CVAE4CP: Conditional VAE for Channel Prediction

• The paper introduces CVAE4CP, which employs a conditional VAE framework to integrate historical channel data and side information for future channel prediction.
• It uses latent variables and normalizing flows to effectively model multi-modal, heavy-tailed channel statistics under high Doppler spreads.
• CVAE4CP demonstrates superior NMSE performance compared to RNN baselines, ensuring reliable predictions in rapidly time-varying OTFS communication systems.

Conditional Variational Autoencoder for Channel Prediction (CVAE4CP) is a machine learning framework designed for channel prediction in high-mobility orthogonal time frequency space (OTFS) communication systems. CVAE4CP specifically addresses the challenge of rapidly time-varying channels subject to high Doppler spreads and complex multipath propagation. Conventional pilot-based estimation methods frequently suffer from outdated channel state information (CSI) and excessive pilot overhead under such conditions. CVAE4CP introduces a probabilistic generative modeling approach that leverages observed historical channel frames and explicit conditioning on physical system and mobility parameters to predict future channel realizations, while robustly quantifying channel uncertainty via a latent representation (Kazemian et al., 6 Jan 2026).

1. Problem Setting and Mathematical Formulation

In OTFS, the physical channel is mapped onto an $M \times N$ delay–Doppler (DD) grid, resulting in a set of complex coefficients $h[i,l]\in\mathbb{C}$, indexed by DD bin and delay tap ($i=0,\ldots,MN-1,\, l=0,\ldots,L-1$). The total channel coefficient vector is $\mathbf{h}\in\mathbb{C}^D$, with $D=MN\!L$. For modeling purposes, channel coefficients are real-stacked:

$$\mathbf{X} = \bigl[\Re\{\mathbf{h}\}^{T},\,\Im\{\mathbf{h}\}^{T}\bigr]^T \in \mathbb{R}^{2D}$$

At time $t$, given $T_p$ frames of observed past channel coefficients $$Y_{\text{past}} = \{\mathbf{X}^{(t-T_p+1)},\ldots,\mathbf{X}^{(t)}\}$$, the goal is to predict channel coefficients at $T_f$ frames ahead, $Y_{\text{future}} = \mathbf{X}^{(t+T_f)}$. A crucial modeling feature is the inclusion of a conditioning vector $s\in\mathbb{R}^{d_s}$ (with $d_s=20$ in reported experiments), encapsulating physical and mobility parameters such as Doppler spread, delay spread, SNR, carrier frequency, subcarrier spacing, CP length, antenna configuration, and UE speed.

The channel prediction target is the conditional distribution:

$p(Y_{\text{future}}\,|\,Y_{\text{past}},s)$

that characterizes the uncertainty and correlations in the DD-domain channel evolution, accounting for available side information.

2. Generative Model and Training Objective

CVAE4CP frames channel prediction as conditional generative modeling via a latent variable $z\in\mathbb{R}^{d_z}$ ($d_z=48$), intended to absorb unmodeled channel variability and environmental factors. The model structure is:

• Decoder (generative model):

$$p_\theta(Y_{\text{future}}\,|\,z,\,Y_{\text{past}},\,s)$$

• Encoder (variational posterior):

$q_\phi(z\,|\,Y_{\text{past}},\,s)$

• Latent prior: Gaussian, $p(z)=\mathcal{N}(0, I)$, optionally enhanced with normalizing flows (NF).

Training is performed by maximizing the conditional evidence lower bound (ELBO):

$$\mathcal{L}(\theta, \phi) = \mathbb{E}_{q_\phi(z\,|\,Y_{\text{past}}, s)} \left[ \log p_\theta(Y_{\text{future}}\,|\,z, Y_{\text{past}}, s) \right] - \mathrm{KL}\left[ q_\phi(z\,|\,Y_{\text{past}}, s) \,\|\, p(z) \right]$$

When normalizing flows are incorporated to enrich the latent space, the prior $p(z)$ is replaced by a transformed prior $p_\theta(z_K\,|\,s)$ with NF length $K$ and the ELBO is modified accordingly:

$$\mathcal{L}_{\mathrm{nf}} = \mathbb{E}_{q_\phi(z_0\,|\,Y_{\text{past}}, s)} \left[ \log p_\theta(Y_{\text{future}}\,|\,z_K, s) \right] - \mathrm{KL}\left( q_\phi(z_0\,|\,Y_{\text{past}}, s)\,\|\, p_\theta(z_K\,|\,s)\right)$$

This construction enables CVAE4CP to accurately model multi-modal and heavy-tailed distributions in fast-fading regimes, while capturing the temporal structure of DD coefficients (Kazemian et al., 6 Jan 2026).

