Direction of Arrival (DOA) Estimation (with MATLAB + Simulator)

Direction of Arrival (DOA) Estimation

Direction of Arrival (DOA) estimation is a fundamental problem in array signal processing. It involves determining the angle at which one or more incident signals arrive at a sensor array. This information is critical in applications such as radar, sonar, wireless communications, and acoustic source localization.

1. Signal Model

Consider a uniform linear array (ULA) of $ M $ sensors receiving a narrowband signal $ s(t) $ from a direction $ \theta $. The received signal at the array can be modeled as:

$$ \mathbf{x}(t) = \mathbf{a}(\theta) s(t) + \mathbf{n}(t) $$

where:

$ \mathbf{x}(t) = [x_1(t), x_2(t), \dots, x_M(t)]^T $ is the received signal vector.
$ \mathbf{a}(\theta) $ is the steering vector of the array, representing the relative phase shifts across sensors:

$$ \mathbf{a}(\theta) = \begin{bmatrix} 1 \\ e^{-j 2 \pi \frac{d}{\lambda} \sin \theta} \\ e^{-j 2 \pi \frac{2d}{\lambda} \sin \theta} \\ \vdots \\ e^{-j 2 \pi \frac{(M-1)d}{\lambda} \sin \theta} \end{bmatrix} $$

Here, $ d $ is the sensor spacing, and $ \lambda $ is the wavelength of the incident signal.

$ \mathbf{n}(t) $ is additive noise (assumed zero-mean, spatially uncorrelated).

For a vector sensor (e.g., x, y, z components), the model generalizes to 3D:

$$ \mathbf{x}(t) = \begin{bmatrix} x(t) \\ y(t) \\ z(t) \end{bmatrix} = \mathbf{a}(\theta, \phi) s(t) + \mathbf{n}(t) $$

where $ \theta $ is the elevation and $ \phi $ is the azimuth angle.

2. Covariance Matrix

Assuming the signal $ s(t) $ is stationary and ergodic, the sample covariance matrix of the received signal is:

$$ \mathbf{R} = \mathbb{E}[\mathbf{x}(t) \mathbf{x}^H(t)] \approx \frac{1}{N} \sum_{t=1}^{N} \mathbf{x}(t) \mathbf{x}^H(t) $$

where $ N $ is the number of snapshots and $ (\cdot)^H $ denotes the Hermitian transpose. The covariance matrix captures the signal subspace and noise subspace, which are central to subspace-based DOA methods like MUSIC.

3. Classical DOA Estimation (Beamforming)

A simple beamforming approach estimates the DOA by scanning a range of angles and computing the output power:

$$ P_{\text{BF}}(\theta) = \mathbf{a}^H(\theta) \mathbf{R} \mathbf{a}(\theta) $$

The DOA is the angle $ \theta $ that maximizes $ P_{\text{BF}}(\theta) $.

Pros: Simple and intuitive.
Cons: Resolution limited by array aperture.

4. Subspace-based DOA: MUSIC Algorithm

The MUltiple SIgnal Classification (MUSIC) algorithm leverages the orthogonality between the signal subspace and the noise subspace:

Eigen-decompose the covariance matrix:
$$ \mathbf{R} = \mathbf{E}_s \mathbf{\Lambda}_s \mathbf{E}_s^H + \mathbf{E}_n \mathbf{\Lambda}_n \mathbf{E}_n^H $$
- $ \mathbf{E}_s $ → eigenvectors corresponding to signal eigenvalues.
- $ \mathbf{E}_n $ → eigenvectors corresponding to noise eigenvalues.
Construct the MUSIC pseudo-spectrum:
$$ P_{\text{MUSIC}}(\theta) = \frac{1}{\mathbf{a}^H(\theta) \mathbf{E}_n \mathbf{E}_n^H \mathbf{a}(\theta)} $$
- Peaks of $ P_{\text{MUSIC}}(\theta) $ correspond to the DOA of incoming signals.
- Advantages: High resolution even with closely spaced sources.

5. DOA Estimation with Vector Sensors

Vector sensors measure both pressure and particle velocity components (x, y, z). This allows:

3D DOA estimation (azimuth and elevation).
Improved signal-to-noise ratio through MRC (Maximal Ratio Combining) of the sensor components:

$$ r_{\text{MRC}} = w_x x + w_y y + w_z z $$

where $ w_i = \frac{|h_i|^2}{|h_x|^2 + |h_y|^2 + |h_z|^2} $ are weights based on channel strengths.

6. Summary

DOA estimation is the process of finding the angle of incidence of signals at a sensor array.
Beamforming gives coarse estimates; subspace methods like MUSIC provide high-resolution estimates.
Vector sensors improve estimation by providing multi-component measurements.
Accurate DOA estimation requires proper calibration, non-zero signals, and enough snapshots for covariance estimation.

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