Time-Series Channel Charting

• Time-series channel charting is a method that reconstructs low-dimensional radio maps by preserving both temporal evolution and spatial geometry.
• It utilizes temporal dissimilarity metrics and deep learning architectures to capture dynamic channel variations and enforce embedding smoothness.
• The approach enhances applications like localization, mobility analysis, and resource scheduling in complex, non-stationary wireless environments.

Time-series channel charting comprises dimensionality-reduction methodologies that reconstruct spatio-temporal radio maps from temporally indexed Channel State Information (CSI). Unlike purely geometric CC approaches, time-series charting explicitly exploits the temporal evolution of wireless channels to enforce embedding smoothness, topological stability, and global geometry recovery, crucial for localization, mobility analysis, and resource scheduling in dynamic wireless environments.

1. Theoretical Foundations and Motivation

Channel charting (CC) seeks a mapping $$\mathcal{C}_\theta:\mathbb{C}^{B\times N_{\rm sub}} \to \mathbb{R}^2$$ that projects high-dimensional, time-varying CSI sequences into low-dimensional coordinates, $\hat{\mathbf{x}}^{(l)}$, while preserving spatial geometry and temporal structure. Time-series channel charting expands beyond static manifold learning by integrating temporal dependencies, either through explicit time-indexed dissimilarity metrics or deep sequential models, thereby regularizing the chart topology across evolving radio environments (Gao et al., 26 Jan 2026, Stephan et al., 2023).

The motivation stems from shortcomings of static CC: embedding instability, inability to capture time-varying multipath effects, and discontinuities in user trajectories. Empirical evidence demonstrates that for mobile scenarios—industrial halls, robot navigation, 6G networks—purely static metrics (e.g., Isomap on CSI or timestamp-only triplets) lead to inferior localization accuracy, chart warping, and drift under non-stationary channel conditions (Gao et al., 26 Jan 2026, Stahlke et al., 2022).

2. Temporal Dissimilarity Metrics

Time-series charting utilizes several metrics that encode temporal information:

• Timestamp-based losses: The temporal separation $d_{\rm time}(i,j)=|t^{(i)}-t^{(j)}|$ directly penalizes chart points for temporal distance. When fused with CSI-derived metrics, charting improves local continuity and reconstructs complex (Ω-shaped) trajectories beyond the reach of CSI-only embeddings (Stephan et al., 2023, Ferrand et al., 2020).
• Angle-delay profile (ADP) fusion: By measuring angular-domain cosine dissimilarity over antennas/taps and fusing with rescaled timestamp differences ($\min\{d_{\rm ADP}, \gamma d_{\rm time}\}$), temporal and spatial neighborhood graphs enhance global topology recovery—even under NLoS conditions (Stephan et al., 2023).
• Doppler-based dissimilarity: Temporal phase-difference accumulation through the Doppler effect, $\Delta\varphi_{b_1,b_2}^{(l_1,l_2)}$, associates frequency changes to spatial movement, requiring only frequency (not phase/time) synchronization. This metric instantiates a self-supervised loss that is expressed directly in global distances and obviates post-hoc affine correction (Euchner et al., 2024).
• Time-distance (TDOA/ToF) metrics: In time-synchronized, high-bandwidth SISO setups, delay differences between CIRs correlate linearly with Euclidean movement. Geodesic distance approximation through nearest-neighbor graphs supports robust chart construction and indoor localization (Stahlke et al., 2022).

3. Deep Learning Architectures for Temporal Charting

Several architectures have been proposed for time-series channel charting:

• LSTM-AE-channel charting: Sequences of $(T \times D)$ CSI samples $X_t$ are fed into an LSTM encoder, whose final hidden state $h_T$ becomes the source for a latent embedding $z_t$ and subsequent 2D chart coordinates $y_t$. Decoder LSTM reconstructs the input sequence from $y_t$. The overall loss couples a reconstruction term and a topology-preserving distance alignment, with gradient balancing to avoid domination of either geometric or temporal terms (Gao et al., 26 Jan 2026).

  Component	Function	Output
  LSTM Encoder	Captures temporal dependencies in CSI	$h_T$ (sequence state)
  Dense, Chart Layer	Projects $h_T$ to latent space and chart coords	$z_t$, $y_t \in \mathbb{R}^2$
  LSTM Decoder	Reconstructs CSI sequence from $y_t$	$\hat{X}_t$

• Siamese networks for geodesic alignment: Two identical parametric branches embed CSI pairs. The loss matches the charted Euclidean distance to a precomputed geodesic graph (constructed via local time-distance metrics), producing globally consistent charts without explicit position labels or costly ground truth (Stahlke et al., 2022, Euchner et al., 2024).

