CRFCAN: A Complex-Valued Cross-Domain Residual Network for Joint Channel and Phase Noise Estimation in Sub-THz OFDM Systems

I. INTRODUCTION

• Sub-THz bandwidth and severe phase noise complicate reliable joint estimation, while conventional methods face overhead, computational cost, or limited ability to model coupled time–frequency effects.Existing deep-learning approaches are also described as relying on cascaded or hybrid designs and often overlooking complex phase relationships and physical constraints.Source: §I · pp. 1–2
• CRFCAN combines complex-valued cross-domain processing, multiplicative phase-rotation modeling, and physics-aware constraints for end-to-end joint channel and phase-noise estimation.Its FFT/IFFT-based architecture processes frequency-selective fading and time-varying phase rotations, while a soft-normalization tail constrains the output.Source: §I · p. 2

A. OFDM System Model

• The OFDM system model treats effective phase noise as a multiplicative time-domain impairment whose frequency-domain spreading induces common phase error and inter-carrier interference.The received block follows receiver filtering, sampling, downsampling, and cyclic-prefix removal.Source: A. OFDM System Model · p. 3

B. Sub-THz Channel Model

• The SISO TDL-C model represents non-line-of-sight propagation with moderate delay spreads and is specified for carrier frequencies from 0.5 to 100 GHz.The tap delays are scaled by the target RMS delay spread to represent deployment scenarios.Source: B. Sub-THz Channel Model · p. 3
• Each tap has a zero-mean, wide-sense stationary narrowband complex Gaussian gain corresponding to Rayleigh fading, with Doppler variation following the Jakes spectrum.The maximum Doppler frequency is given by fd = vfc/c.Source: B. Sub-THz Channel Model · p. 3
• The normalized tap delays use an exponential power delay profile, and the discrete-time channel incorporates multipath, Doppler spread, and pulse-shaping-induced inter-symbol interference.The discrete-time channel is obtained by combining the physical channel with the transmit and receive pulse-shaping response and sampling.Source: B. Sub-THz Channel Model · p. 3

C. Phase Noise Model

• An implementation-specific calibration factor ∆cal adjusts the generalized 3GPP model toward practical hardware characteristics, including PLL noise suppression and front-end thermal noise.The model relates single-sideband phase-noise PSD L(f) to linear-scale PSD Sϕ(f).Source: C. Phase Noise Model · p. 4
• The PLL-based oscillator phase-noise spectrum is modeled as a transfer function with fractional products describing oscillator noise and PLL tracking behavior.Its zeros and poles represent compensation mechanisms and oscillator or loop-filter bandwidth limitations, respectively.Source: C. Phase Noise Model · p. 4
• At carrier frequencies other than the 29.55 GHz reference, the model scales the white noise floor to account for frequency multiplication and oscillator figure-of-merit degradation.The model also preserves the standardized multi-segment slope characteristics while reflecting increased phase instability at higher carrier frequencies.Source: C. Phase Noise Model · p. 4

A. Network Architecture

• CRFCAN uses a complex-convolution front end, alternating IFFT and FFT residual groups, and two tail modules to estimate channel response and phase noise jointly.Its input pairs the received signal with a pilot grid whose data positions are zero-filled; the comb-type pilots shift cyclically across consecutive OFDM symbols.Source: A. Network Architecture · p. 5

1) Complex-Valued Neural Network:

• Complex-valued processing preserves phase relationships across time and frequency, which CRFCAN's FFT/IFFT transitions and multiplicative phase-rotation modules rely on.Its complex-valued formulation represents features as z = p + jq, unlike real-valued networks that process in-phase and quadrature components as independent channels.Source: 1) Complex-Valued Neural Network · p. 5

2) Cross-Domain Signal Flow:

• Sub-THz phase noise is temporally correlated, while the channel has structured frequency-selective fading; in frequency, multiplicative phase noise induces inter-carrier interference.These distinct signal properties motivate joint time- and frequency-domain reasoning for CSI–PN estimation.Source: 2) Cross-Domain Signal Flow · p. 5
• CRFCAN alternates frequency-domain refinement followed by IFFT projection with time-domain refinement followed by FFT mapping back to frequency.Each cross-domain stage learns a complex-valued residual mapping with a learnable scaling coefficient, retaining identity connections and focusing on incremental refinement.Source: 2) Cross-Domain Signal Flow · pp. 5–6
• At runtime, CRFCAN uses single-shot, non-iterative cross-domain processing with fixed, predictable computational complexity.The authors identify this property as attractive for low-latency processing in wideband sub-THz receivers.Source: 2) Cross-Domain Signal Flow · p. 6

