Learned Narrowband Angle-of-Arrival Estimation#

Goal#

Train a portable single-source AoA estimator for an eight-element half-wavelength uniform linear array. The returned model receives noisy complex snapshots only. Source angles are offline labels and are unavailable at runtime.

The paired comparison holds angles, array snapshots, and AWGN seeds fixed across Bartlett, MUSIC, and the returned learned estimator.

CLI training summary#

(
  set -euo pipefail
  ROOT="$(git rev-parse --show-toplevel)"
  BUNDLE="$ROOT/.noema/training_exports/aoa_estimation"
  cd "$ROOT"

  uv sync --extra onnx
  uv run --project "$ROOT" --extra onnx noema differentiable export \
    "$ROOT/recipes/aoa_adapter_ula_baseline.yaml" \
    --training-plan "$ROOT/demo_trainings/aoa_estimation_covariance_mlp/training_plan.yaml" \
    --out "$BUNDLE" --force
  uv run --project "$ROOT" --extra onnx python \
    "$ROOT/demo_trainings/prepare_example.py" aoa-estimation "$BUNDLE" \
    --project-root "$ROOT"

  uv run --project "$ROOT" --extra onnx noema dataset-capture run \
    "$BUNDLE/capture_train_recipe.yaml" --out "$BUNDLE/data/train" --force
  uv run --project "$ROOT" --extra onnx noema dataset-capture run \
    "$BUNDLE/capture_validation_recipe.yaml" --out "$BUNDLE/data/validation" --force
  uv run --project "$ROOT" --extra onnx noema dataset-capture run \
    "$BUNDLE/capture_test_recipe.yaml" --out "$BUNDLE/data/test" --force

  cd "$BUNDLE"
  uv run --project "$ROOT" --extra onnx python validate_contract.py
  uv run --project "$ROOT" --extra onnx python train_demo.py
  uv run --project "$ROOT" --extra onnx python evaluate_demo.py
  cd "$BUNDLE/reference_training"
  uv run --project "$ROOT" --extra onnx python build_benchmark.py
  cd "$ROOT"
  uv run --project "$ROOT" --extra onnx noema benchmark validate \
    "$BUNDLE/reference_training/benchmark_pack.yaml"
  uv run --project "$ROOT" --extra onnx noema benchmark run \
    "$BUNDLE/reference_training/benchmark_pack.yaml"
)

1. Export and capture from Workbench#

Open Localization / Sensing > ULA angle-of-arrival estimation, then select Train/replace for Estimator. Keep:

  • snapshots from array_observation.observation;

  • angles_deg from array_observation.truth.

Use 4,096 records, the default three-way split, and the training plan’s 0, 5, 10, 15, 20 dB capture sweep. The captured snapshot tensor preserves the complex array and time axes; the exporter presents it to ONNX as a final real/imaginary axis.

2. Train and return the estimator#

The starter converts each frame to a normalized complex sample covariance, obtains a 0.25°-grid Bartlett estimate, and trains a bounded residual head for sub-grid and noise corrections. The last layer starts at zero, so training starts from the physical estimator rather than an arbitrary MLP. Validation angle MSE selects the checkpoint; held-out angles remain sealed until evaluate_demo.py.

The returned aoa_estimator.onnx implements:

  • input snapshots_ri: [batch, antenna, snapshot, 2];

  • output angles_deg: [batch].

The schema-v2 artifact records the exact trained antenna and snapshot sizes and binds the model to model.aoa_estimator_adapter.

3. Compare estimators#

The post-training pack compares Bartlett, MUSIC, and the learned estimator at 0 and 15 dB over three fresh paired seeds. Use aoa.rmse_deg as the primary metric and inspect aoa.mae_deg as a companion measure.

Completed benchmark result#

The completed paired campaign contains 18 runs: Bartlett, MUSIC, and the learned estimator at two held-out SNRs over three fresh seeds. Points are means over the three seeds; bands are two-sided Student-t 95% confidence intervals.

Note

Three paired seeds make this a compact workflow result, not a publication-strength population claim.

The physics-informed learned estimator reaches 0.345° mean RMSE at 0 dB, compared with 0.356° for Bartlett and 0.364° for MUSIC. At 15 dB all three are effectively tied near 0.093°. This is a small low-SNR point-estimate improvement, and the three-seed confidence intervals overlap. It is not a claim that learning generally dominates classical single-source array processing.

Paired benchmark summary#

SNR (dB)

Method

Role

Mean RMSE (degree)

Primary 95% CI low

Primary 95% CI high

0

Bartlett reference

baseline

0.355609497958

0.112940089193

0.598278906723

0

MUSIC

baseline

0.363952401628

0.132107910709

0.595796892548

0

Learned Bartlett-residual estimator

candidate

0.345006895503

0.0744081549881

0.615605636017

15

Bartlett reference

baseline

0.0933945370652

0.085953806086

0.100835268044

15

MUSIC

baseline

0.0933945370652

0.085953806086

0.100835268044

15

Learned Bartlett-residual estimator

candidate

0.0934483725855

0.0788052559106

0.10809148926

Download the run-level projection, chart data, or provenance manifest.

This is deliberately a far-field, single-source, narrowband, calibrated-array protocol. Multiple sources, coherent multipath, calibration error, near-field propagation, and wideband beam squint need separate contracts and should not be inferred from this demo.