Efficient Adaptive Matrix Spatial Filter with Nulling

Abstract

Beamspace adaptive matrix spatial filters have been extensively studied for their superior nulling performance and mathematical elegance. However, a major drawback of current spatial filters is the high computational cost—often reaching $\mathcal{O}\left(M^{4.5}\right)$—due to their formulation as second-order cone programming (SOCP) problems that rely on iterative interior-point methods. This paper proposes a robust and efficient matrix filtering framework with adaptive nulling capabilities to suppress interference. The proposed method is formulated as a convex optimization problem that admits a non-iterative, closed-form solution, thereby reducing the complexity to $\mathcal{O}\left(M^3\right)$. Consequently, it can be efficiently implemented on resource-constrained embedded platforms. Furthermore, the algorithm incorporates an explicit passband flatness constraint, which significantly improves compatibility with downstream Direction-of-Arrival (DOA) estimation modules. To achieve even greater efficiency, we introduce a novel dual sequential rank-1 update strategy, further lowering the overall computational complexity to $\mathcal{O}\left(M^2\right)$.

1. Introduction

Beamspace processing [1,2,3,4,5,6,7,8] has emerged as a compelling paradigm in array signal processing, primarily because it facilitates dimensionality reduction and alleviates the computational complexity inherent in DOA estimation algorithms. Moreover, transforming data into the beamspace domain offers substantial benefits in sharpening source resolution, enhancing robustness against array geometry perturbations, and mitigating DOA estimation bias.

Numerous beamspace design methods have been proposed in the literature; however, many overlook the impact of out-of-sector interference. In [9], an adaptive beamforming algorithm with sidelobe control was introduced. Subsequently, ref. [10] proposed a matrix filter with nulling (MFN) capabilities to adaptively suppress out-of-sector signals through deep stopband attenuation. This framework was further refined in [11] by incorporating a column orthogonality constraint, which preserves the white noise characteristics at the preprocessor output. To address potential signal degradation, ref. [12] introduced explicit constraints to minimize passband distortion.

Despite these advancements, a significant limitation persists: these methods [9,10,11,12] typically formulate the nulling adaptation as a second-order cone programming (SOCP) problem, often solved via the SeDuMi toolbox [13] or modern solvers such as MOSEK [14] and SDPT3 [15]. Solving these SOCPs typically necessitates iterative interior-point algorithms with a computational complexity of approximately $\mathcal{O}\left(L{M}^{3.5}\right)$, where M represents the problem dimension and L denotes the iteration count [16,17]. In many practical scenarios, L scales proportionally with the dimension M. Consequently, to facilitate a unified comparison with non-iterative methods based on the array size, the effective complexity is characterized as $\mathcal{O}\left(M^{4.5}\right)$. Furthermore, while [10] achieves deep nulls, it suffers from significant passband ripples—a drawback only partially mitigated by the additional flatness constraints in [12]. Ultimately, the high computational overhead of these iterative solvers significantly restricts their deployment on resource-constrained embedded platforms. Fan et al. [18] alleviated computational complexity by proposing a truncated nuclear norm regularized matrix spatial filter solved via a two-layer APGL algorithm. Chu et al. [19] enhanced robustness against strong interference through covariance tapering and cascaded sparse Bayesian learning, while Wang et al. [20] introduced a SWN-MUSIC framework using covariance square-root inversion for jammer suppression. Zhang et al. [21] further eliminated iterative matrix inversion by integrating an MVDR-DL beamformer with fast sequential sparse Bayesian learning, achieving low-latency, high-resolution wideband DOA estimation under intense interference.

In this paper, we develop an efficient adaptive spatial filtering algorithm. By formulating the matrix filtering with the nulling process as a constrained convex optimization problem, we derive a closed-form solution with a computational complexity of only $\mathcal{O}\left(M^3\right)$. Building on this, we design a novel spatial bandpass filter incorporating simplified constraints to minimize passband ripple. The proposed algorithm is non-iterative, yet it achieves deep nulls in the directions of out-of-sector interferences while maintaining a flat response within the sector of interest. These attributes make it highly suitable for real-time beamspace preprocessing. To further alleviate the computational burden, we introduce a novel dual sequential method that reduces the complexity to $\mathcal{O}\left(M^2\right)$, facilitating its deployment on resource-constrained platforms.

