Propagation Path Token

Each propagation path token encodes a unique multipath component characterized by three fundamental physical parameters:

• Delay (τ): The time offset of the path's arrival relative to the line-of-sight, representing the physical distance traveled
• Doppler shift (ν): The frequency offset caused by relative motion between transmitter, receiver, or scatterers
• Complex gain (α): A complex-valued coefficient capturing both amplitude attenuation and phase rotation

This parameterization maps directly to the delay-Doppler domain representation of the channel, where the wireless environment is expressed as a sparse set of resolvable paths rather than a dense impulse response.

Raw physical parameters are projected into a high-dimensional learnable embedding space before being processed by the transformer:

• Delay and Doppler values are encoded using sinusoidal or learned positional encodings that preserve their continuous, real-valued nature
• The complex gain is decomposed into magnitude and phase components or processed using complex-valued linear projections
• The resulting token vector fuses all three parameters into a unified representation that the attention mechanism can attend to

This embedding step is critical because it allows the model to learn semantic relationships between paths—for example, recognizing that two paths with similar delays but different Doppler shifts likely originate from the same physical cluster.

Unlike traditional sequential processing of channel impulse responses, propagation path tokens treat the channel as an unordered set of paths:

• The transformer's self-attention mechanism is permutation-equivariant, meaning the order of tokens does not affect the learned relationships
• This naturally accommodates channels with varying numbers of multipath components—a sparse rural channel may have 3 tokens while a dense urban channel has 20
• Padding tokens or learned aggregation strategies handle variable-length token sequences within a batch

The set-based approach mirrors the physical reality that multipath components arrive from arbitrary directions with no inherent ordering, making it more physically consistent than grid-based representations.

The transformer's self-attention mechanism computes pairwise interactions between all propagation path tokens, enabling the model to learn complex inter-path relationships:

• Path clustering: The model can group tokens belonging to the same physical scattering cluster based on correlated parameters
• Interference modeling: Attention weights can capture how one strong path may mask or distort a weaker path arriving at a similar delay
• Global context: Each token is updated based on the full set of paths, allowing the model to reason about the entire multipath structure when predicting channel characteristics

These learned interactions replace hand-crafted clustering algorithms like K-means or Gaussian mixture models traditionally used for multipath component tracking.

Propagation path tokens serve as an intermediate representation between raw channel estimates and downstream tasks:

• Input: A conventional channel estimator (e.g., least squares or minimum mean square error) produces an initial impulse response or delay-Doppler profile
• Tokenization: A peak detection or super-resolution algorithm extracts dominant paths and converts them into token embeddings
• Transformer processing: The token sequence is processed by a transformer encoder to refine estimates, predict missing paths, or compress the representation
• Output: The processed tokens can be decoded into an enhanced channel estimate, used directly for beamforming weight prediction, or compressed for CSI feedback

This modular design allows the learned token representation to augment rather than replace existing signal processing blocks.

To ensure the learned token representations remain physically meaningful, several constraints and inductive biases are incorporated:

• Delay positivity: Output projections for delay values are passed through a softplus or exponential activation to enforce τ > 0
• Doppler boundedness: Doppler shifts are constrained to [-f_max, f_max] based on the maximum expected velocity in the deployment scenario
• Sparsity regularization: Loss functions include L1 penalties on token activations to encourage the model to use only as many tokens as physically necessary
• Reciprocity awareness: For time-division duplex systems, the model can be trained to recognize that uplink and downlink paths share the same physical delays

These constraints prevent the model from learning non-physical representations that would fail to generalize to real-world deployments.

A learned vector representation that encodes the delay and Doppler shift characteristics of a propagation path. This embedding serves as the positional encoding for propagation path tokens, allowing the transformer to understand a path's location in the delay-Doppler domain rather than relying on a sequential order.

• Encodes continuous delay and Doppler values into a fixed-dimensional vector
• Often implemented using sinusoidal functions or learned lookup tables
• Enables the model to reason about relative path distances and velocity-induced shifts

A mechanism that uses cross-attention to fuse information from two distinct signal representations. In the context of propagation path tokens, cross-attention can fuse a path-based channel representation with a frequency-domain spectrum representation, allowing the model to jointly reason about multipath structure and spectral occupancy.

• Query vectors from one modality attend to key/value vectors from another
• Enables multi-modal physical layer processing within a single architecture
• Used for tasks like joint channel estimation and interference classification

An extension of the standard attention mechanism that operates natively on complex numbers, preserving the magnitude and phase relationships inherent in wireless channel responses. When propagation path tokens carry complex-valued gains, complex-valued attention ensures the model respects the algebraic structure of the physical layer.

• Attention scores computed using complex inner products
• Preserves phase coherence across propagation paths
• Critical for tasks like beamforming and coherent combining

A transformer model that performs channel estimation by processing received pilot signals. Instead of treating the channel as a grid of time-frequency responses, this architecture can operate on a set of propagation path tokens, using self-attention to refine estimates of each path's delay, Doppler, and complex gain.

• Self-attention captures inter-path correlations for improved estimation
• Naturally handles sparse channel representations common in mmWave systems
• Outperforms conventional MMSE estimators in high-mobility scenarios

A transformer network that predicts optimal beamforming weights directly from channel state information. When fed with propagation path tokens, the model can learn to steer beams toward dominant paths and place nulls toward interferers by attending to the spatial signatures encoded in each token.

• Maps a set of path tokens to a complex weight vector per antenna
• Self-attention identifies dominant propagation clusters
• Replaces iterative optimization with a single forward pass

The application of Rotary Position Embedding (RoPE) to RF signal tokens, encoding relative temporal or frequency offsets through rotation in the complex plane. This is particularly well-suited for propagation path tokens, as the relative delay and Doppler between two paths can be naturally expressed as a complex phase rotation.

• Encodes relative offsets rather than absolute positions
• Mathematically consistent with the physics of wave propagation
• Enables the model to generalize to unseen path configurations