# Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators (poster)

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 (poster). EUSIPCO 2024 - 32nd European Signal Processing Conference, Aug 2024, Lyon, France. . hal-04669621

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# Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators

Thomas Aussaguès*,†, Anne Ferréol*,†, Alice Delmer* & Pascal Larzabal†
*Thales, †SATIE, Université Paris-Saclay, UMR CNRS 8029

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EUSIPCO 2024

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## ABSTRACT

Sparse Direction-of-Arrival (DOA) estimators depend on the regularization parameter $\lambda$ 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, $\lambda$ 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 $\lambda$ choice is $\theta$-invariant, only requiring an upper bound on the number of sources. Furthermore, it guarantees the global minimum of the sparse $l_0$-regularized criterion to be the ML solution. Numerical experiments confirm that, for the proposed $\lambda$, sparse and ML estimators yield the same statistical performance.

## INTRODUCTION

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

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

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

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

<figure><img src="image_7.png" /></figure>

### Numerous methods have been proposed : Capon's beamformer [2], MUSIC [3], ML [4]:

* MUSIC can not handle coherent sources

* ML is intractable for large number of sources

* Limited number of identifiable sources

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

### The vectorized covariance matrix model

* High number of identifiable sources

* Enhanced performances in severe scenarios

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

### DOA estimation with the vectorized covariance matrix model : sparse methods

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

### Sparse methods rely on the minimization of a regularized criterion parametrized by $\lambda$ an hyperparameter which is often empirically tuned

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

**Contribution:** we propose a novel $\theta$-invariant regularization parameter choice that ensures equivalence between ML and sparse DOA estimators

<figure><img src="image_12.png" /><figcaption>Source 1</figcaption></figure>

<caption>Source 2</caption>

## THE VECTORIZED COVARIANCE MATRIX MODEL

Let $M$ sources of directions $\Theta = \{\theta_1, ..., \theta_M\}$ impinge on an array of $N$ antennas

**Classical model:**

Steering matrix

Emitted signals

$$x(t) = A(\Theta)s(t) + n(t)$$

Circular Gaussian noise
$C\mathcal{N}(0, \sigma^2 I_N)$

with covariance matrix $R_x = A(\Theta)R_sA(\Theta) + \sigma^2 I_N$

* Number of identifiable sources limited to $N-1$ sources

**The vectorized covariance matrix model [5] (for uncorrelated sources):**

Virtual array steering matrix

Sources powers vector :

$\gamma_{\Theta} = \text{diag}(R_s)$

$$r = \text{vec}(R_x - \sigma^2 I) = B(\Theta)\gamma_{\Theta} + \delta$$

Non-Circular Gaussian noise $C\mathcal{N}(0, \Gamma, C)$

* Allows the processing of at most $N^2 - 1$ sources with increased resolution

**Maximum Likelihood estimation on the vectorized covariance matrix model:**

$$\text{Pre-whitening } y = Wr = B_w(\Theta)\gamma_{\Theta} + \delta_w \quad \text{with } W = \Gamma^{-1/2}, B_w(\Theta) = WB(\Theta), \delta_w = W\delta$$

$$\text{ML estimation } \hat{\Theta} = \arg \min_{\Theta} J_{\text{ML}}(\Theta) \quad \text{with} \quad J_{\text{ML}}(\Theta) = y^H \Pi^{\perp}(\Theta) y$$

* Minimizing $J_{\text{ML}}$ requires $M$-dimensional non-convex optimization with numerous local minima

<figure><img src="image_13.png" /></figure>

**Objective** : solve the ML through sparse DOA estimators

## SPARSE MODELING AND SPARSE ESTIMATION

### Sparse modeling

<figure><img src="image_14.png" /></figure>

### Let $\Phi = \{\varphi_1, ..., \varphi_G\}$ a grid of $G$ pre-defined directions such that $\Theta \subset \Phi$

<figure><img src="image_15.png" /></figure>

### $B_w(\Phi) = W[b(\varphi_1), ..., b(\varphi_G)]$ : $N^2 \times G$ matrix containing the virtual array response $b(\varphi_g)$ in each of the $G$ grid directions

