SPATIALLY FILTERED SPARSE BAYESIAN
LEARNING FOR DIRECTION-OF-ARRIVAL
ESTIMATION WITH LEAKY-WAVE ANTENNAS

Rida Maydani, Yide Wang, Julien Sarrazin, Biyun Ma

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Rida Maydani, Yide Wang, Julien Sarrazin, Biyun Ma. SPATIALLY FILTERED SPARSE BAYESIAN LEARNING FOR DIRECTION-OF-ARRIVAL ESTIMATION WITH LEAKY-WAVE ANTENNAS. International Conference on Acoustics, Speech, and Signal Processing (ICASSP), IEEE, May 2026, Barcelone, Spain. Paper #5836. hal-05471983

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
SPATIALLY FILTERED SPARSE BAYESIAN LEARNING FOR DIRECTION-OF-ARRIVAL
ESTIMATION WITH LEAKY-WAVE ANTENNAS

R.Maydani¹ and Y. Wang¹, J. Sarrazin², B. Ma³,

¹Nantes Université, CNRS, IETR, UMR 6164, F-44000 Nantes, France

²Université Paris-Saclay, CentraleSupélec, CNRS, GeePs, 91192, Gif-sur-Yvette, France

³ School of Electronics and Information Engineering, South China University of Technology,
510640 Guangzhou, China

ABSTRACT

Direction-of-arrival (DoA) estimation with leaky-wave antennas (LWAs) offers a compact and cost-effective alternative to conventional antenna arrays but remains challenging in the presence of coherent sources. To address this issue, we propose a spatially filtered sparse Bayesian learning (SF-SBL) framework. Firstly, the field of view (FoV) is divided into angular sectors according to the frequency beam-scanning property of LWAs, and Bayesian inverse problems are then solved within each sector to improve efficiency and reduce computational cost. Both on-grid SBL and off-grid SBL formulations are developed. Simulation results show that the proposed approach achieves robust and accurate DoA estimation, even with coherent sources.

***Index Terms***— Leaky-wave antennas, Direction-of-arrival estimation, Spatial filtering, On-grid SBL, Off-grid SBL

# 1. INTRODUCTION

Direction-of-arrival (DoA) estimation is essential in radar, wireless communications, and sensing, but conventional implementations typically require many radio-frequency (RF) channels, which increases hardware cost and system complexity [1, 2]. Frequency beam-scanning leaky-wave antennas (LWAs) [3] provide a hardware-efficient alternative by generating frequency-dependent beams without phase shifters or complex feeding networks. In particular, single-beam LWAs can cover a wide field of view (FoV) with a single aperture, making them attractive for compact DoA estimation systems [4, 5, 6].

Since their introduction in DoA estimation [7], LWAs have attracted growing attention as cost-effective candidates for high-resolution angle finding. For single-beam LWAs, subspace-based algorithms such as MUSIC can provide accurate estimation for incoherent sources [8, 9]. However, they fail to handle coherent sources because the rank of the source covariance matrix collapses, and the LWA steering matrix does not exhibit a Vandermonde structure [10]. This prevents

the use of traditional techniques such as spatial smoothing (SSP) to restore the rank. More recently, the Spatially Filtered Interpolation (SFI) method has been introduced [10], which sectorizes the FoV according to the LWA frequency beam-scanning capability and applies the spatial filtering (SF) method to suppress out-of-sector interference before interpolation. While effective, SFI still suffers from numerical instability. These limitations have motivated alternative strategies, and recent work by Zhu et al. [11] has highlighted the potential of Bayesian learning approaches for LWA-based DoA estimation.

Sparse Bayesian Learning (SBL) has emerged as a robust alternative for DoA estimation, since it does not rely on a Vandermonde model and can resolve multiple closely spaced or coherent sources even with limited snapshots [12, 13]. For LWAs, SBL directly processes the frequency-dependent steering response [11]. Two main variants are typically used, on-grid SBL, which estimates source activity on a fixed angular grid, and off-grid SBL, which improves accuracy by modeling small angular deviations through a first-order Taylor expansion of the steering vector [14, 15]. In contrast, gridless approaches such as atomic norm minimization remain inapplicable to LWAs because they require a Vandermonde structure [12].

