# Low-Cost Maximum Likelihood for Direction-of-Arrival Estimation: Analysis & Implementation

Thomas Aussaguès

► To cite this version:

Thomas Aussaguès. Low-Cost Maximum Likelihood for Direction-of-Arrival Estimation: Analysis & Implementation. Journée des doctorants du laboratoire SATIE, Jun 2025, Paris, France. hal-05102825

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https://hal.science/hal-05102825v1

Submitted on 7 Jun 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# 1. WHY DIRECTION-OF-ARRIVAL ESTIMATION?

* **DoA estimation:** a classical signal processing problem with critical applications (radar, telecommunications...)

* Numerous methods have been proposed: Capon's beamformer, MUSIC, Maximum Likelihood (ML) [1]:

* MUSIC can not handle coherent sources

* ML is intractable for large number of sources

* Limited number of identifiable sources

<figure><img src="image_2.png" /></figure>

**Sparse estimators with the Vectorized Covariance Matrix Model [2]**

* ✓ High number of identifiable sources

* ✓ Enhanced performances in severe scenarios

<figure><img src="image_3.png" /></figure>

* **Equivalence between ML and sparse DoA estimators after a pre-whitening transform**

## Contributions

The pre-whitening is shown to:

<figure><img src="image_4.png" /></figure>

* Significantly improve the sparse problem conditioning

<figure><img src="image_5.png" /></figure>

* Simplify the implementation of ML-DoA estimation using a sparse representation

<figure><img src="image_6.png" /></figure>

**Objective:** estimate the sources directions of arrival $\{\theta_1, ..., \theta_M\}$

<figure><img src="image_7.png" /><figcaption>Source 1</figcaption></figure>

<caption>Source 2</caption>

# 3. SPARSE ESTIMATION PROBLEM

## Sparse modeling

<figure><img src="image_8.png" /></figure>

* **Hypothesis:** the sources directions $\{\theta_1, ..., \theta_M\}$ are comprised within a grid of pre-defined directions $\{\varphi_1, ..., \varphi_G\}$

<figure><img src="image_9.png" /></figure>

$$y = \gamma_{\theta_1} \mathbf{b}(\theta_1) + \gamma_{\theta_2} \mathbf{b}(\theta_2) + \gamma_{\theta_3} \mathbf{b}(\theta_3) + \delta_w$$

$$y = \gamma_{01} \times \mathbf{b}(\varphi_1) + \gamma_{02} \times \mathbf{b}(\varphi_2) + \gamma_{10} \times \mathbf{b}(\varphi_1) + \gamma_{11} \times \mathbf{b}(\varphi_2) + \dots + \gamma_{10} \times \mathbf{b}(\varphi_{10}) + \gamma_{01} \times \mathbf{b}(\varphi_1) + \delta_w$$

* **For calibration purpose**, recording of the **array response** $\mathbf{b}(\varphi_g)$ in all the directions $\varphi_g$ comprised in a pre-defined grid

* These recordings can also be used for sparse estimation!

$$y = \mathbf{B}_w \gamma + \delta_w$$

* **Sparse?** $y$ is obtained using *few* vectors of $\mathbf{B}_w$

* **DoA** $\theta_1, ..., \theta_M$: indices of the $M$ non-null components of $\gamma$

## Sparse estimation problem formulation

* **G** $\gg$ $N^2$: the problem is ill-posed $\Rightarrow$ Use of the sparsity prior: regularized problem formulation

$$\hat{\gamma} = \arg \min_{\gamma} \mathcal{J}_{\Psi}(\lambda, \gamma) \quad \text{with} \quad \mathcal{J}_{\Psi}(\lambda, \gamma) = \frac{1}{2} \left\| \mathbf{y} - \mathbf{B}_w(\mathbf{\Phi})\gamma \right\|_2^2 + \underbrace{\mathbf{\Psi}(\lambda, \gamma)}_{\text{Penalty}}$$

* **Penalty function** $\ell_0$-norm, $\Psi$: Continuous Exact $\ell_0$ CEL0 [4] ($\ell_0$-norm approximation), $\ell_1$-norm

* **Regularization parameter** $\lambda$: balances the penalty term $\Phi$ towards the data fidelity term $\Rightarrow$ **controls the sparse criterion global min. position**

* $\mathcal{J}_{\Psi}$ is minimized using the *Proximal Gradient Algorithm* [5] applied to the CEL0 penalty

# 4. WHITENING EFFECTS ON THE DICTIONARY MATRIX

<figure><img src="image_10.png" /></figure>

* After pre-whitening transform, the dictionary becomes $\mathbf{B}_w = \mathbf{W} \times \mathbf{B}$

<figure><img src="image_11.png" /></figure>

What are the effects of the pre-whitening transform on the sparse estimator?

* Spatial correlation function of a dictionary matrix $E$: $r_E(\varphi_i, \varphi_j) = \frac{|\mathbf{e}^H(\varphi_i)\mathbf{e}(\varphi_j)|}{\|\mathbf{e}(\varphi_i)\|_2\|\mathbf{e}(\varphi_j)\|_2}$

* Measures spatial correlation between pairs of directions $\{\varphi_i, \varphi_j\}$

* Pre-whitening: **reduced mainlobe width** + **zero correlation in the sources directions** $\{\theta_1 = 180^\circ, \theta_2 = 195^\circ\}$

<figure><img src="image_12.png" /><figcaption>Projections of the sparse criterion in directions $\{\theta_1, \theta_2\}$.</figcaption></figure>

The pre-whitening:

* **Significantly improves** the sparse problem **conditioning** by transforming elliptical contour lines into circular contour lines

* **Simplifies the implementation** of sparse ML-DoA estimation using a sparse representation: the estimator **outperforms** MUSIC and **achieves** the CRLB

<figure><img src="image_13.png" /><figcaption>Probability of detection and Root Mean Square Error (RMSE).</figcaption></figure>

<footnote>[1] B. Ottersten et al. "Exact and Large Sample Maximum Likelihood Techniques for Parameter Estimation and Detection in Array Processing". In: Radar Array Processing. 2009. Joseph S. Picard and Anthony J. Weiss. "Direction finding of multiple emitters by spatial sparsity and linear programming". In: 2009 9th International Symposium on Communications and Information Technology. 2009. Thomas Aussaguès et al. "Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators". In: 2024 32th European Signal Processing Conference (EUSIPCO). 2024. Emmanuel Soubies, Laure Blanc-Féraud, and Gilles Aubert. "A Continuous Exact 10 penalty (CELO) for least squares regularized problem". In: SIAM Journal on Imaging Sciences 8.3 (July 2015), pp. 1607-1639 (33 pages). Patrick L. Combettes and Jean-Christophe Pesquet. Proximal Splitting Methods in Signal Processing. 2010. arXiv: 0912.3522 [math.OC].</footnote>