Introducing Combined Effects of Filtering and ASE Noise in Optical Links Supposing Different Equalization Algorithms

Technical Summary: Introducing Combined Effects of Filtering and ASE Noise in Optical Links Supposing Different Equalization Algorithms

Problem Statement
Metro and access optical networks are increasingly required to support high-capacity coherent transmission while maintaining accurate and computationally efficient quality-of-transmission (QoT) estimates. In these shorter-link scenarios, linear impairments, particularly cascaded optical filtering from ROADMs and other filtering elements, often dominate over nonlinear propagation effects. Each filtering element reshapes both the signal and the accumulated Amplified Spontaneous Emission (ASE) noise, creating a "filtering penalty" that depends on the sequence of filters, the position of noise sources, and the receiver's digital signal processing (DSP) strategy.

Existing optical network digital twin (DT) models struggle to incorporate these filtering effects in a disaggregated and mathematically consistent manner, especially when accounting for the relative position of ASE noise injection and the specific equalization algorithms used in commercial receivers. Furthermore, while infinite-length equalizer models exist, they often fail to capture the performance penalties associated with the finite tap lengths found in practical DSP implementations.

Methodology
The authors develop a discrete-time modeling framework to jointly analyze the impact of cascaded optical filtering, distributed ASE noise, and transceiver noise on coherent optical links. The methodology proceeds through the following steps:

1. System Abstraction: The optical link is modeled with arbitrary filter transfer functions and colored noise spectra. The model assumes dual-polarization coherent transmission and neglects nonlinear interference (NLI), focusing instead on linear impairments typical of metro-access networks.
2. Transceiver Characterization: A measurement-based model maps experimental back-to-back Bit Error Rate (BER) data to an equivalent transceiver Signal-to-Noise Ratio (SNR). This allows transceiver impairments to be integrated into the same SNR formulation as optical link penalties.
3. Noise Modeling: The framework explicitly treats noise as "colored." ASE noise injected at different points along the link experiences different downstream filtering, resulting in a non-flat noise Power Spectral Density (PSD) at the receiver. Transceiver noise is modeled as a generic colored PSD concentrated at the receiver.
4. Analytical Derivation: Using D-transforms and noise whitening techniques, the authors derive analytical expressions for the post-equalization SNR under five distinct scenarios:
   • Matched Filter Bound (MFB)
   • Zero-Forcing Equalization (ZFE)
   • Minimum Mean Square Error Equalization (MMSE)
   • Fractionally Spaced Equalization (FSE)
   • Finite-Length Equalizers (FLE), with and without explicit colored-noise treatment (Whitened FLE vs. standard FLE).
5. Validation:
   • Time-Domain Simulations: The analytical models are compared against simulations using Least Mean Squares (LMS) equalization under severe filtering conditions, varying tap lengths, and different ASE noise injection positions.
   • Experimental Validation: Measurements using commercial transceivers and ROADMs are used to validate the MMSE and FSE models against realistic filter shapes and noise conditions.

Key Contributions

• Unified SNR Framework: The paper provides a single SNR-based formulation that simultaneously accounts for cascaded filtering, distributed ASE noise, transceiver impairments, and various equalization strategies.
• Disaggregated Penalty Modeling: The derivation allows for the calculation of filtering penalties for individual noise sources (ASE and transceiver), enabling a scalable approach for optical network digital twins.
• Finite-Length Analysis: Unlike many previous works focusing on infinite-length equalizers, this paper derives and validates models for finite-length equalizers (FLE). It demonstrates that finite-length models are necessary to accurately predict the "limited-tap penalty" observed in practical DSP.
• Colored Noise Treatment: The framework explicitly handles the colored nature of noise resulting from cascaded filtering, offering a more rigorous approach than models assuming white noise or single-stage filtering.

Results

• Simulation Validation: Time-domain simulations confirm that infinite-length models (MMSE, FSE) accurately describe asymptotic behavior but fail to predict performance when the number of equalizer taps is low (e.g., 4 or 8 taps). The Finite-Length Equalizer (FLE) model, particularly the version without noise whitening, shows the highest accuracy (maximum absolute error of 0.29 dB with 4 taps) in matching simulation results.
• Tap Length Impact: Increasing the number of LMS taps causes the results of finite-length models to converge with infinite-length models. However, for practical tap counts (e.g., 32 taps), the FLE model remains superior in capturing the specific penalties of limited resources.
• Noise Position: The results highlight that noise injected closer to the receiver suffers a higher filtering penalty because it undergoes less downstream filtering before the equalizer, whereas noise injected earlier passes through more filters, reducing the equalizer's need to enhance noise.
• Experimental Validation: Experimental results with commercial transceivers and ROADM filters show good agreement with the analytical MMSE and FSE models. The models exhibit a slightly conservative margin, attributed to conservative filter modeling and the difference between ideal infinite-length assumptions in the theory and practical DSP limits.
• Model Comparison: The Zero-Forcing Equalizer (ZFE) provides a simple, conservative estimate suitable for cases where filtering is not extreme. The MMSE and FSE models yield identical performance curves in the absence of sampling-phase errors, as expected theoretically.

Significance and Claims
The paper claims that the proposed framework provides a "tractable basis" for quality-of-transmission estimation and the implementation of optical-network digital twins. It argues that filtering penalties cannot be represented merely by a scalar bandwidth loss; they are dependent on filter shape, noise-source position, receiver noise, and equalizer architecture.

By offering a range of models from simple (ZFE) to complex (FLE), the work enables network planners to select the appropriate level of complexity based on their specific application needs. The framework allows for the accurate evaluation of filtering penalties without relying on full time-domain simulations for every lightpath, thereby supporting the development of efficient, impairment-aware system-level planning tools. The authors note that future work should extend experimental validation to characterize the actual DSP tap configurations of commercial transceivers and test a wider variety of filter shapes.