Physics-Informed Transformer for Multi-Band Channel Frequency Response Reconstruction

Abstract: Wideband channel frequency response (CFR) estimation is challenging in multi-band wireless systems, especially when one or more sub-bands are temporarily blocked by co-channel interference. We present a physics-informed complex Transformer that reconstructs the full wideband CFR from such fragmented, partially observed spectrum snapshots. The interference pattern in each sub-band is modeled as an independent two-state discrete-time Markov chain, capturing realistic bursty occupancy behavior. Our model operates on the joint time-frequency grid of $T$ snapshots and $F$ frequency bins and uses a factored self-attention mechanism that separately attends along both axes, reducing the computational complexity to $O(TF^2 + FT^2)$. Complex-valued inputs and outputs are processed through a holomorphic linear layer that preserves phase relationships. Training uses a composite physics-informed loss combining spectral fidelity, power delay profile (PDP) reconstruction, channel impulse response (CIR) sparsity, and temporal smoothness. Mobility effects are incorporated through per-sample velocity randomization, enabling generalization across different mobility regimes. Evaluation against three classical baselines, namely, last-observation-carry-forward, zero-fill, and cubic-spline interpolation, shows that our approach achieves the highest PDP similarity with respect to the ground truth, reaching $ρ\geq 0.82$ compared to $ρ\geq 0.62$ for the best baseline at interference occupancy levels up to 50%. Furthermore, the model degrades smoothly across the full velocity range, consistently outperforming all other baselines.

• The paper introduces CFRTransformer, a physics-informed complex Transformer architecture that accurately reconstructs missing multi-band CFR data using holomorphic processing.
• It leverages factored self-attention and composite loss functions to ensure amplitude, phase, and temporal fidelity even under high interference and mobility.
• Empirical evaluations demonstrate CFRTransformer's superior performance over traditional methods, achieving PDP similarity factors up to 0.87 in diverse scenarios.

Physics-Informed Transformer for Multi-Band Channel Frequency Response Reconstruction

Introduction

The paper presents CFRTransformer, a physics-informed complex-valued Transformer architecture for reconstructing wideband channel frequency responses (CFR) from time–frequency grids with missing fragments due to realistic, bursty interference. The model targets practical applications in integrated wireless sensing and communication systems, such as passive localization, user detection, and physical-layer sensing in 5G NR and IEEE 802.11bf, where full-band CFR is unavailable because of co-channel interference. Unlike conventional gap-filling or CNN-based approaches, the design emphasizes physical constraints, complex-valued processing, and efficient 2D sequence modeling, incorporating domain knowledge through a composite training objective.

Problem Formulation and Modeling Assumptions

The multi-band CFR is observed across $N_b$ non-overlapping sub-bands, yielding $F = N_b F_b$ frequency bins, with observations captured over $T$ snapshots. Physical effects—multipath, Doppler shifts, and nonstationary scatterer behavior—are modeled via a sum-of-paths CIR, with Doppler incorporated as randomized per-sample velocity and phase increments. Co-channel interference is staged as a two-state Discrete-Time Markov Chain (DTMC) binary mask, realistic for dynamic spectrum sharing and occupancy, affecting entire sub-bands over contiguous time windows.

CFRTransformer Architecture

Masked Channel and Input Representation

Inputs to the model are three per-grid features: real and imaginary parts of the masked CFR and the interference mask.

Figure 1: The masked channel, which is the input tensor to the CFRTransformer.

These are embedded via a holomorphic ComplexLinear layer, applying weight tying that enforces the Cauchy-Riemann conditions to guarantee preservation of amplitude–phase relationships critical for accurate path profiling in delay and Doppler domains. This approach avoids the breakdowns seen when treating real–imaginary parts independently.

Frequency Positional Encoding

Absolute spectral location is provided via sinusoidal encoding along the frequency axis, ensuring that each bin's spatial context is unambiguous and that the model is not invariant to band permutation—crucial in multi-band, fragmented spectra.

Factored Self-Attention

The architecture uses a two-pass, axis-wise (factored) self-attention block rather than a fully flattened $TF$-length attention, drastically reducing computational complexity from $O((TF)^2)$ to $O(TF^2 + FT^2)$:

• Frequency attention: Models spectral correlations within a time slice, enabling inference of missing bands from their neighbors.
• Time attention: Learns Doppler-coherent channel evolution and captures multi-snapshot interference patterns.

