lead to interference between different frequency components and degrade performance. The proposed FDBE packs all frequency components into frequency - dependent blocks of a specified size B and defines an adaptive equalization matrix to compensate for multiple frequency components simultaneously, aiming to improve the adaptation speed and symbol error rate (SER) performance of the equalizer, effectively reducing the impact of MD and MDL on the system and enhancing the overall performance of the MDM system.

2.The key parameters for designing the FDBE mainly include the following

• Frequency block size B: It determines the number of frequency components included in each block equalization. As described in the section 3.2 paragraph 2 of the revised manuscript, “…we pack the entire frequency components into  frequency dependent blocks within a specified size , where represents the operation of floor function.” The size of B affects the performance of the FDBE. From the simulation results, different values of B show different performances in adaptation time and SER performance. For example, under different MDL and SNR conditions, “…Simulation results suggest that the Adam algorithm with  and  has the best SER performance for dB and dB, respectively.” This indicates that the selection of B needs to comprehensively consider factors such as the system's MDL and SNR to achieve optimal performance.
• Algorithm parameters: Different adaptive algorithms have their own key parameters. As described in the section 5.2 paragraph 1 of the revised manuscript, “For the LMS algorithm, we choose the step size for dB and  for . For the RLS algorithm, the forgetting factor  = 0.999 is defined. For the Adam algorithm, we define the step size , ,  for  and  for , respectively, …” The adjustment of these parameters affects the convergence speed and performance of the algorithm. Specifically, the values of hyperparameters for the LMS and RLS algorithms are derived from empirical values [4], whereas the values of hyperparameters for the ADAM algorithm are obtained through optimization using training sequences as follows:

Figure 1. Hyperparameters optimization for the ADAM algorithm.

• System - related parameters: These parameters include the number of modes, transmission distance, symbol rate, sampling rate, etc. In the section 5.1 paragraph 1-2 of the revised manuscript, “…a 12-mode MDM transmission system is studied... The receiver filter is a fifth-order Butterworth low-pass filter with -dB cutoff frequency  [20].” These parameters represent typical settings of actual MDM systems. They affect the channel response and noise characteristics of the system, and thus have an impact on the design and performance of the FDBE. For example, the transmission distance and the number of modes affect the degree of mode coupling and MD, thereby affecting the delay spread that the equalizer needs to compensate. The symbol rate and sampling rate are related to the spectral characteristics and sampling accuracy of the signal, which are of great significance for the design and performance evaluation of the equalizer.

 

Author’s action:

1.section 1

“…Simulation results indicate that the FDBE with the Adam adaption algorithm has the fastest adaption time and the best SER performance. This suggests that, in comparison to traditional FDEs, the proposed FDBE effectively mitigates the impact of MD and MDL, thereby enhancing the overall performance of the MDM system.…”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript reported an adaptive frequency-domain block equalizer (FDBE) using the Adam algorithm to compensate for multiple frequency components to achieve fast adaption and high performance in MDM optical fiber communication systems. I recommend the publication of the manuscript in Photonics with a minor revision, after the authors have properly addressed my comments below:

(1) The process flow with different algorithms should be given, rather than just the algorithm steps. 

(2) The important parameter values for the processing algorithm should be provided in the manuscript.

 (3) The title of Figure1 should be revised. 

Author Response

Comment 2.1:

The process flow with different algorithms should be given, rather than just the algorithm steps.

Response 2.1:
  Thank you for the comment. Please kindly note that the important parameter values for the processing algorithm are mentioned in in Chapter 2 System Model and Figure 1 of the revised manuscript, the process flow with different algorithms can be detailed as follows:

“Figure 1. Complex baseband model of MDM transmission with MIMO equalizer.”