<figure><img src="image_16.png" /></figure>

### Sparse vectorized covariance matrix model

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

### DOAs corresponds to the directions $\varphi_G$ of $\gamma_0$ non zeros components

### Sparse estimation

* $G \gg N^2$ : the problem is ill-posed

<figure><img src="image_19.png" /></figure>

### Use of the sparsity prior : regularized problem formulation

<figure><img src="image_20.png" /></figure>

### Penalty function $\Psi : l_0$ norm, Continuous Exact $l_0$ (CEL0) [6]

$$\hat{\gamma} = \arg \min_{\gamma} J_{\Psi}(\gamma, \gamma) \quad \text{with} \quad J_{\Psi}(\gamma, \gamma) = \frac{1}{2} \|y - B_{w}(\Phi)\gamma\|_{2}^{2} + \underbrace{\Psi(\gamma, \gamma)}_{\text{Data fidelity}}$$

<figure><img src="image_21.png" /></figure>

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

$$y = \gamma_{01} \times b(\varphi_1) + \gamma_{02} \times b(\varphi_2) + \gamma_{03} \times b(\varphi_3) + \gamma_{10} \times b(\varphi_{10}) + \gamma_{11} \times b(\varphi_{11}) + \delta_w$$

## REGULARIZATION PARAMETER $\lambda$ CHOICE

<figure><img src="image_22.png" /></figure>

### $\lambda$ balances data fidelity towards the solution sparsity

Example : influence of the regularization parameter on the criterion minima

Contour lines of $J_{l_0}: y = \begin{bmatrix} 1 \\ 0.5 \end{bmatrix}, B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $\gamma_0 = \begin{bmatrix} 0 \\ 0.5 \end{bmatrix}$.

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

**Equivalence:** both ML and sparse criterion have the same global minimizer

<figure><img src="image_25.png" /></figure>

### Delmer's [7] interval $[\lambda^-, \lambda^+]$ ensures the equivalence but the statistics of $\lambda^-, \lambda^+$:

* Are difficult to compute

* Depend on the directions

<figure><img src="image_26.png" /></figure>

**Contributions** [8]:

<figure><img src="image_27.png" /></figure>

* The min. of the ML, $\epsilon$, criterion belongs to the interval $[\lambda^-, \lambda^+]$

<figure><img src="image_28.png" /></figure>

* $\epsilon$ follows, asymptotically, a $\chi_2$ distribution with $N^2 - M$ degrees of freedom

<figure><img src="image_29.png" /></figure>

* $\theta$-invariant distribution

## SIMULATIONS

* The proposed $\lambda$ choice belongs to the interval $[\lambda_-, \lambda_+]$ : it ensures equivalence with the ML

* Numerical simulations confirm that both ML and sparse estimators have the same statistical performance

<figure><img src="image_30.png" /></figure>

## CONCLUSION

* Novel $\theta$-invariant regularization parameter choice

* Ensuring equivalence between sparse and ML DOA estimators

* Efficient implementation of the ML

[1] Hamid Krim and Mats Viberg. "Two Decades of Array Signal Processing Research: The Parametric Approach". In: Signal Processing Magazine, IEEE 13 (Aug. 1996), pp. 67–94.

[2] J. Capon. "High-resolution frequency-wavenumber spectrum analysis". In: Proceedings of the IEEE 57.8 (1969), pp. 1408–1418.

[3] G. Bienvenu and L. Kopp. "Optimality of high resolution array processing using the eigensystem approach". In: IEEE Transactions on Acoustics, Speech, and Signal Processing 31.5 (1983), pp. 1235–1248.

[4] B. Ottersten et al. "Exact and Large Sample Maximum Likelihood Techniques for Parameter Estimation and Detection in Array Processing". In: Radar Array Processing. Ed. by Simon Haykin, John Litva, and Terence J. Shepherd. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993, pp. 99–151.

[5] 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, pp. 1258–1262.

[6] Emmanuel Soubies, Laure Blanc-Féraud, and Gilles Aubert. "A Continuous Exact l0 penalty (CEL0) for least squares regularized problem". In: SIAM Journal on Imaging Sciences 8.3 (July 2015), pp. 1607–1639 (33 pages).

[7] Alice Delmer, Anne Ferréol, and Pascal Larzabal. "On regularization Parameter for L0-Sparse Covariance Fitting Based DOA Estimation". In: ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). 2020.

[8] Thomas Aussaguès et al. "Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators". In: 2024 32nd European Signal Processing Conference (EUSIPCO). 2024.