Building on these advances, this work proposes a sector-wise SF-based method that integrates on-grid and off-grid SBL for DoA estimation with single-beam LWAs. The FoV is divided into angular sectors based on the frequency-angle mapping, and only the frequency components whose main beams fall within a sector are selected, which reduces basis mismatch and computational load. Subsequently, in each sector, on-grid SBL estimates the source directions, while an off-grid refinement step compensates for residual grid mismatch to enhance resolution. The method combines the robustness of SBL with the hardware efficiency of LWAs, enabling accurate coherent-source estimation without interpolation or subspace decomposition.

The paper is organized as follows. Section 2 introduces the LWA model, while Section 3 presents the system model.
The proposed SF-based method integrating on-grid and off-grid SBL is described in Section 4. Simulation results are reported in Section 5, and finally, conclusions are drawn in Section 6.

## 2. LEAKY-WAVE ANTENNA MODEL

In this work, we consider a 1D periodic unidirectional leaky-wave waveguide of physical length $l_a$ and spatial period $p$. The LWA supports a complex propagation mode that radiates energy into free space, producing a far-field pattern that varies with frequency. The far-field response of the LWA at frequency $f$ and observation angle $\theta$ is modeled as [3]:

$$a_f(\theta) = l_a e^{-j(k_z - k_0 \sin \theta)^{l_a/2}} \text{sinc} \left[ (k_z - k_0 \sin \theta)^{l_a/2} \right], \quad (1)$$

where $k_0 = \frac{2\pi f}{c}$ is the free-space wavenumber, $c$ is the speed of light, and $k_z = \beta - j\alpha$ is the complex wavenumber along the waveguide. Here, $\alpha$ accounts for leakage loss (i.e., radiation), and $\beta$ represents the guided-mode phase constant. Assuming the modulation of the waveguide generates a single fast spatial harmonic (typically the -1st), the guided-mode phase constant can be expressed as:

$$\beta = \beta_0 - \frac{2\pi}{p}, \qquad (2)$$

where $\beta_0$ is the unperturbed phase constant of the dielectric-filled waveguide. Under fundamental mode operation, $\beta_0$ can be approximated by:

$$\beta_0 = k_0 \sqrt{\epsilon_r} \sqrt{1 - (f_c/f)^2}, \qquad (3)$$

with $\epsilon_r$ denoting the relative permittivity of the dielectric medium and $f_c = \frac{c}{2W_a\sqrt{\epsilon_r}}$, representing the waveguide's cutoff frequency, determined by its width $W_g$.

For radiation to occur, the condition $-k_0 \le \beta \le k_0$ must hold. The main beam direction as a function of frequency is then given by [3]:

$$\theta_0(f) = \sin^{-1} \left( \frac{\beta}{k_0} \right), \qquad (4)$$

showing how the radiation angle $\theta_0$ varies with frequency. This frequency-to-angle mapping enables beam steering over a wide field of view using a single physical antenna element.

In this study, the LWA is designed with a relative permittivity $\epsilon_r = 10.2$, waveguide width $W_g = 2.1$ mm, modulation period $p = 5.5$ mm, total length $l_a = 20$ cm, and leakage factor $\alpha/k_0 = 0.01$. These parameters enable full angular scanning from -90° to +90° as the frequency sweeps from $f_{min} = 24$ GHz to $f_{max} = 27.29$ GHz. The 3-dB beamwidth is approximately 10° near broadside and gradually broadens toward end-fire. This frequency-dependent beam-steering property forms the foundation of the SF-based method described in the following sections.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Normalized radiation pattern of the LWA.</figcaption></figure>