 

1. Initialization

• Define the system parameters, including the number of modes D, the total number of sections ​, the section length , the symbol rate , the receiver oversampling rate ​, the SNR, the MDL, etc, according to the section 5.1 paragraph 1-2 of the revised manuscript, “…a 12-mode MDM transmission system is studied... The receiver filter is a fifth-order Butterworth low-pass filter with -dB cutoff frequency  [20].”
• Set the initial parameters for the adaptive algorithms according to the section 5.2 paragraph 1 of the revised manuscript, “For the LMS algorithm, we choose the step size for dB and  for . For the RLS algorithm, the forgetting factor  = 0.999 is defined. For the Adam algorithm, we define the step size , ,  for  and  for , respectively, …”
• Initialize the equalization matrix for each frequency k as a D×D zero matrix and  for each frequency block b as a  zero matrix to prepare for subsequent iterative updates.

2. Equalization Matrix Update

• Randomly generate training sequences  for each mode d. Perform pulse shaping, electrical-to-optical conversion , and transmission through the MDM channel . At the receiver, perform optical-to-electrical conversion and electrical filtering  to obtain the received signal. Sample the received signal at equally spaced frequencies to obtain the frequency-domain representation Y​[k].
• Pack the frequency components into frequency-dependent blocks of size B. For each block b, compute the adaptive equalization matrix using the Adam, LMS, RLS and Adam Algorithms: Follow the update equations in Algorithm 1 of section 3.2 of the manuscript to update .
• LMS Algorithm: Follow the update equation (10) to update .
• RLS Algorithm: Follow the update equations (11) and (12) to update .
• Adam Algorithm: Follow the Algorithm 2 of section 3.2 of the manuscript to update .

Apply the equalization matrix  to the packed input block to obtain the equalized output.

3. Performance Evaluation

  Evaluate the symbol error rate (SER). Adjust the adaptive algorithm parameters to optimize the performance. The optimized parameters are updated in step “1. Initialization”.

4. Iteration and Convergence

  Repeat the block equalization process for subsequent training blocks of received signals. Monitor the SER and convergence behavior of the adaptive algorithms. Stop the iteration when the SER reaches a predefined threshold (for example, ) or the maximum number of training blocks is reached (for example,  in Figure 4 of the manuscript).

 

Comment 2.2:

The important parameter values for the processing algorithm should be provided in the manuscript.

Response 2.2:

  Thank you for the comment. Please kindly note that the important parameter values for the processing algorithm are mentioned in the section 5.2 paragraph 1 of the revised manuscript, “…The parameters of different algorithms were tuned in order to improve performance of the algorithms. For the LMS algorithm, we choose the step size  for dB and  for . For the RLS algorithm, the forgetting factor  = 0.999 is defined. For the Adam algorithm, we define the step size , ,  for  and  for , respectively, …” The adjustment of these parameters affects the convergence speed and performance of the algorithm. Specifically, the values of hyperparameters for the LMS and RLS algorithms are derived from empirical values Ref[20] of the manuscript, whereas the values of hyperparameters for the ADAM algorithm are obtained through optimization using training sequences as follows:  

Fig. 1. Hyperparameters optimization for the ADAM algorithm.

 

Author’s action:

1.section 5.1 and 5.2

“…a long-haul MDM transmission system is studied, which is described by the system model of Section 2…”

“…We select the 12-mode FMFs with low uncoupled GD spread, relying on strong mode coupling [20]...”

“The total frequency block length  is  to ensure fast adaption.”

“…The parameters of different algorithms were initially optimized using training sequences, in order to improve performance of the algorithms…”

2.Add reference 31

31. Ip, Ezra, and Joseph M. Kahn. "Digital equalization of chromatic dispersion and polarization mode dispersion." Journal of Lightwave Technology 25.8 (2007): 2033-2043.

 

 

Comment 2.3:

The title of Figure1 should be revised

Response 2.3:

  Thank you for the suggestions. The title of Figure1 was an oversight on our part, and we sincerely apologize for not having carefully checked it. As suggested, we have revised the title of Figure1 to " Complex baseband model of MDM transmission with MIMO equalizer."

  Author’s action: The title of Figure1 of chapter 2 System Model in the revised manuscript.

“Figure 1. Complex baseband model of MDM transmission with MIMO equalizer.”

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Thank you very much for the author's work. 

