Delay-Doppler Embedding

Encodes a wireless channel as a compact set of discrete tokens, one for each significant propagation path. This sparsity is a natural fit for the self-attention mechanism.

• Token per Path: Each token represents a single scatterer with a specific delay and Doppler shift.
• Efficient Attention: The number of tokens scales with the number of paths, not the number of OFDM subcarriers, making attention computationally tractable.
• Physical Sparsity: Exploits the inherent sparsity of the wireless channel in the delay-Doppler domain, where energy is concentrated in a few peaks.

Unlike a raw IQ sample or a subcarrier symbol, a delay-Doppler embedding directly captures the underlying geometric relationship between transmitter, receiver, and scatterers.

• Range and Velocity: Delay maps directly to a scatterer's distance (range), and Doppler shift maps to its relative velocity.
• Invariant Representation: The embedding is invariant to the arbitrary phase of the carrier, focusing only on the physical path parameters.
• Radar Heritage: This representation is borrowed directly from radar signal processing, where the delay-Doppler matrix is the standard tool for target detection.

A critical distinction exists between classical delay-Doppler maps and learned embeddings used as transformer tokens.

• Classical Map: A 2D grid computed via a matched filter or Fourier transform, often with high sidelobes and limited resolution.
• Learned Embedding: A dense vector produced by a neural network, trained end-to-end to extract features from the delay-Doppler domain that are optimal for a downstream task like detection or channel estimation.
• Super-Resolution: Learned embeddings can implicitly perform super-resolution, distinguishing paths closer than the Fourier limit defined by the time-frequency resource grid.

The set of path tokens is processed by a transformer encoder, allowing the model to learn interactions between scatterers.

• Cross-Path Attention: The model can learn that two paths with similar delays but different Dopplers might originate from the same physical cluster.
• Permutation Invariance: Self-attention over a set of tokens is naturally permutation invariant, meaning the model's output is unaffected by the arbitrary ordering of the detected paths.
• Global Context: Each path token can attend to all other paths, building a holistic understanding of the multipath environment for superior equalization.

Delay-Doppler embeddings provide a stable representation even in high-mobility scenarios where traditional time-frequency methods break down.

• OTFS Foundation: This is the core principle behind Orthogonal Time Frequency Space (OTFS) modulation, where symbols are multiplexed in the delay-Doppler domain.
• Quasi-Static Tokens: While the complex gain of a path may change rapidly, its delay and Doppler shift remain relatively stable over a much longer observation window.
• Predictable Evolution: A transformer can learn to predict the evolution of a path's complex gain from its delay-Doppler coordinates, enabling robust channel aging prediction.

The geometric nature of the embedding makes it a natural pivot point for fusing RF sensing data with other modalities.

• Common Coordinate System: A scatterer's delay-Doppler coordinates can be directly mapped to a point in a 3D spatial coordinate system (with additional angle-of-arrival information).
• Joint Communication and Sensing (JCAS): The same delay-Doppler embedding can be used simultaneously for data demodulation (communication) and for building a radar map of the environment (sensing).
• Sensor Alignment: A transformer can use cross-attention to align delay-Doppler tokens from an RF front-end with point cloud tokens from a LiDAR sensor for robust autonomous navigation.

A discrete, learnable token representing an individual multipath component, characterized by its delay, Doppler shift, and complex gain. This tokenization allows a transformer to process a wireless channel as a set of paths rather than a continuous impulse response.

• Encodes the physical parameters of a single scatterer
• Enables attention mechanisms to model inter-path relationships
• Serves as the direct output of a Delay-Doppler Embedding layer

A transformer-based model designed to process Channel State Information (CSI) matrices directly. It leverages self-attention to capture spatial and frequency correlations for superior channel estimation and feedback compression in massive MIMO systems.

• Replaces traditional compressed sensing with learned attention
• Models correlations across antenna arrays and subcarriers
• Often uses Delay-Doppler Embedding as its input tokenization strategy

An extension of the standard attention mechanism that operates natively on complex numbers, preserving the magnitude and phase relationships inherent in IQ baseband signals.

• Attention weights computed using complex-valued dot products
• Preserves phase information critical for coherent combining
• Essential for processing the complex gains stored in Delay-Doppler tokens

The application of Rotary Position Embedding (RoPE) to RF signal tokens, encoding relative temporal or frequency offsets through rotation in the complex plane.

• Naturally suited for complex-valued signal representations
• Encodes relative delay and Doppler shifts as rotations
• Provides a mathematically elegant way to inject physical channel parameters into the attention mechanism

A preprocessing module that converts a raw time-series signal into a sequence of tokens representing localized time-frequency patches. This enables a standard transformer backbone to process spectral content efficiently.

• Bridges the gap between raw waveforms and transformer inputs
• Delay-Doppler Embedding is a specialized form of time-frequency tokenization
• Captures both temporal evolution and spectral occupancy simultaneously

A fully learned neural receiver architecture that replaces the entire traditional signal processing chain—including channel estimation, equalization, and demapping—with a single end-to-end deep learning model.

• Often uses transformer backbones with Delay-Doppler tokenization
• Learns optimal processing jointly rather than through isolated blocks
• Demonstrates the practical deployment of Delay-Doppler Embedding in real receiver systems