Delay-Doppler Domain CSI

The Delay-Doppler domain is accessed via the Zak transform, a unitary transformation that maps time-domain signals onto a two-dimensional lattice. Unlike the time-frequency domain, which spreads a high-mobility channel's energy across many symbols, the Zak transform compacts the channel into a sparse, quasi-static grid. This sparsity is the key to efficient estimation and prediction, as the entire multipath profile is captured by a small set of delay and Doppler shift coordinates.

A defining characteristic of the Delay-Doppler domain is the inherent sparsity of the channel representation. Physical propagation paths naturally resolve into a limited number of distinct points, each defined by a specific delay tap and Doppler shift. This separability means that complex, time-varying multipath in the time-frequency domain becomes a static, sparse interaction pattern in Delay-Doppler. This property drastically reduces the number of parameters a predictive model must track, enabling highly efficient neural network architectures for CSI forecasting.

Delay-Doppler domain CSI is the native channel representation for Orthogonal Time Frequency Space (OTFS) modulation. In OTFS, data symbols are multiplexed directly on the Delay-Doppler grid. This means the channel acts as a simple 2D convolution with a sparse kernel. Predicting CSI in this domain is therefore equivalent to predicting the convolution kernel itself, allowing the receiver to perform a single, uniform equalization step that is robust to the Doppler-induced inter-carrier interference that plagues OFDM systems in high-mobility scenarios.

While the time-frequency domain experiences rapid, complex fading, the Delay-Doppler representation evolves much more slowly. The physical geometry of scatterers (delay and Doppler shift) changes on a timescale far longer than the phase of a subcarrier. This quasi-static nature makes the domain ideal for predictive algorithms. A model can forecast the sparse Delay-Doppler CSI for hundreds of future time slots with high accuracy, as it only needs to track the birth, death, and slow drift of physical paths rather than rapid phase rotations.

Each point in the Delay-Doppler grid corresponds directly to a physical scatterer or reflector in the environment. The coordinates explicitly represent a path's round-trip distance (delay) and relative velocity (Doppler). This direct mapping provides unparalleled interpretability for AI models. A predicted change in the CSI is not an abstract matrix update but a physically meaningful event, such as a new reflector appearing or an existing one accelerating, allowing engineers to validate model outputs against physical laws.

The primary advantage of operating in the Delay-Doppler domain is inherent resilience to Doppler spread. In conventional OFDM, high velocity causes subcarrier orthogonality to break down. By predicting CSI in the Delay-Doppler domain, the system explicitly models and resolves Doppler shifts rather than treating them as destructive interference. This allows for reliable communication and accurate channel prediction even in extreme mobility environments like high-speed rail, low-earth orbit satellites, and supersonic aviation, where time-frequency domain methods fail.

Channel aging occurs when CSI becomes stale between estimation and transmission due to node mobility. Delay-Doppler domain CSI inherently mitigates this because the channel representation is stable over a much longer duration compared to time-frequency domain CSI. The Doppler spread is resolved as a discrete parameter, allowing predictive algorithms to forecast channel evolution over multiple transmission time intervals without requiring millisecond-level updates.

Wireless channels in the Delay-Doppler domain exhibit inherent sparsity because physical propagation paths correspond to a small number of discrete delay and Doppler shift pairs. This sparsity enables:

• Compressed sensing techniques for pilot reduction
• Low-rank matrix completion for CSI recovery
• Efficient neural network architectures that exploit the sparse structure for prediction tasks

The Symplectic Finite Fourier Transform (SFFT) is the discrete implementation used in practical OTFS systems to convert between the time-frequency and Delay-Doppler domains. It consists of a pair of 2D Fourier transforms: one along the time axis and one along the frequency axis. This transform preserves the symplectic structure of the channel, ensuring that the input-output relationship remains a straightforward 2D convolution rather than a complex time-varying operation.