The remainder of this paper is organized as follows. Section 2 formulates the problem and presents the core derivation of the proposed efficient matrix filtering algorithm. In Section 3, the performance of the algorithm is validated and compared with several state-of-the-art methods through extensive simulations. Section 4 demonstrates the applicability of the proposed method on a 2D planar random array using experimental data. Finally, Section 5 concludes the paper and discusses potential directions for future research.

2. Methods

2.1. System Modeling

Consider a uniform linear array (ULA) of M omnidirectional sensors with unit gains. There are S signals of interest and I signals of out-of-sector interference emitted from far-field. The array signals received can be modeled as [1,2,3]:

$$\mathit{x}\left(t\right)={\mathbf{A}}_s\left({\mathit{\theta }}_s\right)\mathbf{s}\left(t\right)+{\mathbf{A}}_i\left({\mathit{\theta }}_i\right)\mathit{i}\left(t\right)+\mathit{n}\left(t\right),$$

(1)

where ${\mathbf{A}}_s\left({\mathit{\theta }}_s\right)\in {\mathbb{C}}^{M\times S}$ denotes the steering matrix corresponding to the signal of interest (SOI) $\mathit{s}\left(t\right)$, and ${\mathit{\theta }}_s={[{\theta }_1,\dots ,{\theta }_S]}^T$ represents the $S\times 1$ vector of their respective directions of arrival (DOAs). The term $\mathit{s}\left(t\right)$ refers to the vector of signal waveforms. Similarly, ${\mathbf{A}}_i\left({\mathit{\theta }}_i\right)\in {\mathbb{C}}^{S\times I}$ is the steering matrix corresponding to I out-of-sector interferences with DOAs defined by ${\mathit{\theta }}_i={[{\theta }_1,\dots ,{\theta }_I]}^T$. The additive measurement noise is represented by the $M\times 1$ vector $\mathit{n}\left(t\right)$. The signal steer matrix is defined by:

$${\mathbf{A}}_s\left({\mathit{\theta }}_s\right)=[\mathit{a}\left({\theta }_{s_1}\right),\dots ,\mathit{a}\left({\theta }_{s_S}\right)],$$

(2)

and the interference steer matrix is formulated by:

$${\mathbf{A}}_i\left({\mathit{\theta }}_i\right)=[\mathit{a}\left({\theta }_{i_1}\right),\dots ,\mathit{a}\left({\theta }_{i_S}\right)],$$

(3)

where the steer vector is defined as:

$$\mathit{a}\left(\theta \right)={\displaystyle \frac{1}{M}}{\left[1,e^{\mathrm{j}{\displaystyle \frac{2\pi f}{c}}dsin\theta },\dots ,e^{\mathrm{j}{\displaystyle \frac{2\pi f}{c}}(M-1)dsin\theta }\right]}^T,$$

(4)

and $\theta$ denotes the direction of arrival relative to the array broadside, f is the frequency, d represents the inter-element spacing, c is the propagation speed of wave, and the operator ${[·]}^T$ indicates the matrix transpose. Furthermore, we assume that the desired signals, interferences and noise are mutually uncorrelated. The noise is modeled as a zero-mean spatially white Gaussian process with variance ${\sigma }^2$. The cross-spectrum matrix (CSM), denoted by ${\mathbf{R}}_{\mathit{x}}$, is defined as:

$$\begin{array}{cc}\hfill {\mathbf{R}}_x& \triangleq \mathbb{E}\left\{\mathit{x}\left(t\right){\mathit{x}}^H\left(t\right)\right\}\hfill \\ \hfill & ={\mathbf{A}}_s\left({\mathit{\theta }}_s\right){\mathbf{R}}_s{\mathbf{A}}_s^H\left({\mathit{\theta }}_s\right)+{\mathbf{A}}_i\left({\mathit{\theta }}_i\right){\mathbf{R}}_i{\mathbf{A}}_i^H\left({\mathit{\theta }}_i\right)+{\sigma }^2\mathbf{I},\hfill \end{array}$$