B. FFT-Powered Cross-Domain Residual Groups

• FFT-powered residual groups iteratively refine joint CSI and PN estimates by alternating feature representations between the time and frequency domains.Each group applies an intra-group nonlinear transformation and a domain-specific FFT or IFFT operator, with a group-level skip connection for incremental refinement.Source: B. FFT-Powered Cross-Domain Residual… · p. 6, 1) Motivation: Limitations of… · p. 6

1) Motivation: Limitations of Depth-Only Designs:

• Depth-only scaling can yield diminishing returns and training instability in structured estimation tasks, despite the greater representation power of deeper residual stacks.The paper motivates residual groups as a means of stable deep refinement through group-level residual learning.Source: 1) Motivation: Limitations of… · p. 6

2) Intra-Group Processing: Convolutional Blocks and Temporal Modeling:

• Each residual group uses CRCAB for complex-valued feature refinement and PRCAB for learned I/Q coupling and phase rotation.CRCAB applies channel attention; PRCAB applies a learned complex-valued phase rotation.Source: 2) Intra-Group Processing:… · p. 6

3) Asymmetric Block Configuration for IFFT and FFT Residual Groups:

• The IFFT-RG applies five successive CRCABs to refine frequency-domain channel features before projecting them into the time domain.Its linear-heavy layout targets frequency selectivity and inter-subcarrier correlation.Source: 3) Asymmetric Block Configuration for… · p. 6
• The FFT-RG places one PRCAB between four CRCABs to model time-domain amplitude variations and phase rotation before FFT projection.The surrounding CRCABs capture amplitude variations and approximately linear correlations; the central PRCAB adds a feature-dependent phase-rotation offset.Source: 3) Asymmetric Block Configuration for… · p. 6
• FFT-powered residual groups combine complex-valued convolution for local correlation with domain mapping for physical consistency.Source: 3) Asymmetric Block Configuration for… · p. 6

1) CRCAB:

• CRCAB applies two complex convolutions and channel-wise attention to intermediate features before residual addition.Its attention uses a bottleneck gating network to generate channel weights.Source: 1) CRCAB · p. 7

• PRCAB models phase noise as multiplicative rotation rather than additive feature perturbation.Source: 2) PRCAB · p. 7
• PRCAB concatenates real and imaginary input components for phase-aware feature learning and regression.The real-valued processing pipeline uses convolutional layers and channel attention.Source: 2) PRCAB · p. 7
• A lightweight convolutional projection estimates a phase feature that is mapped to a unit-modulus complex phasor.The learnable parameter α controls the maximum allowable phase deviation.Source: 2) PRCAB · p. 7
• PRCAB applies the phase-rotated residual through a learnable gate η that controls the phase-rotation update step size.The residual formulation is used to support stable training.Source: 2) PRCAB · pp. 7–8

• The channel tail projects fused frequency-domain features to a CSI estimate using complex-valued convolutional kernels.The kernels progressively reduce the channel dimension from C to 1.Source: D. Physically Constrained Dual Output · p. 8
• The PN branch estimates the complex multiplicative impairment coefficient from fused time-domain features, rather than estimating phase angle directly.Its tail shares the channel tail's architectural form but is dedicated to PN estimation.Source: D. Physically Constrained Dual Output · p. 8
• A soft magnitude constraint accommodates bounded amplitude fluctuations in the effective PN coefficient instead of enforcing unit magnitude.The nominal magnitude and allowable variation are controlled by μ and σ, respectively.Source: D. Physically Constrained Dual Output · p. 8
• Softly constraining PN magnitude reduces the PN branch's effective degrees of freedom and helps stabilize joint optimization.The output-stage constraint is intended to preserve phase information while gently regularizing magnitude.Source: D. Physically Constrained Dual Output · p. 8

A. Simulation Setup

• The network receives a complex tensor containing the processed received signal and pilot grid, and jointly outputs channel response and effective multiplicative phase noise.Source: A. Simulation Setup · p. 8
• The simulation assumes a quasi-static channel across consecutive OFDM symbols and applies continuous-time phase noise before receiver processing.For the 100 GHz setting, three PN models are calibrated using Δ_cal = −10 dB with reference to measured hardware noise levels.Source: A. Simulation Setup · pp. 8–9

