Uncertainty-Weighted Multi-Task CNN for Joint DoA and Rain-Rate Estimation Under Rain-Induced Array Distortions

We investigate joint direction-of-arrival (DoA) and rain-rate estimation for a uniform linear array operating under rain-induced multiplicative distortions. Building on a wavefront fluctuation model whose spatial correlation is governed by the rain-rate, we derive an angle-dependent covariance formulation and use it to synthesize training data. DoA estimation is cast as a multi-label classification problem on a discretized angular grid, while rain-rate estimation is formulated as a multi-class classification task. We then propose a multi-task deep CNN with a shared feature extractor and two task-specific heads, trained using an uncertainty-weighted objective to automatically balance the two losses. Numerical results in a two-source scenario show that the proposed network achieves lower DoA RMSE than classical baselines and provides accurate rain-rate classification at moderate-to-high SNRs.

💡 Research Summary

This paper addresses the challenging problem of jointly estimating the direction‑of‑arrival (DoA) of multiple sources and the ambient rain rate when a uniform linear array (ULA) is subject to rain‑induced multiplicative distortions. The authors first build on a physics‑based wavefront fluctuation model: the complex electric‑field perturbation is modeled as a zero‑mean circularly symmetric complex Gaussian process whose spatial correlation coefficient α depends on rain rate, propagation distance, sensor spacing, and carrier frequency. An empirical expression for α (Eq. 4) captures how stronger rain reduces inter‑sensor correlation, and the coefficient is angle‑dependent because the effective spacing on the wavefront varies with the incident angle (d(θ)=d₀ cosθ).

Using this model, the received signal at each sensor is expressed as the product of the distortion‑free steering vector and a multiplicative distortion vector b(θ,t). Lifting the observation model to the covariance domain yields the overall covariance matrix R_y = Σₙ R_x(θₙ) ⊙ R_b(θₙ) + R_n, where R_b(θₙ) encodes the rain‑rate information through α. Consequently, the covariance matrix alone contains sufficient statistics for both DoA and rain‑rate inference.

To generate training data, the authors synthesize rain‑distorted covariance samples for six rain‑rate classes (0, 2, 5, 10, 25, 50 mm/h) and five SNR levels (0–20 dB). For each class they compute R_b using Eq. 14‑15, draw complex Gaussian distortion samples via a Cholesky factorization, and form sample covariance matrices \hat R_y from simulated snapshots. Each \hat R_y is converted into a three‑channel real tensor X (real part, imaginary part, phase) of size M×M×3.

The proposed deep learning architecture consists of a shared CNN encoder (four convolutional blocks with 256 3×3 filters, batch normalization, ReLU, and a 2×2 max‑pool after the second block) followed by two task‑specific heads. The DoA head outputs G logits (G=61 for a 1° grid over –30° to +30°) and is trained with a multi‑label binary cross‑entropy loss (sigmoid activation). The rain‑rate head outputs R logits (R=6) and uses a standard softmax cross‑entropy loss.

A key contribution is the use of uncertainty‑based loss weighting. Each task is assigned a learnable log‑variance s (s_DoA, s_rain). The total loss is L = e^{‑s_DoA}·L_DoA + e^{‑s_rain}·L_rain + (G/2)·s_DoA + (1/2)·s_rain, following Kendall et al. (2018). This formulation automatically balances the tasks during training, preventing one loss from dominating due to scale differences.

Experiments are conducted on a fixed two‑source scenario (N=2) with range R=200 m. The network is trained on 100 k samples (90 % train, 10 % validation) and evaluated on 10 k test samples. Results show that the uncertainty‑weighted network consistently outperforms classical DoA estimators (MUSIC, root‑MUSIC, ESPRIT, MVDR) across all SNRs, achieving the lowest RMSE, especially in low‑SNR regimes where the adaptive weighting yields a 2–5 dB RMSE reduction compared to fixed weighting (α=1 or α=0.05). Rain‑rate classification accuracy remains above 94 % for all SNRs and exceeds 99 % for SNR ≥ 10 dB.

The paper acknowledges limitations: only two sources, fixed array geometry, and discretized rain‑rate classes were considered; real‑world validation with measured radar data is absent. Nevertheless, the integration of a physics‑driven data synthesis pipeline with uncertainty‑aware multi‑task deep learning provides a compelling framework for robust array processing under adverse weather conditions.