(5)

where ${\mathbf{R}}_s\triangleq \mathbb{E}\left\{\mathit{s}\left(t\right){\mathit{s}}^H\left(t\right)\right\}$ and ${\mathbf{R}}_i\triangleq \mathbb{E}\left\{\mathit{i}\left(t\right){\mathit{i}}^H\left(t\right)\right\}$ represent the source covariance matrix and the interference covariance matrix, respectively. Here, $\mathbf{I}$ denotes the identity matrix, $\mathbb{E}\{·\}$ is the statistical expectation operator, and ${[·]}^H$ signifies the conjugate transpose (Hermitian) operator. In practical scenarios, the CSM is typically estimated from N available snapshots as:

$${\widehat{\mathbf{R}}}_x={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^N}\mathit{x}\left(t\right){\mathit{x}}^H\left(t\right),$$

(6)

where N represents the number of snapshots utilized for the sample covariance estimation.

2.2. Beamspace Matrix Spatial Filtering

A beamspace matrix $\mathbf{B}\in {\mathbb{C}}^{M\times M_r}$ serves as a linear operator that transforms the original element-space snapshot vector $\mathbf{x}\left(t\right)$ into a reduced-dimension beamspace vector $\mathbf{y}\left(t\right)$, defined as:

$$\mathit{y}\left(t\right)={\mathbf{B}}^H\mathit{x}\left(t\right),$$

(7)

where the number of beamspace channels $M_r$ is typically much smaller than the number of physical sensors M (i.e., $M_r<M$). By performing direction-of-arrival (DOA) estimation on the lower-dimensional vector $\mathbf{y}\left(t\right)$, the overall computational efficiency of the algorithm is significantly enhanced without compromising the essential spatial information.

The objective is to design a beamspace-based spatial filter capable of separating desired signals from interferences through a precisely synthesized transformation matrix $\mathbf{B}$, as explored in [1,2,3,4,5,6,7,8]. Ideally, the signals of interest are preserved within the filter’s passband, while interferences are suppressed in the stopband. However, these conventional data-independent approaches often fail to account for the dynamic characteristics of the received data. Consequently, in scenarios featuring strong interferences, the spatial filtering performance deteriorates significantly, leading to insufficient interference rejection and potential signal distortion.

In this section, we formulate the matrix filter design as a convex optimization problem. First, a quiescent matrix filter is synthesized using the discrete prolate spheroidal sequence (DPSS) method [1,2]. To characterize the spatial energy concentration within the passband ${\mathrm{\Theta }}_p$, we construct a positive-definite matrix $\mathbf{C}$ as follows:

$$\mathbf{C}={\int }_{{\mathrm{\Theta }}_p}\mathit{a}\left(\theta \right){\mathit{a}}^H\left(\theta \right)d\theta ,$$

(8)

where $\mathit{a}\left(\theta \right)$ represents the array steering vector corresponding to the spatial angle $\theta$. The beamspace transformation matrix $\mathbf{B}\in {\mathbb{C}}^{M\times M_r}$ is formed by the $M_r$ principal eigenvectors of $\mathbf{C}$:

$$\mathbf{B}=[{\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_{M_r}],$$

(9)

where ${\mathit{v}}_m$ denotes the eigenvector associated with the m-th largest eigenvalue. This selection ensures that the spatial power within the passband is maximally preserved in the beamspace domain. The convex optimization problem can be formulated as:

$$\underset{\mathbf{B}}{min}Tr\left\{{\mathbf{B}}^H{\mathbf{R}}_x\mathbf{B}\right\}+\alpha {∥\mathbf{B}-{\mathbf{B}}_q∥}_F^2,$$

(10)

where $Tr\{·\}$ denotes the matrix trace operator, ${∥·∥}_F$ represents the Frobenius norm, and $\alpha$ is the hyperparameter to control the weights of constraints. The output power of the beamspace processor is characterized by the first term of (10), expressed as:

$$Tr\left\{{\mathbf{B}}^H{\mathbf{R}}_x\mathbf{B}\right\}=Tr\left\{\mathbb{E}\left\{\mathit{y}\left(t\right){\mathit{y}}^H\left(t\right)\right\}\right\}=\mathbb{E}\left\{{\mathit{y}}^H\left(t\right)\mathit{y}\left(t\right)\right\}.$$

The beamspace matrix $\mathbf{B}$ is further constrained by the second term of (10), which regulates the spatial filter’s gain within the sector of interest (SOI). For instance, in the design of a quiescent beamspace matrix, the passband gain is typically normalized to unity [12] to ensure signal preservation.

