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# Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators

Thomas Aussaguès, Anne Ferréol, Alice Delmer, Pascal Larzabal

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Thomas Aussaguès, Anne Ferréol, Alice Delmer, Pascal Larzabal. Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators. Science Talks, 2025, 14, pp.100449. 10.1016/j.sctalk.2025.100449. hal-05033137

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Submitted on 14 Apr 2025

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# Looking for equivalence between maximum likelihood and sparse DOA estimators

Thomas Aussaguès<sup>a,b,*</sup>, Anne Ferréol<sup>a,b</sup>, Alice Delmer<sup>a</sup>, Pascal Larzabal<sup>b</sup>

ARTICLE INFO

Keywords:

DOA estimation

Sparse estimation

Regularization parameter

Maximum likelihood

A B S T R A C T

Sparse Direction-of-Arrival estimators depend on the regularization parameter λ, which is often empirically tuned. In this work, conducted under the vectorized covariance matrix model, we are looking for theoretical equivalence between the Maximum Likelihood (ML) and sparse estimators. We show that under mild conditions, λ can be chosen thanks to the distribution of the minimum of the ML criterion in the case of two impinging sources. We derive this distribution under complex non-circular Gaussian noise. The corresponding λ choice is 0-invariant, only requiring an upper bound on the number of sources. Furthermore, it guarantees the global minimum of the sparse / 0-regularized criterion to be the ML solution.

Numerical experiments confirm that, for the proposed λ, sparse and ML estimators yield the same statistical performance.

Video to this article can be found online at https://doi.org/10.1016/j.sctalk.2025.100449.

<footnote><sup>a</sup> THALES, 4 avenue des Louvresses, Gennevilliers, France</footnote>

<footnote><sup>b</sup> SATIE, UMR CNRS 8029, 4 avenue des sciences, Gif-sur-Yvette, France</footnote>

<footnote>* Corresponding author at: THALES, 4 avenue des Louvresses, Gennevilliers, France.
E-mail address: thomas.aussagues@ens-paris-saclay.fr (T. Aussaguès).</footnote>
Figures and tables

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Example of DoA estimation with two impinging sources of directions $\theta_1$ and $\theta_2$. The array is located on an airplane. The objective is to compute estimates (respectively denoted $\hat{\theta}_1$ and $\hat{\theta}_2$) of the true directions.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Illustration of the incoming wavefront from direction $\theta_1$ on a 3-elements array.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** 6-element array (with 5 antennas distributed around a circle of radius $0.8\lambda_0$ and one central antenna) and the corresponding virtual array.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Sparse representation of the vectorized array covariance matrix observation **r** using the vectors contained within the dictionary matrix **B**. 3 sources of directions $\theta_1, \theta_3, \theta_3$ with corresponding power $\gamma_{\theta_1}, \gamma_{\theta_2}, \gamma_{\theta_3}$ are considered such that $\mathbf{r} = \mathbf{B}(\theta)\gamma_{\theta} + \delta$ where $\mathbf{B}(\theta) = [\mathbf{b}(\theta_1), \mathbf{b}(\theta_2), \mathbf{b}(\theta_3)]$ is the virtual array steering matrix with $\mathbf{b}(\theta_m)$ the steering vector in direction $\theta_m, \gamma_{\theta} = [\gamma_{\theta_1}, \gamma_{\theta_2}, \gamma_{\theta_3}]^T$ is the sources powers vector and $\delta$ a noise vector due to error caused by the finite number of samples used to estimate the array covariance matrix. Assuming that the sources directions $\theta_1, \theta_3, \theta_3$ lie within a pre-defined grid of directions $\varphi = \{\varphi_1, \varphi_2, \dots, \varphi_G\}$, the previous model can be rewritten using the dictionary matrix $(\mathbf{B}(\varphi) = [\mathbf{b}(\varphi_1), \mathbf{b}(\varphi_2), \dots, \mathbf{b}(\varphi_G)]$ as $\mathbf{r} = \mathbf{B}(\varphi)\gamma_0 + \delta$ where $\gamma_0$ is a sparse vector having only 3 non-null components associated to the 3 sources.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Illustration of role of the regularization parameter $\lambda$ for a one-dimensional criterion. Our objective is to select $\lambda$ such that the global minimizer of the $l_0$ criterion, denoted $\mathcal{J}_{l_0}$, corresponds to the ML global minimizer $\gamma_{\text{ML}}$. On the left subfigure, $\lambda$ is too high leading to a global minimizer located in 0. On the right subfigure, the choice of $\lambda$ is adequate as the global minimizer of $\mathcal{J}_{l_0}$ is the ML global minimizer.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** (Top subfigure) Cumulative Distribution Functions of the interval bounds $\lambda_-$, $\lambda_+$ and the minimum of the maximum likelihood criterion. (Bottom subfigure) Root Mean Square Error (RMSE) as a function of the Signal to Noise Ratio (SNR) for direction $\theta_1$. All subfigures are obtained for a scenario with two sources of directions $\theta_1 = 180^\circ$ and $\theta_2 = 200^\circ$ with $K = 200$ snapshots. The top subfigure displays the CDF of the interval bounds $\lambda_-$, $\lambda_+$ and the minimum of the maximum likelihood criterion for SNR = 10 dB. It shows that the minimum of the ML criterion belongs to the interval $[\lambda_- , \lambda_+]$ thus enabling equivalence between sparse and ML DoA estimators. The bottom subfigure presents the performance of the MUSIC method, the ML and the sparse estimator for which we choose $\lambda_+$ using the distribution of the minimum of the ML criterion. The equivalence between sparse and ML DoA estimators is verified as they both have the performance for SNR $\ge -10$ dB. Furthermore, they both outperform MUSIC in the low SNR region.</figcaption></figure>

## Funding

This research was funded by Agence Nationale de la Recherche et de la Technologie (2023/0662).

## CRediT authorship contribution statement

**Thomas Aussaguès:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Anne Ferréol:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. **Alice Delmer:** Writing – review & editing, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. **Pascal Larzabal:** Writing – review & editing, Validation, Supervision, Methodology, Conceptualization.

## Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

No data was used for the research described in the article.

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