Technical Summary: Joint Single-Shot TOA and DOA Estimation for VAA-Based BLE Ranging with Phase Ambiguity

1. Problem Statement

Accurate Direction-of-Arrival (DoA) and Time-of-Arrival (ToA) estimation is critical for wireless localization. While classical super-resolution methods (e.g., MUSIC) perform well on physical multi-antenna arrays, they are often impractical for size- and cost-constrained Bluetooth Low Energy (BLE) devices. Virtual Antenna Array (VAA) techniques offer a solution by synthesizing spatial diversity from single-antenna measurements taken along a receiver trajectory.

However, applying VAA to BLE ranging faces two primary challenges:

1. Two-Way Channel Frequency Response (CFR) Limitations: BLE ranging typically constructs a two-way CFR to cancel Local Oscillator (LO) and synchronization offsets via a Hadamard product. While this removes phase offsets, it exacerbates multipath effects and reduces the Signal-to-Noise Ratio (SNR).
2. Binary Phase Ambiguity: Recovering the one-way CFR from the two-way CFR requires an element-wise square root operation. This introduces a binary $\{ \pm 1 \}$ phase ambiguity that, if unresolved, severely degrades the performance of ToA and DoA estimation.
3. Non-Uniform Geometry: Unlike uniform linear arrays, VAAs formed by user trajectories often have irregular geometries, complicating traditional array signal processing and rendering existing ambiguity resolution heuristics (designed for uniform arrays) ineffective.

Existing approaches often assume multiple snapshots or uniform array structures and fail to address the specific constraints of BLE's single-shot measurements and the inherent binary ambiguity of the square-root recovery.

2. Methodology

The authors propose a unified framework that integrates VAA modeling with BLE two-way CFR processing, coupled with a deep learning-based phase recovery mechanism.

System Model

The system considers a Single-Input Single-Output (SISO) multi-carrier BLE scenario where a User Equipment (UE) moves along a trajectory to form a VAA.

• Signal Model: The received signal is modeled as a sum of multipath components, where the channel coefficient includes amplitude, initial phase, and a spatial phase shift dependent on the UE's displacement ($d_n$) and direction ($\phi_n$).
• Two-Way CFR Construction: To cancel LO offsets, the system forms a two-way CFR ($\hat{Y}$) by taking the Hadamard product of the forward ($Y^{(R)}$) and reverse ($Y^{(I)}$) channel responses. This operation cancels the LO phase matrix $\Phi$ but introduces a squared channel term and noise.
• Ambiguity Introduction: The one-way CFR is recovered via $(\hat{Y})^{1/2} = Y \circ N$, where $N$ is a binary matrix with elements $\{1, -1\}$ representing the unknown phase ambiguity.

Deep Learning-Based Phase Recovery

To resolve the binary ambiguity matrix $N$, the paper proposes a Double Neural Network Algorithm with a voting mechanism:

1. Dual Predictors: Two U-Net architectures are employed:
   • A row-wise network ($f_{row}$) predicts signs along the spatial dimension (antenna positions).
   • A column-wise network ($f_{col}$) predicts signs along the frequency dimension (subcarriers).
   • Both networks output probabilities for the sign consistency relative to the first element of their respective row or column.
2. Training: The networks are trained using supervised learning with cross-entropy loss, using ground-truth labels derived from the known sign relationships in the simulated data.
3. Voting Mechanism:
   • The row-wise predictor is empirically identified as more reliable due to the fixed subcarrier spacing versus the random VAA trajectory.
   • The algorithm enforces consistency with the row-wise predictions.
   • For each row $n$, the sign of the first element ($N_{n,1}$) is determined by aggregating candidate estimates from all columns using majority voting.
   • Once the first element of each row is resolved, the entire ambiguity matrix $N$ is reconstructed using the row-wise predictions.

Parameter Estimation

Once the ambiguity matrix $N$ is recovered, the corrected one-way CFR ($\tilde{Y}$) is obtained. A 2D MUSIC algorithm is then applied to this corrected data to estimate the ToA and DoA by computing the pseudospectrum and identifying peaks.

3. Key Contributions

• Unified VAA-BLE Model: The paper establishes a unified signal model that combines VAA spatial diversity with the BLE two-way CFR construction, explicitly accounting for the binary phase ambiguity introduced during one-way CFR recovery.
• Neural Voting Framework: It introduces a novel deep learning approach that decomposes the ambiguity resolution task into row-wise and column-wise prediction, fusing them via a voting mechanism to robustly handle non-uniform array geometries.
• Single-Shot Capability: The proposed method enables joint ToA and DoA estimation from single-shot measurements, addressing a limitation of prior works that relied on multiple snapshots.
• Performance under Non-Uniformity: The approach is specifically designed to function effectively under irregular VAA geometries where traditional continuity-based methods fail.

4. Simulation Results

The authors evaluated the method in a 2D scenario (40m $\times$ 40m) with a UE moving along a trajectory to form a VAA of $N=16$ elements and $M=80$ subcarriers. The channel included 1 Line-of-Sight (LoS) and 1–3 Non-Line-of-Sight (NLoS) paths.

• Comparisons: The proposed method was compared against:
  1. Two-way CFR MUSIC (without ambiguity resolution).
  2. A phase-continuity based approach for ambiguity resolution (designed for uniform arrays).
  3. The theoretical Cramér–Rao Lower Bound (CRLB).
• Performance Metrics: Mean Squared Error (MSE) for DoA and ToA estimation across varying SNRs.
• Findings:
  • The proposed method significantly outperformed both baselines across all SNR levels.
  • The phase-continuity approach failed completely under non-uniform geometries, resulting in large estimation errors.
  • The proposed estimator's MSE closely approached the CRLB at SNR $\geq 5$ dB, demonstrating statistical efficiency.
  • The method achieved superior ToA and DoA accuracy, particularly in low-to-moderate SNR regimes.

5. Significance and Claims

The paper claims that by integrating VAA modeling with a learning-assisted neural voting strategy, it is possible to resolve the binary phase ambiguity inherent in BLE two-way CFR recovery. This enables the application of super-resolution algorithms like MUSIC on resource-constrained, single-antenna BLE devices. The authors assert that their approach provides a cost-efficient, high-fidelity alternative to physical multi-antenna arrays, achieving near-optimal estimation accuracy even in the presence of multipath and non-uniform antenna geometries.