As (10) is formulated as an unconstrained convex problem, a closed-form solution exists. In the following, we delineate the derivation process.

The straightforward idea is to take the gradient of the objective function

$$f\left(\mathbf{B}\right)=Tr\left\{{\mathbf{B}}^H{\mathbf{R}}_x\mathbf{B}\right\}+\alpha {∥\mathbf{B}-{\mathbf{B}}_q∥}_F^2,$$

(11)

and set it to zero: ${\nabla }_{{\mathbf{B}}^{*}}f\left(\mathbf{B}\right)=\mathbf{0}$.

Rather than employing Wirtinger derivatives, we simplify the derivation by transforming the original problem. The objective function is thus rewritten as follows:

$$f\left(\mathit{b}\right)=\parallel ({\mathbf{G}}_{\mathit{x}}^T\otimes \mathbf{I}){\mathit{b}\parallel }_2^2+\alpha {\parallel \mathit{b}-{\mathit{b}}_q\parallel }_2^2.$$

(12)

Here ${\mathbf{G}}_{\mathit{x}}\triangleq {\mathbf{R}}_{\mathit{x}}^{1/2}\in {\mathbb{C}}^{M\times M}$ is the Hermitian square root of the covariance matrix. The vectorized forms are defined as $\mathit{b}\triangleq vec\left\{{\mathbf{B}}^H\right\}$ and ${\mathit{b}}_q\triangleq vec\left\{{\mathbf{B}}_q^H\right\}$. Furthermore, ⊗, ${\parallel ·\parallel }_2$, and $vec\{·\}$ denote the Kronecker product, the Euclidean norm, and the column-wise vectorization operator, respectively. The following identity is employed in the derivation from (11) to (12):

$$vec\left\{\mathbf{XYZ}\right\}=({\mathbf{Z}}^T\otimes \mathbf{X})vec\left\{\mathbf{Y}\right\},$$

(13)

where $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ are matrices of compatible dimensions.

Setting the gradient of $f\left(\mathit{b}\right)$ in (12) with respect to $\mathit{b}$ to zero yields the following closed-form solution:

$$\mathit{b}=\alpha {\left({({\mathbf{G}}_x^T\otimes {\mathbf{I}}_{M_r})}^H({\mathbf{G}}_x^T\otimes {\mathbf{I}}_{M_r})+\alpha {\mathbf{I}}_{M{M}_r}\right)}^{-1}{\mathit{b}}_q.$$

(14)

The solution in (14) requires the inversion of an $\left(M{M}_r\right)\times \left(M{M}_r\right)$ matrix. To reduce the computational burden and provide a more efficient implementation, the expression is reformulated as:

$$\begin{array}{cc}\hfill {\left({({\mathbf{G}}_{\mathit{x}}^T\otimes {\mathbf{I}}_{M_r})}^H({\mathbf{G}}_{\mathit{x}}^T\otimes {\mathbf{I}}_{M_r})+\alpha {\mathbf{I}}_{M{M}_r}\right)}^{-1}& ={\left(\left({{\mathbf{G}}_{\mathit{x}}^T}^H{\mathbf{G}}_{\mathit{x}}^T\right)\otimes {\mathbf{I}}_{M_r})+\alpha {\mathbf{I}}_{M{M}_r}\right)}^{-1}\hfill \\ \hfill & ={\left({\mathbf{R}}_{\mathit{x}}^T\otimes {\mathbf{I}}_{M_r}+\alpha {\mathbf{I}}_{M{M}_r}\right)}^{-1}\hfill \\ \hfill & ={({\mathbf{R}}_{\mathit{x}}^T+\alpha {\mathbf{I}}_M)}^{-1}\otimes {\mathbf{I}}_{M_r},\hfill \end{array}$$

(15)

note that the loaded matrix ${\mathbf{R}}_{\mathit{x}}^T+\alpha {\mathbf{I}}_M$ is guaranteed to be non-singular (positive definite) because ${\mathbf{R}}_{\mathit{x}}^T$ is Hermitian positive semi-definite.