3. Network Architecture and Conditioning

The encoder $q_\phi$ and decoder $p_\theta$ are instantiated as feed-forward multilayer perceptrons (MLPs):

• Encoder input: concatenation of past frames and conditioning

$$[Y_{\text{past}};\,s] \in \mathbb{R}^{T_p \cdot 2D + d_s}$$

• Encoder network: two hidden layers (width 1024, ReLU), outputting $\mu_\phi\in\mathbb{R}^{d_z}$ and $\log \sigma^2_\phi\in\mathbb{R}^{d_z}$.
• Sampling: latent code via reparameterization $z_0 = \mu_\phi + \sigma_\phi \odot \varepsilon$, $\varepsilon \sim \mathcal{N}(0, I)$.
• Normalizing flows: $z_k = f_k(z_{k-1}; s)$, with simple planar or affine flows, neural network parameters conditioned on $s$.
• Decoder: input $[z_K;\,Y_{\text{past}};\,s]$, two hidden layers of width 1024, final linear output describes Gaussian likelihood for future channel coefficients, i.e., $\mathcal{N}(\hat Y_{\text{future}}, \sigma^2 I)$.
• Conditioning: Side information $s$ is concatenated to every hidden layer input. No feature-wise linear modulation (FiLM) is used in baseline.

This compositional structure ensures effective fusion of temporal and physical context, facilitating expressive and reliable predictive distributions for future channel states (Kazemian et al., 6 Jan 2026).

4. Dataset Generation, Implementation, and System Parameters

CVAE4CP utilizes synthetic Monte Carlo channel realizations:

• Dataset size: 1000 samples ($M=32$, $N=32$, $L=6$ yields $D=6144$ complex or $2D=12,288$ real-valued features).
• Conditioning vector $s$: 20 dimensions, sampled uniformly or categorically over ranges reflecting operational variability (SNR, speed, carrier frequency, bandwidth, subcarrier spacing, CP length, antenna config).
• Train/test split: 800/200.
• Optimization: Adam ($\alpha=10^{-3}$), batch size 16, 50 epochs.

This simulation regime permits robust benchmarking under extreme channel dynamics and varied mobility profiles, crucial for high-Doppler OTFS scenarios (Kazemian et al., 6 Jan 2026).

5. Evaluation Metrics and Empirical Findings

Performance is measured via normalized mean squared error (NMSE):

$$\mathrm{NMSE} = \frac{1}{N_{\mathrm{te}}} \sum_{i=1}^{N_{\mathrm{te}}} \frac{\left\| \mathbf{h}_i - \hat{\mathbf{h}}_i \right\|_2^2} {\left\| \mathbf{h}_i \right\|_2^2}$$

Major findings, illustrating the efficacy of CVAE4CP:

Scenario	CVAE4CP NMSE	RNN Baseline NMSE
$f_D = 5$ kHz, $\Delta = 1$	$\approx 4\times 10^{-3}$	$\approx 5\times 10^{-2}$
$f_D$ high, $\Delta = 10$	$\approx 2\times 10^{-3}$	$\approx 2\times 10^{-1}$

CVAE4CP outperforms the RNN baseline [15] by more than an order of magnitude at high Doppler and long prediction horizons, indicating significant robustness against rapid channel decorrelation (Kazemian et al., 6 Jan 2026).

6. Modeling Principles, Advantages, and Directions for Extension

CVAE4CP leverages conditional probabilistic modeling to represent the full uncertainty and multi-modality inherent in fast-fading DD channels. Latent variables $z$ capture unobserved or unresolved scattering phenomena, while normalizing flows allow the prior to represent non-Gaussian, heavy-tailed behaviors observed in high-mobility environments. Incorporation of explicit side information $s$ facilitates data-efficient adaptation to diverse system configurations.

Identified limitations include focus on single-user, single-antenna settings; scaling to multi-antenna (MIMO) or multi-user interference contexts will necessitate structured latent priors and potentially convolutional flow architectures. Real-world channel evolution (e.g., statistical drift) raises the prospect of meta-learning or continual adaptation strategies. Computational overhead of normalizing flows may be mitigated through lightweight transformations or alternative expressive priors.

CVAE4CP thus enables flexible, conditional generative modeling for channel prediction, yielding superior NMSE performance under severe mobility and Doppler, and supporting robust operation in anticipated 6G OTFS deployments (Kazemian et al., 6 Jan 2026).

7. Relation to Variational Methods in Channel Estimation

Conditional variational autoencoders have been previously leveraged for channel estimation tasks distinct from prediction, including approximate MMSE estimation in pilot-based SIMO receivers (Baur et al., 2022). In such approaches, the latent variable $z$ is employed to parameterize posterior channel statistics, with the VAE decoder outputting conditional first and second moments for LMMSE plug-in estimation. Multiple variants balance benchmark accuracy against practical access to clean CSI.

While CVAE4CP focuses on predictive generative modeling of future DD-domain channel realizations, both approaches substantiate the utility of VAEs and CVAEs for robust, uncertainty-aware learning in communication systems with complex and nonstationary channel statistics (Kazemian et al., 6 Jan 2026, Baur et al., 2022).