• Triplet-based DNNs and curriculum learning: Anchor-positive-negative triplets are sampled based on temporal proximity, enforcing that temporally close CSI points remain close in the chart, while distant points are separated. Both margin-hinge and softmax losses have been used, as well as time-series meta-samples (stacked sequence inputs) (Ferrand et al., 2020).
• Deep timestamp estimators: DNN $\mathcal{D}_\Theta$ predicts timestamp difference from paired CSI features. This approach combines temporal prediction with classical manifold embedding, facilitating charting in scenarios where timestamped data mediates spatial-temporal relationships (Stephan et al., 2023).

4. Optimization, Training, and Practical Constraints

Common practices in time-series channel charting involve large-scale sampling:

• Data formation: Sliding windows of CSI sequences, real/imag/log-magnitude features, inverse DFT for CIR tap extraction, and per-sample CFO/phase correction (Euchner et al., 2024, Gao et al., 26 Jan 2026).
• SGD-based training: Large mini-batches (e.g., $30\,000$ pairs per epoch), adaptive learning rates, and augmentation by overlapping windows or random pair/triplet selection ensure coverage across temporal dynamics (Gao et al., 26 Jan 2026, Ferrand et al., 2020).
• Synchronization requirements: Doppler-based and time-distance metrics are feasible under frequency or time synchronization only; phase and angle-of-arrival alignment are not strictly necessary. Single-antenna BS setups and moderate UE speeds suffice for accurate charting (Euchner et al., 2024, Stahlke et al., 2022).
• Inference and mapping to physical space: Once trained, parametric networks (Siamese or LSTM-AE) embed new CSI samples via forward passes. Affine fitting can convert the learned chart to mapped coordinates if anchor positions are available (Stahlke et al., 2022).

5. Evaluation Metrics and Experimental Outcomes

The benchmarking of time-series channel charting employs topology and localization measures:

• Topology continuity (CT) and trustworthiness (TW): Correlation and neighbor overlap between ground-truth locations and chart embeddings, with values approaching unity representing high chart fidelity (Stephan et al., 2023, Gao et al., 26 Jan 2026, Ferrand et al., 2020).
• Kruskal’s stress (KS) and Rajski's Distance (RD): Quantifies distance-preservation quality and mutual information between manifolds. Lower values indicate better geometric consistency (Stephan et al., 2023).
• Mean Absolute Error (MAE): Localization error between predicted and true physical positions. LSTM-AE-CC achieves MAE $= 0.0619$ m; Siamese+G-fuse on massive MIMO data yields MAE $= 0.54$ m; and Doppler-CC obtains MAE $= 1.235$ m (train) / $1.346$ m (test) (Gao et al., 26 Jan 2026, Stephan et al., 2023, Euchner et al., 2024).
• Robustness to NLoS and dynamic scenarios: Fused temporal/CSI metrics and temporal DNN architectures sustain high performance under missing arrays, complex trajectories, and non-line-of-sight conditions (Stephan et al., 2023, Gao et al., 26 Jan 2026).

6. Limitations, Advantages, and Future Directions

Time-series channel charting achieves global geometric recovery, improved stability, and continuity without extensive ground-truth annotation. Key advantages include reduced annotation costs, resilience to environmental changes, and real-time adaptability (at inference stage). However, several limitations exist:

• The necessity for synchronization (frequency in Doppler-CC; time in time-distance CC) can constrain network deployment (Euchner et al., 2024, Stahlke et al., 2022).
• Model transfer across heterogeneous environments and online adaptation remain open challenges (Ferrand et al., 2020, Gao et al., 26 Jan 2026).
• The lack of explicit temporal regularizers in current architectures suggests the potential for further improving chart smoothness by adding penalties such as $L_{\rm smooth} = \beta \sum_t \|z_t - z_{t-1}\|^2$ (Gao et al., 26 Jan 2026).
• Extensions toward joint angle/delay metrics, reinforcement-learning-guided charting, and multimodal feature fusion are plausible next steps (Stephan et al., 2023, Gao et al., 26 Jan 2026).

7. Comparative Table: Principal Methods and Metrics

Method	Temporal Mechanism	MAE (m)
LSTM-AE-CC (Gao et al., 26 Jan 2026)	LSTM, AE, pairwise topology loss	0.0619
Doppler-CC (Euchner et al., 2024)	Doppler phase-difference loss	1.346
Siamese+G-fuse (Stephan et al., 2023)	Fused angle-delay/timestamp, geodesic	0.54
Triplet-DNN (Ferrand et al., 2020)	Timestamp-based triplets	~0.97
Time-distance Siamese (Stahlke et al., 2022)	Time-difference geodesic	0.72 (UWB), 1.46 (5G)

The table reflects published MAEs and mechanisms directly traceable to article sources.

---

Time-series channel charting synthesizes temporal and geometric information to recover richly structured, robust, and globally consistent radio maps across dynamic environments, enabling high-accuracy localization and adaptive wireless systems. The integration of temporal modeling—via explicit metrics or sequential DNN architectures—underpins the superiority of such approaches over classical, static channel charting methods.