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CNN-Transformer with Sigmoid Label for High-Resolution Sub-Degree DOA Estimation
Lifeng Wang, Xintong Li, Aifei Liu
TL;DR
DL-based DOA estimation uses a discrete angular grid, which constrains peak-based estimates. This paper proposes a Sigmoid label with a derivative readout; under a unified backbone, it reports better accuracy and small-separation resolution than the compared labels.
Problem
Peak-based DL readout confines DOA estimates to grid points, limiting accuracy even with a perfectly trained network.
Source: §1 · p. 1Method
The method trains a CNN-Transformer with a label formed by summing angle-shifted monotonic sigmoids, then differentiates its output to locate sharp source-angle peaks.
Source: §1 · p. 1Results
The Sigmoid label with MSE outperforms the Gaussian, hard-label MSE, and hard-label BCE baselines; at 2° separation its high-SNR RMSE is about 0.2°, and it resolves 1° at 5 dB with probability 1.
Source: §1 · p. 1, §5.3 · p. 4, §5.4 · p. 4Takeaways & Limitations
With a 0.25° grid, the derivative readout recovers source angles at sub-degree resolution, breaking the 1° quantization limit.
Source: §1 · p. 1Naively refining the grid from 1° to 0.1° increases class count tenfold and raises classification difficulty, network-capacity needs, and training-data demands.
Source: §3 · p. 2
Abstract
from arXiv · showhide
This paper addresses the grid quantization error of deep-learning (DL)-based direction-of-arrival (DOA) estimation. Classification- based DOA methods discretize the continuous angular space into a grid, so that with a peak-based read-out, the accuracy and resolution are limited by the grid step and by the width of the label peak. To address this issue, we propose a sigmoid label that encodes the sources' DOAs as the sum of angle-shifted monotonic sigmoid functions. Differentiating the output of a CNN-Transformer network trained with the sigmoid label produces sharp spectral peaks at the source DOAs. Combined with a moderately refined grid step of 0.25 degrees, the CNN-Transformer with the proposed sigmoid label attains sub-degree accuracy and a 1-degree resolution, breaking the limit of the grid quantization error. Under the unified CNN-Transformer backbone, we compare the proposed Sigmoid label with the Gaussian label, the 0-1 hard label, and the binary cross-entropy (BCE) baseline. Numerical results demonstrate that the Sigmoid label attains the highest accuracy and superior resolution compared to other labels.
1. INTRODUCTION
Grid-confined peak readout limits DL-based DOA accuracy, while the proposed Sigmoid label and derivative readout recover source angles beyond that quantization limit. Under a unified backbone, the Sigmoid label with MSE outperforms the compared Gaussian and hard-label alternatives.
1. INTRODUCTION
- Peak-based decoding confines DL-based DOA estimates to grid points, limiting accuracy even when a network is perfectly trained.Gaussian soft labels smooth the target but retain the grid-confined peak readout.Source: §1 · p. 1
- The Sigmoid label sums monotonic sigmoid functions for source angles, and differentiating the predicted label produces sharp peaks at the source DOAs.With a 0.25° grid, the derivative readout attains sub-degree resolution; the comparison uses the unified HMC-ViT backbone.Source: §1 · p. 1
- Under the unified backbone, the Sigmoid label with MSE outperforms Gaussian-label MSE, hard-label MSE, and hard-label BCE.Source: §1 · p. 1
2. BACKGROUND
The paper models array observations and uses their sample covariance as the CNN-Transformer input. Its fixed HMC-ViT backbone combines convolutional local features with multiple transformer class tokens, while label designs use different output-grid steps.
2.1. Array received signal
- A ULA receives K far-field narrowband signals; the sample covariance matrix is formed by averaging over N snapshots and used as the CNN-Transformer input.The signal description defines the sensor-output vector, Gaussian noise, steering vectors, and source samples.Source: §2.1 · pp. 1–2
2.2. HMC-ViT
- HMC-ViT combines a CNN for local feature extraction with multiple ViT class tokens that focus on angular sub-regions; the backbone remains unchanged.The class tokens jointly attend to spatial features.Source: §2.2 · p. 2
- The Sigmoid and Gaussian labels use 0.25° output-grid steps, whereas the hard and BCE labels use 1° steps.Source: §2.2 · p. 2
3. PROBLEM STATEMENT
Peak-based DL methods return discrete grid points, so their accuracy is structurally bounded by grid spacing. Refining the grid raises classification and training-data demands, motivating a label and readout that avoid the naive refinement trade-off.