The authors propose an adaptive FDBE using the Adam algorithm to compensate for multiple frequency components to achieve fast adaption and high performance in MDM optical fiber communication system.

I think the author needs to compare with more other people's work to show the innovation and importance of this work. Overall, the article can be accepted.

Author Response

Comment 3.1:

I think the author needs to compare with more other people's work to show the innovation and importance of this work. Overall, the article can be accepted.

Response 3.1:
  Thank you for the comment. Below is a comparison of our work with some non-cited references:

1. Shi and B. C. Thomsen, “Sparse Adaptive Frequency Domain Equalizers for Mode-Group Division Multiplexing” (2015):

• Novelty: This work focuses on sparse adaptive frequency domain equalizers for mode-group division multiplexing. It aims to reduce the computational complexity by exploiting sparsity in the equalizer coefficients.
• Comparison: Our work also aims to improve performance and reduce complexity but takes a different approach by using frequency-dependent blocks and the Adam algorithm for adaptive equalization. The Adam algorithm is known for its fast convergence and robustness, which is not explicitly discussed in the referenced work.

2. Lee et al., “A Sparsity Managed Adaptive MIMO Equalization for Few-Mode Fiber Transmission” (2016):

• Novelty: This study proposes a sparsity-managed adaptive MIMO equalization for few-mode fiber transmission, aiming to handle various differential mode delays. It emphasizes sparsity as a means to manage computational complexity.
• Comparison: Our work introduces the concept of frequency-domain block equalization, which tackles the interference between different frequency components due to modal dispersion and mode-dependent loss. Additionally, the use of the Adam algorithm for adaptation, not specifically mentioned in this reference, provides faster convergence and improved performance compared to traditional algorithms.

3. K. Shi et al., “SLM-Based Mode Division Multiplexing System With 6×6 Sparse Equalization” (2015):

• Novelty: This work presents an SLM-based MDM system with a 6×6 sparse equalizer, exploiting sparsity to reduce computational complexity.
• Comparison: Our approach focuses on frequency-domain block equalization using the Adam algorithm, which offers distinct advantages in terms of adaptation speed and symbol error rate performance. The referenced work does not discuss the use of the Adam algorithm or block-based equalization strategies.

4. J. Zhou et al., “An Equalization Initialization Procedure for MDM Systems Based on Orthogonal Matching Pursuit” (2017):

• Novelty: This study proposes an equalization initialization procedure using orthogonal matching pursuit for MDM systems. It aims to improve the convergence speed of the equalizer through an efficient initialization strategy.
• Comparison: While our work also aims to improve performance, we do so by proposing an adaptive frequency-domain block equalizer using the Adam algorithm. The block-based approach and the utilization of the Adam algorithm represent the main novelties of our work, which are not addressed in this reference.

5. FARUK, M. S.; KIKUCHI, K., “Adaptive frequency-domain equalization in digital coherent optical receivers” (2011).

• Novelty: The research mainly focuses on adaptive frequency domain equalization technology for digital coherent optical receivers.
• Comparison: Our work focuses on mode-division multiplexing systems, while FARUK and KIKUCHI's paper mainly focuses on frequency domain equalization in single-mode optical fiber transmission. We propose an efficient frequency domain block equalization method by packing all frequency components into frequency-dependent blocks of a specific size and defining an adaptive equalization matrix. This matrix is iteratively calculated using advanced optimization algorithms such as ADAM, RLS, and LMS to achieve faster convergence and better performance. Through simulation experiments, we found that the adaptive frequency domain block equalizer using the ADAM algorithm is better than using the LMS and RLS algorithms in terms of adaptation speed and symbol error rate (SER) performance.

  In summary, our manuscript introduces a novel adaptive frequency-domain block equalizer for MDM systems, leveraging the Adam optimization algorithm for fast convergence and improved performance. This approach differs from the cited references, which mainly focus on sparsity-exploiting adaptive equalizers or initialization procedures. The combination of block-based equalization and the Adam algorithm represents a unique contribution that aims to mitigate the effects of modal dispersion and mode-dependent loss in MDM transmission systems.

Author Response File: Author Response.pdf