A position-wise feed-forward network with residuals and normalization bridges each block, and the output head is another holomorphic linear projection to the complex domain.

Physics-Informed Training and Loss Design

The composite loss is central to enforcing physical plausibility:

• Spectral fidelity ($\mathcal{L}_\mathrm{CFR}$): Enforces amplitude and phase accuracy between estimates and ground truth.
• PDP fidelity ($\mathcal{L}_\mathrm{PDP}$): Minimizes the MSE over the power delay profile—essential for temporal resolution, evaluated via a normalized similarity metric.
• CIR sparsity ($\mathcal{L}_\mathrm{sparse}$): Encourages reconstructions with realistic (physically motivated) sparse path structures.
• Temporal smoothness ($\mathcal{L}_\mathrm{temp}$): Penalizes nonphysical, abrupt fluctuations in CFR, parameterized by a velocity-dependent smoothness cost.

Randomization of velocity per training instance ensures coverage of multipath-Doppler regimes from pedestrian to vehicular, preventing domain overfitting and fostering robustness to mobility.

Evaluation and Results

Qualitative Analysis

CFRTransformer reconstructs both CFR amplitudes and PDPs with high fidelity, sharply localizing delay taps and minimizing spurious power smear compared to historical, zero-fill, or spline interpolation strategies.

Figure 2: CFR magnitude (top) and PDP (bottom) for a single representative trace at 30% interference occupancy; CFRTransformer preserves both major delay components and amplitude structure relative to heuristic reconstructions.

PDP Similarity Across Interference Occupancy

The model sustains a mean PDP similarity factor $F = N_b F_b$0 for interference probabilities up to 0.5, markedly exceeding the best baseline (historical fill at $F = N_b F_b$1 and spline at $F = N_b F_b$2) under realistic spectrum fragmentation.

Figure 3: Mean PDP similarity factor $F = N_b F_b$3 vs. interference probability $F = N_b F_b$4; CFRTransformer exhibits superior robustness as spectrum occupancy increases.

Mobility Robustness

Dynamic evaluation over a 60× velocity range verifies that performance degrades smoothly from $F = N_b F_b$5 (quasi-static) to $F = N_b F_b$6 (vehicular) with CFRTransformer, while baselines lose coherence or do not exploit temporal redundancy.

Figure 4: Mean PDP similarity factor $F = N_b F_b$7 vs. UE velocity at three levels of channel complexity; CFRTransformer maintains a gradual and limited performance drop, even as Doppler spread and multipath richness increase.

Sub-band Count and Spectral Diversity

Performance increases with the number of sub-bands, validating that greater spectral observations supply richer context for cross-band inference. Models trained with more sub-bands consistently achieve higher $F = N_b F_b$8.

Figure 5: Mean PDP similarity factor $F = N_b F_b$9 for CFRTransformer as a function of (a) interference occupancy and (b) UE velocity across different numbers of sub-bands, highlighting the benefit of increased spectral diversity.

Effect of Training Regimes

Ablation experiments show that training with per-sample velocity randomization is essential for maintaining performance across mobility regimes and precludes the domain shift issues observed with fixed-velocity models.

Theoretical and Practical Implications

From a practical standpoint, CFRTransformer provides a deployable tool for wideband wireless systems confronted with unpredictable band occupancy, enhancing the reliability of downstream physical-layer sensing (e.g., localization, Doppler radar, or gesture recognition). Theoretically, the approach sets a precedent in combining holomorphic neural designs, physics-consistent multi-objective loss functions, and efficient axis-wise attention to balance scalability and physical expressiveness. The design is adaptable to arbitrary numbers of bands, time windows, or mobility levels—critical for next-generation communication systems deploying in heterogeneous, highly dynamic environments.

The physics-informed composite loss and factored architecture present a unified recipe for addressing other multidimensional reconstruction tasks in applied signal processing where both computational and physical constraints are stringent.

Conclusion

CFRTransformer establishes a new standard for wideband CFR reconstruction in multi-band, interference-limited environments, surpassing classical interpolation or extrapolation baselines by leveraging complex-valued modeling, factored attention, and physics-informed learning objectives. Its robustness to co-channel interference and mobility variations is validated empirically and supported by ablation analysis. Future work may extend these results to real-field wireless data and further reduce inference latency, opening avenues for robust, real-time, spectrum-adaptive physical layer intelligence.