## 3. SIGNAL MODEL AND PROBLEM FORMULATION

Consider $K$ narrowband plane wave signals impinging on the LWA from distinct directions $\theta_k$, where $k = 1, \dots, K$. Each source transmits a multi-carrier signal (e.g., OFDM) spanning $N$ uniformly spaced frequency points between $f_{min}$ and $f_{max}$. We assume the source amplitude is constant across subcarriers, i.e., $s_{k_n} = s_k$ for all $n$ and fixed $k$. The frequency vector is defined as:

$$\mathbf{f} = [f_{\min}, f_{\min} + \Delta f, \dots, f_{\max}]^T, \qquad (5)$$

where $\Delta f = \frac{f_{max} - f_{min}}{N-1}$ and $f_n$ is the $n^{th}$ element of the frequency vector $\mathbf{f}$. The following frequency-domain system model can be used to express the received signal [16]:

$$\mathbf{y}[t] = \mathbf{A}\mathbf{x}[t] + \mathbf{n}[t], \qquad (6)$$

where $\mathbf{y}[t] \in \mathbb{C}^{N \times 1}$ is the received data vector, with $t = 1, 2, \dots, T$ denoting the $t$-th snapshot among $T$. The steering matrix is $\mathbf{A} = [\mathbf{a}(\theta_1), \dots, \mathbf{a}(\theta_K)] \in \mathbb{C}^{N \times K}$, with steering vectors $\mathbf{a}(\theta) = [a_{f_1}(\theta), a_{f_2}(\theta), \dots, a_{f_N}(\theta)]^T$, where $a_{f_n}(\theta)$ is the LWA response at frequency $f_n$ given in (1). The source vector is $\mathbf{x}[t] \in \mathbb{C}^{K \times 1}$, and the additive white Gaussian noise (AWGN) vector is $\mathbf{n}[t] \in \mathbb{C}^{N \times 1}$ with independent entries, zero mean, and covariance $\sigma^2 \mathbf{I}_N$. The columns of $\mathbf{A}$ represent the LWA responses to incoming plane waves at directions $\theta_i$, $i = 1, \dots, K$. We assume $N > K$.

The covariance matrix of the received signal vector (6) can be written as:

$$\mathbf{R} = E[\mathbf{y}\mathbf{y}^H] = \mathbf{A}\mathbf{R}_x\mathbf{A}^H + \sigma^2\mathbf{I}_N, \qquad (7)$$

For coherent sources, subspace-based methods fail due to rank deficiency and the non-Vandermonde structure of the LWA response [10]. SBL methods can overcome these issues by directly processing the LWA steering matrix [12], except gridless SBL methods, which still require a Vandermonde structure. Consequently, we propose a sector-wise SF-based method combined with SBL to enable robust DOA estimation.

## 4. SBL WITH SPATIALLY-FILTERED

Following [10], we propose an SF-SBL framework that exploits the LWA frequency-angle mapping to divide the
FoV into L angular sectors indexed by $\ell = 1, \dots, L$. For each sector, a set of P candidate DoAs is defined as $\Theta^{(\ell)} = [\theta_1^{(\ell)}, \theta_1^{(\ell)} + \Delta\theta^{(\ell)}, \dots, \theta_P^{(\ell)}]$, where $\Delta\theta^{(\ell)}$ is the angular step in sector $\ell$. To spatially filter out sources outside sector $\ell$, a frequency selection is applied by exploiting the LWA frequency-angle mapping. Specifically, a subset of $N_s$ frequency samples is extracted from the received vector y as

$$ \mathbf{y}_s^{(\ell)} = [y_{i^{(\ell)}}, \dots, y_{i^{(\ell)} + N_s - 1}]^T, \quad (8) $$

where $i^{(\ell)}$ is the $i^{(\ell)}$-th frequency component $f_{i^{(\ell)}}$ that corresponds to the LWA beam direction $\theta_1^{(\ell)}$ within sector $(\ell)$. And, the frequency component $f_{i^{(\ell)} + N_s - 1}$ corresponds to the LWA beam direction $\theta_P^{(\ell)}$, based on the relation given in (4).