3. PROBLEM STATEMENT
- Peak-based DL methods decode a spatial spectrum by locating its peaks, returning a grid point and confining estimates to the discrete angular grid.The hard label places a 1 at the true-angle grid point; the Gaussian label spreads a bump around that angle, but both use peak-based readout.Source: §3 · p. 2
- For a 1° grid, peak-based quantization error for a single source lies in [−ρin/2, ρin/2], regardless of perfect training.The limit follows from the grid-point readout, not insufficient training.Source: §3 · p. 2
- Using a 0.1° grid increases the number of classes by a factor of 10, dramatically raising classification difficulty, required network capacity, and the amount of training data.Source: §3 · p. 2
- The authors identify label design and readout—not grid step alone—as the bottleneck, motivating derivative peaks for sub-degree resolution without naive 0.1° dataset growth.Source: §3 · p. 2
4. PROPOSED SIGMOID LABEL AND ITS DERIVATIVE
The section defines a cumulative Sigmoid target and a derivative-based readout: source angles are encoded as sigmoid centers, then recovered as peaks in the derivative of the predicted curve.
4.1. Sigmoid label design
- Soft labels use ρin=1° and ρout=0.25°, while hard and BCE labels use 1° for both grid steps.Source: §4.1 · p. 2
- The Sigmoid target sums source-centered monotonic sigmoid functions into a CDF, with w=0.2° controlling each transition’s sharpness.This differs from Gaussian and hard labels, which form peaks around each source.Source: §4.1 · p. 2
- A sigmoid output activation keeps predictions within [0, 1] and suppresses large oscillations so the derivative remains unimodal at each sigmoid.Source: §4.1 · pp. 2–3
4.2. First-derivative-based readout
- The readout computes the predicted curve’s central-difference derivative and recovers source angles from its local maxima.It retains the K tallest peaks, maps their grid indices to angles, and smooths the output before differentiation to suppress non-monotonic fluctuations.Source: §4.2 · p. 3
- Derivative peaks have widths set by sigmoid transition width w rather than grid spacing, enabling sub-degree resolution on the 0.25° grid.The peaks are narrower and taller than Gaussian bumps, so closely spaced sources produce distinct peaks at their true angles.Source: §4.2 · p. 3
5. SIMULATION RESULTS
The simulations compare the Sigmoid label with three label baselines and MUSIC under a unified CNN-Transformer backbone, examining spectra, estimation accuracy, and angular resolution.
5.1. Simulation Setup
- Simulations use a 16-element half-wavelength ULA, with 0.25° output grids for Sigmoid and Gaussian labels and 1° grids for hard and BCE labels.Testing uses 1,000 Monte Carlo trials with 1,000 snapshots each; RMSE is evaluated from −20 to 5 dB, and resolution at 0 and 5 dB.Source: §5.1 · p. 3
5.2. Spatial Spectra
- At 2° separation and 5 dB, the Sigmoid derivative produces sharp, well-separated peaks at the source angles, unlike the broad Gaussian, hard, BCE, and MUSIC spectra.The Gaussian spectrum has poorly separated sub-peaks biased away from the true angles; the other baseline spectra have broader peaks and higher sidelobes.Source: §5.2 · pp. 3–4, Figure 2 · p. 4
5.3. Performance versus SNR
- At 2° separation, the Sigmoid label attains about 0.2° RMSE at high SNR, while MUSIC has RMSE of tens of degrees; differences are small at 20° separation.The Sigmoid label has the lowest RMSE among the compared methods in the small-separation case.Source: Figure 3 · p. 4, §5.3 · p. 4
5.4. Angular Resolution
- At 5 dB, the Sigmoid label resolves a 1° separation with probability 1, while the Gaussian, hard, BCE, and MUSIC methods fail.At 0 dB, none resolves 1°; from 2° onward, the Sigmoid, Gaussian, and hard labels begin resolving sources, while BCE and MUSIC require larger separations.Source: Figure 4 · p. 4, §5.4 · p. 4
6. CONCLUSION
The conclusion presents a Sigmoid label and derivative readout for deep-learning DOA estimation. Simulations report higher accuracy and better resolution than the compared labels, including unique resolution of a 1° separation at 5 dB.
6. CONCLUSION
- At 5 dB, the CNN-Transformer with the Sigmoid label uniquely resolves two sources separated by 1°, while the other labels and MUSIC fail.The method combines a cumulative Sigmoid label with a derivative readout that produces sharp peaks at source DOAs, achieving sub-degree resolution beyond the 1° quantization limit.Source: §6 · p. 4