Substituting (15) into (14) yields:

$$\begin{array}{cc}\hfill \mathit{b}& =vec\left\{{\mathbf{B}}^H\right\}\hfill \\ \hfill & =vec\left\{\alpha {\mathbf{B}}_q^H{\left[{({\mathbf{R}}_{\mathit{x}}^T+\alpha {\mathbf{I}}_M)}^{-1}\right]}^T\right\},\hfill \end{array}$$

(16)

which can be further simplified to:

$$\mathbf{B}={\left\{\alpha {\mathbf{B}}_q^H{\left[{({\mathbf{R}}_{\mathit{x}}^T+\alpha {\mathbf{I}}_M)}^{-1}\right]}^T\right\}}^H=\alpha {({\mathbf{R}}_{\mathit{x}}+\alpha {\mathbf{I}}_M)}^{-1}{\mathbf{B}}_q.$$

(17)

To further improve stopband attenuation, an additional regularization term is introduced to the objective function to suppress the out-of-sector response. The reformulated problem is given by:

$$\underset{\mathbf{B}}{min}Tr\left\{{\mathbf{B}}^H{\mathbf{R}}_{\mathit{x}}\mathbf{B}\right\}+\alpha {∥\mathbf{B}-{\mathbf{B}}_q∥}_F^2+\beta {\parallel {\mathbf{B}}^H{\mathbf{A}}_s\parallel }_F^2,$$

(18)

where ${\mathbf{A}}_s\in {\mathbb{C}}^{M\times s}$ is the steering matrix formed by s steering vectors uniformly sampled across the stopband, and $\beta$ denotes the regularization weight.

Following a similar derivation as in (17), the corresponding solution is obtained as:

$$\mathbf{B}={\left\{\alpha {\mathbf{B}}_q^H{\left[{({\mathbf{R}}_{\mathit{x}}^T+\alpha {\mathbf{I}}_M+\beta \mathbf{S})}^{-1}\right]}^T\right\}}^H=\alpha {({\mathbf{R}}_{\mathit{x}}+\alpha {\mathbf{I}}_M+\beta \mathbf{S})}^{-1}{\mathbf{B}}_q,$$

(19)

where $\mathbf{S}\triangleq {\left({\mathbf{A}}_s{\mathbf{A}}_s^H\right)}^T$ is a structured matrix related to the stopband steering manifold.

Unlike iterative methods, the proposed algorithm yields a closed-form solution with a computational complexity of only $\mathcal{O}\left(M^3\right)$. This deterministic efficiency makes the approach highly suitable for resource-constrained embedded platforms where real-time processing is critical.

2.3. Further Reducing Computational Complexity via the Dual-Update RLS Algorithm

To enhance computational efficiency, we employ the Recursive Least Squares (RLS) algorithm for snapshot-wise updates. However, the constant regularization term $\mathbf{F}\triangleq \alpha {\mathbf{I}}_M$ in (17) breaks the standard recursive structure, precluding the direct application of RLS. We address this challenge by proposing a novel dual-sequential update scheme that decomposes the global optimization into rank-1 recursions. This approach successfully incorporates the constant regularization while maintaining the $\mathcal{O}\left(M^2\right)$ complexity advantage, making it ideal for real-time processing.

Our objective is to recursively compute the inverse of the diagonally loaded covariance matrix

$$\mathbf{R}\left(n\right)\triangleq {\mathbf{R}}_{\mathit{x}}\left(n\right)+\alpha {\mathbf{I}}_M,$$

(20)

at each time index n as new snapshots arrive. By employing an exponential window with a forgetting factor $\lambda$, the recursive diagonal loading (RDL) formulation is given by:

$$\mathbf{R}\left(n\right)\triangleq \lambda \mathbf{R}(n-1)+\mathit{x}\left(n\right){\mathit{x}}^H\left(n\right)+\alpha (1-\lambda ){\mathbf{I}}_M,$$

(21)

where the residual term $\alpha (1-\lambda ){\mathbf{I}}_M$ is effectively assimilated into the preceding state $\lambda \mathbf{R}(n-1)$. This formulation ensures that the update at each step is structured as a rank-one modification (plus a constant diagonal term), thereby facilitating the use of the matrix inversion lemma.