## 4.1. Sector width selection

The 3-dB beamwidth of the LWA at frequency $f$ can be approximated by [17, 18]

$$ B_{\theta}(f) \approx \frac{180}{\pi} \frac{\lambda(f)}{l_{\alpha} \cos \theta_{0}(f)}, \quad (9) $$

where $\lambda(f) = c/f$ is the wavelength, $l_\alpha$ is the aperture length, and $\theta_0(f)$ is the main beam direction. The maximum beamwidth across the band is

$$ B_{\theta, \max} = \max_{f} B_{\theta}(f). \quad (10) $$

To ensure that each sector fully contains the main lobe at all relevant frequencies, the sector width $w_\theta$ is chosen such that

$$ w_{\theta} \geq B_{\theta, \max}. \quad (11) $$

This condition ensures that the main-lobe energy of each frequency is fully contained within a single sector, avoiding leakage across adjacent sectors. In practice, we use

$$ w_{\theta} = \gamma B_{\theta, \max}, \quad (12) $$

with $\gamma > 1$ a design parameter trading robustness against complexity.

## 4.2. SF-based On-Grid SBL

After applying spatial filtering, we adopt the SBL framework. In this section, we apply the on-grid SBL method to each sector, where the data model is given by:

$$ \mathbf{Y}_s^{(\ell)} = \mathbf{A}_s^{(\ell)} \mathbf{X}^{(\ell)} + \mathbf{N}^{(\ell)}, \quad (13) $$

where $\mathbf{Y}_s^{(\ell)} \in \mathbb{C}^{N_s \times T}$ contains the $N_s$ frequency samples of sector $\ell$ across $T$ snapshots, $\mathbf{A}_s^{(\ell)} \in \mathbb{C}^{N_s \times P}$ is the steering matrix, $\mathbf{X}^{(\ell)} \in \mathbb{C}^{P \times T}$ is the row-sparse source matrix, and $\mathbf{N}^{(\ell)}$ is noise. Each row $\mathbf{x}_{p,:}^{(\ell)} \sim \mathcal{CN}(0, \gamma_p \mathbf{I}_T)$, with $\gamma_p \ge 0$, while the noise variance $\sigma^2$ is assigned a weak Gamma prior.

The EM updates alternate between posterior inference and hyperparameter refinement. Following [11], the sparsity update is

$$ \gamma_p \leftarrow \frac{1}{2\zeta} \left( \sqrt{T^2 + 4\zeta(\|\mathbf{u}_{p,:}\|_2^2 + T \Sigma_{pp})} - T \right), \quad (14) $$

with $\zeta = 10^{-2}$, $\mathbf{u}_{p,:}$, the $p$th row of $\mathbf{U} = \sigma^{-2}\Sigma\mathbf{A}_s^{(\ell)H}\mathbf{Y}_s^{(\ell)}$, and $\Sigma_{pp}$ the $p$th diagonal element of

$$ \Sigma = \Gamma - \Gamma \mathbf{A}_s^{(\ell)H} \mathbf{C}^{-1} \mathbf{A}_s^{(\ell)} \Gamma, \quad \mathbf{C} = \sigma^2 \mathbf{I}_{N_s} + \mathbf{A}_s^{(\ell)} \Gamma \mathbf{A}_s^{(\ell)H}. $$

The noise variance is updated as

$$ \sigma^2 \leftarrow \frac{\|\mathbf{Y}_s^{(\ell)} - \mathbf{A}_s^{(\ell)} \mathbf{U}\|_F^2 + \sigma^2 T \sum_{p=1}^{P} (1 - \gamma_p^{-1} \Sigma_{pp}) + d}{N_s T + c - 1}, \quad (15) $$

with $(c, d) = (10^{-4}, 10^{-4})$. Iterating (14)–(15) drives inactive rows of $\mathbf{X}^{(\ell)}$ to zero, revealing the DoAs.

