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

Anatolij Zubow, Joana Angjo, Sigrid Dimce, Falko Dressler

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.

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 snapshots and frequency bins and uses a factored self-attention mechanism that separately attends along both axes, reducing the computational complexity to . 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 compared to 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.

Paper Structure

This paper contains 30 sections, 13 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Multi-Band Wideband Channel
  4. DTMC Interference Model
  5. CFRTransformer
  6. Input Features and ComplexLinear Embedding
  7. Frequency Positional Encoding
  8. Factored Attention Blocks
  9. Frequency attention
  10. Time attention
  11. Physics-Informed Training Loss
  12. Spectral fidelity
  13. PDP fidelity
  14. CIR sparsity
  15. Temporal smoothness
  16. ...and 15 more sections

Figures (5)

  • Figure 1: CFR magnitude (top) and PDP (bottom) for a single representative trace at 30% interference occupancy. From left to right: clean full-band reference $H_{\mathrm{full}}$; masked observation $H_{\mathrm{masked}}$ (equivalent to zero-fill); CFRTransformer reconstruction $\hat{H}_{\mathrm{Transformer}}$; historical-fill reconstruction $\hat{H}_{\mathrm{Historical}}$. Dark horizontal bands in the masked CFR indicate blocked sub-bands.
  • Figure 2: Mean PDP similarity factor $\rho$ vs. interference probability $\pi_\mathrm{busy}$.
  • Figure 3: Mean PDP similarity factor $\rho$ vs. UE velocity at fixed $\pi_\mathrm{busy}{=}0.5$. The secondary axis shows maximum Doppler shift $f_d$ [Hz].
  • Figure 4: Mean PDP similarity factor $\rho$ vs. UE velocity evaluated at three channel complexities: $P \in \{2, 6, 10\}$ multipath components ($\pi_{\mathrm{busy}}=0.5$).
  • Figure 5: Mean PDP similarity factor $\rho$ for CFRTransformer as a function of (a) interference occupancy $\pi_{\mathrm{busy}}$ at $v=7$ m/s, and (b) UE velocity at $\pi_{\mathrm{busy}}=0.5$, for $N_b \in \{3, 5, 7, 9\}$ sub-bands.