# Whitening Effects for ML-DoA Estimation using a Sparse Representation of Array Covariance (poster)

Thomas Aussaguès, Anne Ferréol, Alice Delmer, Pascal Larzabal

► To cite this version:

Thomas Aussaguès, Anne Ferréol, Alice Delmer, Pascal Larzabal. Whitening Effects for ML-DoA Estimation using a Sparse Representation of Array Covariance (poster). IEEE 50th International Conference on Acoustics, Speech and Signal Processing (ICASSP 2025), Apr 2025, Hyderabad, India. IEEE, ICASSP 2025 - 2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). hal-05032832

HAL Id: hal-05032832

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Submitted on 14 Apr 2025

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# Whitening Effects for ML-DoA Estimation using a Sparse Representation of Array Covariance

Thomas Aussaguès ¹,², Anne Ferréol¹,², Alice Delmer ¹,² & Pascal Larzabal ¹,²

¹Thales, ²SATIE, Université Paris-Saclay, UMR CNRS 8029

## 2025 IEEE International Conference on
Acoustics, Speech and Signal Processing

## April 06 - 11, 2025 Hyderabad, India

### 1. WHY DIRECTION-OF-ARRIVAL ESTIMATION?

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

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

* 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

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

* High number of identifiable sources

* Enhanced performances in severe scenarios

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

<figure><img src="image_4.png" /><figcaption>Source 1</figcaption></figure>

<caption>Source 2</caption>

### Contributions

The pre-whitening is shown to:

* Significantly improve the sparse problem conditioning

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

### 3. SPARSE ESTIMATION PROBLEM

#### Sparse modeling

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

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

* **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

X $G \gg N^2$: 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 $\Psi$:** Continuous Exact $\ell_0$ CEL0 [4] ($\ell_0$-norm approximation)

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

* $\mathcal{J}_{\Psi}$ is minimized using the *Forward-Backward Splitting* algorithm [5] applied to the CEL0 penalty

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

### 2. 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

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

Emitted signals

Circular Gaussian noise

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

$\mathcal{CN}(0, \sigma^2\mathbf{I}_N)$

$\mathbf{x}(t) = \mathbf{A}(\Theta)\mathbf{s}(t) + \mathbf{n}(t)$
with covariance matrix $\mathbf{R}_x = \mathbf{A}(\Theta)\mathbf{R}_s\mathbf{A}(\Theta) + \sigma^2\mathbf{I}_N$

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

**The vectorized covariance matrix model [2] (for uncorrelated sources):**

Virtual array steering matrix

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

Sources powers vector :

$\gamma_{\Theta} = \text{diag}(\mathbf{R}_s)$

Non-white Gaussian

noise $\mathcal{CN}(0, \Gamma)$

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

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

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

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

**Equivalence between sparse and ML DoA estimators:** obtained under *white Gaussian noise* [3]

<figure><img src="image_12.png" /></figure>

**Pre-whitening: correlated Gaussian noise $\mathcal{CN}(0, \Gamma) \rightarrow$ white Gaussian noise $\mathcal{CN}(0, \mathbf{I}_{N^2})$**

$$y = \mathbf{W}\mathbf{r} = \mathbf{B}_w(\Theta)\gamma_{\Theta} + \delta_w \quad \text{with} \quad \mathbf{W} = \Gamma^{-1/2}, \mathbf{B}_w(\Theta) = \mathbf{W}\mathbf{B}(\Theta), \delta_w = \mathbf{W}\delta$$

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

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

### 4. WHITENING EFFECTS ON THE DICTIONARY MATRIX

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

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

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

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

After the pre-whitening:

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

**Outperforms** both MUSIC and the non-whitened estimator

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

**Achieves the CRLB**

$$r_E(\varphi_i, \varphi_j) = \frac{|\mathbf{e}^H(\varphi_i)\mathbf{e}(\varphi_j)|}{\|\mathbf{e}(\varphi_i)\|_2 \|\mathbf{e}(\varphi_j)\|_2}$$

* 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_18.png" /></figure>

The pre-whitening:

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

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

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

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

<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.</footnote>

<footnote>[2] 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.</footnote>

<footnote>[3] Thomas Aussaguès et al. "Looking for Equivalence between Maximum Likelihood and Sparse DOA Estimators". In: 2024 32th European Signal Processing Conference (EUSIPCO). 2024.</footnote>

<footnote>[4] 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).</footnote>

<footnote>[5] Patrick L. Combettes and Jean-Christophe Pesquet. *Proximal Splitting Methods in Signal Processing*. 2010. arXiv: 0912.3522 [math.OC].</footnote>