## 4.3. SF-based Off-Grid SBL

While the on-grid formulation assumes that the true DoAs coincide with the predefined grid $\{\vartheta_p\}_{p=1}^P$, in practice sources rarely align perfectly, which leads to basis mismatch. To mitigate this, we adopt the off-grid SBL framework [14], in which each direction is parameterized as

$$ \theta_p = \vartheta_p + \beta_p, \quad (16) $$

with small offset $\beta_p$. A first-order Taylor expansion of the steering vector yields

$$ \mathbf{Y}_s^{(\ell)} \approx (\mathbf{A}_s^{(\ell)} + \mathbf{B}_s^{(\ell)} \mathrm{diag}(\beta))\mathbf{X}^{(\ell)} + \mathbf{N}^{(\ell)}, \quad (17) $$

where $\mathbf{B}_s^{(\ell)}$ contains angular derivatives and $\beta = [\beta_1, \dots, \beta_P]^T$.

The EM updates for $\{\gamma_p\}$ and $\sigma^2$ remain as in the on-grid case. The off-grid offsets $\{\beta_p\}$ are then refined following [14] as

$$ \beta_p \leftarrow \frac{\Re\{\mathbf{a}_s^{(\ell)}(\vartheta_p)^H \Sigma \mathbf{b}_s^{(\ell)}(\vartheta_p)\}}{\Re\{\mathbf{b}_s^{(\ell)}(\vartheta_p)^H \Sigma \mathbf{b}_s^{(\ell)}(\vartheta_p)\}}, \quad p = 1, \dots, P, \quad (18) $$

where $\mathbf{a}_s^{(\ell)}(\vartheta_p)$ and $\mathbf{b}_s^{(\ell)}(\vartheta_p)$ are the steering vector and its angular derivative at $\vartheta_p$, and $\Sigma$ is the posterior covariance of the coefficients.

By alternating between hyperparameter updates and $\beta$ refinement, the algorithm converges to sparse estimates $\hat{\beta}_p = \vartheta_p + \hat{\beta}_p$ for all active indices. The complete procedure of the SF-OnGrid-SBL and SF-OffGrid-SBL methods for single beam LWA is outlined in Algorithm 1.

# 5. SIMULATION RESULTS

This section presents the simulation results of the proposed methods. All experiments were carried out on a standard laptop (Intel Core i5, MATLAB implementation). The single-beam LWA parameters are given in Section 2, with the number of frequency samples fixed to $N = 100$. Unless otherwise
**Algorithm 1** Spatially filtered on-/off-grid SBL applied within sector $\ell$

**Require:** Sector $\ell$; $\mathbf{Y}_s^{(\ell)} \in \mathbb{C}^{N_s \times T}$; on-grid steering matrix $\mathbf{A}_s^{(\ell)}$; (off-grid) derivative matrix $\mathbf{B}_s^{(\ell)}$; grid $\{\vartheta_p\}$.

1. Initialize $\gamma_p \leftarrow \gamma_0 > 0$, $\beta_p \leftarrow 0$; noise precision prior $\sigma^{-2} \sim \text{Gamma}(c, d)$ with $c = d = 10^{-4}$ [13].

2. **repeat**

3. **E-step:** form effective dictionary $\mathbf{\Phi}(\beta) \leftarrow \mathbf{A}_s^{(\ell)} + \mathbf{B}_s^{(\ell)} \text{diag}(\beta)$.

4. Update posterior covariance: $\mathbf{\Sigma} \leftarrow (\text{diag}(\gamma)^{-1} + \frac{1}{\sigma^2} \mathbf{\Phi}^H \mathbf{\Phi})^{-1}$.

5. **for** $t = 1, \dots, T$ **do**

6. Compute posterior mean: $\boldsymbol{\mu}_t \leftarrow \frac{1}{\sigma^2} \boldsymbol{\Sigma} \mathbf{\Phi}^H \mathbf{y}_s^{(\ell)} [t]$.

7. **end for**

8. **M-step (sparsity):** update $\gamma_p \leftarrow \frac{1}{T} \sum_{t=1}^T |\mu_{p,t}|^2 + \Sigma_{pp} [12]$.

9. **M-step (noise):** update $\sigma^2$ by evidence maximization with $(c, d) = (10^{-4}, 10^{-4})$ [13].