B. Training Methodology

• Training samples cover Eb/N0 of 20–30 dB, while independently generated testing samples cover 0–30 dB.Training uses 30,000 samples per point and a 70%/30% training-validation split; testing uses 5,000 samples per point.Source: B. Training Methodology · p. 9
• The end-to-end objective jointly applies complex-valued mean squared error to channel and phase-noise estimates, with uncertainty-based weighting to balance the tasks.The weighting uses learnable task-dependent log-variance parameters and avoids manual loss-weight tuning.Source: B. Training Methodology · p. 9
• Training uses cyclic learning-rate warm-up and cosine annealing, progressively decaying maximum and minimum rates across cycles.The schedule is described as supporting coarse-to-fine optimization and stabilizing joint learning.Source: B. Training Methodology · p. 9

1) Iterative LS-Based Estimator:

• The iterative LS benchmark alternates channel and PN estimation, using limited PN samples with DFT-based low-pass interpolation and a truncated channel impulse response.The alternating least-squares refinement requires iterative optimization and has non-negligible computational complexity; simplified interpolation may struggle with highly frequency-selective channels and rapidly varying PN.Source: 1) Iterative LS-Based Estimator · pp. 9–10

2) Cascaded Multi-Network Baseline:

• The cascaded benchmark stages channel refinement, payload-symbol PN estimation and refinement, then data denoising across three neural networks.Conventional estimators provide initial inputs, with the networks acting as nonlinear refiners.Source: 2) Cascaded Multi-Network Baseline · p. 10

3) End-to-End Neural Transceiver:

• DeepSRX jointly mitigates channel and PN effects from received blocks and pilot-derived side information, directly outputting soft bit estimates without explicit channel or PN estimates.It uses a fully convolutional residual architecture.Source: 3) End-to-End Neural Transceiver · p. 10
• CRFCAN and DeepSRX use comb-type pilots, while the iterative and cascaded benchmarks additionally require block-type pilots.All methods share channel realizations, PN models, modulation formats, and Eb/N0 settings; CRFCAN uses lower overall pilot density than the conventional and cascaded benchmarks.Source: 3) End-to-End Neural Transceiver · p. 10

D. Overall Performance Evaluation

• CRFCAN's channel-estimation NMSE and PN-estimation MSE decrease together as Eb/N0 rises across RAN4 and unseen RAN1 Set1 and Set2, without catastrophic degradation or fine-tuning.It is trained only on RAN4; the authors attribute cross-model generalization to FFT/IFFT cross-domain refinement and soft normalization of the estimated PN magnitude.Source: D. Overall Performance Evaluation · p. 10, Figure 7 · p. 11
• CRFCAN achieves the lowest BER among practical schemes under all three PN models, lowers the high-Eb/N0 BER floor, and can outperform Perfect H only at high Eb/N0.It also improves on the cascaded three-network baseline; DeepSRX degrades in the joint CSI–PN setting.Source: D. Overall Performance Evaluation · pp. 10–11, Figure 8 · p. 11

E. Robustness Analysis

• For 4QAM and 16QAM under three PN models, EVM decreases as Eb/N0 rises, while 16QAM has a higher medium-to-high-Eb/N0 floor than 4QAM.The paper relates the higher 16QAM floor to sensitivity to residual PN-induced CPE/ICI and identifies residual PN/ICI as the primary limit once thermal noise diminishes.Source: Figure 9 · p. 11, E. Robustness Analysis · p. 11

F. Ablation Study

• Removing FFT/IFFT transitions causes the largest high-Eb/N0 BER degradation; replacing PRCAB with CRCAB also increases BER, while removing the PN tail causes a smaller decline.The FFT/IFFT-removed variant approaches the combined-ablation error floor; the PN tail is described as a useful regularizer rather than a dominant performance driver.Source: F. Ablation Study · p. 12, Figure 10 · p. 12
• Replacing uncertainty-based loss weighting with equal-weight summation of channel and PN CMSE losses degrades BER performance.The ablation attributes a benefit to adaptive loss balancing under heterogeneous task difficulties.Source: F. Ablation Study · p. 12
• Only full CRFCAN and the no-PN-tail variant achieve BER below Perfect H only at high Eb/N0; removing all three components fails to surpass that bound.The reported ablation configuration uses three pairs of residual groups with five residual blocks per group, selected as a practical performance–complexity balance.Source: F. Ablation Study · pp. 11–12

G. Computational Complexity

• With G=3, B=5, C=64, K=3, Nc=64, and Nt=8, the network contains approximately 4.7 × 10^6 real-valued parameters.The computational cost is dominated by complex-valued convolutions within residual groups.Source: G. Computational Complexity · p. 12

V. CONCLUSION

• The paper concludes that CRFCAN provides an effective practical receiver solution, with single-shot inference, cross-model generalization, and end-to-end training simplicity.Its simulations under three PN models report lower channel NMSE and PN MSE, improved BER and EVM, and a reduced high-Eb/N0 error floor.Source: §V · p. 12