10. **if** off-grid enabled **then**

11. **Off-grid refinement:** update each $\beta_p$ using the OGSBI linearized step [14].

12. **end if**

13. Prune small $\gamma_p$; optionally merge adjacent active components.

14. **until** convergence

15. **Return:** estimated DoAs $\hat{\theta}_p = \vartheta_p + \hat{\beta}_p$ for active indices $p$; deduplicate across overlapping sectors.

stated, the field of view was set to $\Theta_{\text{FoV}} = \{-90^\circ, -90^\circ + \Delta\theta, \dots, 90^\circ\}$ with a grid step of $\Delta\theta = 1^\circ$, used both for coarse search and refinement. Each run used $T = 100$ snapshots, and performance metrics were averaged over $I = 100$ Monte Carlo trials.

The angular domain was divided into $L = 6$ sectors, each spanning $w_\theta = 30^\circ$. This choice ensures compatibility with the maximum 3-dB beamwidth of the antenna, $B_{\theta,\max} = 10.66^\circ$, and corresponds to $\gamma \approx 2.8$. The frequency samples in each sector were assigned according to the frequency-angle mapping inherent to the LWA.

For benchmarking, four algorithms were considered. The proposed approaches are SF-OnGrid-SBL and SF-OffGrid-SBL. As references, the Cropped-SVD SBL method [11] was applied over the entire FoV, and the SFI+SSP+MUSIC framework [10] was implemented with the same $L = 6$ sectors and $30^\circ$ sector width. In the SFI+SSP+MUSIC setup, the virtual array was defined with $V = N_s = 16$ elements, $P = 14$ virtual angles, and an SSP subarray length of $F = 0.6V$.

Fig. 2 shows the RMSE performance for three coherent sources at DoAs [10.3°, 15.7°, 20.7°] across different SNR levels. The proposed SF-OffGrid-SBL consistently achieves the lowest RMSE, showing strong robustness against grid mismatch. The SF-OnGrid-SBL improves as the SNR in-

<figure><img src="image_3.png" /><figcaption>Fig. 2: RMSE performance versus SNR for three coherent sources at DoAs [10.3°, 15.7°, 20.7°].</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: Computational time of the compared algorithms versus SNR for three coherent sources at DoAs [10.3°, 15.7°, 20.7°].</figcaption></figure>

creases, but its performance remains limited at low SNR. The classical SFI+SSP+MUSIC method provides moderate accuracy, with performance stabilizing at higher SNR values. Finally, the Cropped-SVD SBL baseline yields the worst accuracy among all methods, although it performs reasonably well at high SNR. These results confirm the effectiveness and robustness of the proposed approach across a SNR range.

Fig. 3 reports the computational time versus SNR. The Cropped-SVD SBL method requires significantly higher run-time, mainly due to the full FoV processing and large dictionary size. In contrast, both SF-OnGrid-SBL and SF-OffGrid-SBL methods considerably reduce the execution time by exploiting sector-wise processing. The SFI+SSP+MUSIC method remains the most computationally efficient, although at the cost of reduced estimation accuracy. Overall, the results highlight a favorable trade-off achieved by the proposed SF-OffGrid-SBL, combining high accuracy with competitive computational efficiency.

## 6. CONCLUSION

This paper presents a SF-SBL framework for DoA estimation with single-beam LWAs. By exploiting the frequency-angle mapping, the SF-OnGrid-SBL and SF-OffGrid-SBL reduce complexity and avoid numerical interpolation instabilities while handling coherent sources. Simulations show that SF-OffGrid-SBL achieves the lowest RMSE with competitive runtime, outperforming existing methods. Future work will extend the approach to multi-beam LWAs and validate it on measured antenna data.
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Acknowledgment

This work was supported by the ANR BeSensiCom project (Grant ANR-22-CE25-0002) of the French Agence Nationale de la Recherche, the National Key Research and Development Program of China under Grant 2024YFE0105400, and carried out in the framework of COST Action CA20120 INTERACT.