Referring to (21), the additive term $\alpha (1-\lambda ){\mathbf{I}}_M$ is of full rank. If this term is decomposed into M rank-one updates as $\alpha (1-\lambda ){\sum }_{k=1}^M{\mathit{e}}_k{\mathit{e}}_k^H$, where ${\mathit{e}}_k$ denotes the k-th canonical basis vector, then applying the RLS algorithm sequentially for each update would result in a total computational complexity of $\mathcal{O}\left(M^3\right)$. Consequently, the complexity advantage of the recursive approach would be neutralized.

To overcome this complexity bottleneck, we propose a dual-sequential update scheme that leverages the time-multiplexed nature of the regularization term. Specifically, the full-rank regularization is distributed across M successive snapshots, yielding the following recursive formulation:

$$R\left(n\right)\triangleq \lambda \mathbf{R}(n-1)+\mathbf{x}\left(n\right){\mathbf{x}}^H\left(n\right)+\sqrt{M\alpha (1-\lambda )}{\mathit{e}}_k\left(n\right){\mathit{e}}_k^H\left(n\right),$$

(22)

where $k=(nmodM)+1$, and ${\mathit{e}}_k$ denotes the k-th canonical basis vector. This strategy ensures that each step remains a rank-one modification, thereby preserving the $\mathcal{O}\left(M^2\right)$ efficiency.

Let $\mathbf{P}\left(n\right)\triangleq \mathbf{R}{\left(n\right)}^{-1}$ denote the inverse covariance matrix. Each iteration comprises a dual-stage RLS update to ensure recursive consistency. In the initial stage, the standard RLS update [22] is performed using the observation vector $\mathit{x}\left(n\right)$, yielding:

$$\begin{array}{cc}\hfill \mathit{u}\left(n\right)& =\mathit{P}(n-1)\mathit{x}\left(n\right),\hfill \\ \hfill \mathit{k}\left(n\right)& ={\displaystyle \frac{\mathit{u}\left(n\right)}{\lambda +{\mathit{x}}^H\left(n\right)\mathit{u}\left(n\right)}},\hfill \\ \hfill {\mathbf{P}}_{\mathbf{x}}\left(n\right)& ={\lambda }^{-1}\mathbf{P}(n-1)-{\lambda }^{-1}\mathit{k}\left(n\right){\mathit{u}}^H\left(n\right).\hfill \end{array}$$

(23)

Subsequently, a second update is applied to incorporate the time-multiplexed regularization term $\sqrt{M\alpha (1-\lambda )}{\mathit{e}}_{k\left(n\right)}{\mathit{e}}_k^H\left(n\right)$, formulated as:

$$\mathbf{P}\left(n\right)={\mathbf{P}}_{\mathit{x}}\left(n\right)-{\displaystyle \frac{M(1-\lambda )}{1+M(1-\lambda ){\mathbf{P}}_{k,k}}}{\mathit{p}}_k{\mathit{p}}_k^H,$$

(24)

where ${\mathbf{P}}_{k,k}$ and ${\mathit{p}}_k$ denote the $\left(k\right(n),k(n\left)\right)$-th element and the $k\left(n\right)$-th column of ${\mathbf{P}}_{\mathbf{x}}\left(n\right)$, respectively.

The proposed dual-stage RLS algorithm maintains an overall complexity of $\mathcal{O}\left(M^2\right)$. This efficiency stems from the fact that both update phases—incorporating the observation vector and the regularization term, respectively—are formulated as rank-one recursions.

For (19), to incorporate the stopband constraint into the recursive framework, we perform a Cholesky decomposition on the combined regularization matrix:

$$\alpha {\mathbf{I}}_M+\beta \mathbf{S}\triangleq \mathbf{G}=\mathbf{L}{\mathbf{L}}^H,$$

(25)

where $\mathbf{L}$ is a lower triangular matrix. Accordingly, the dual-sequential update policy is reformulated as:

$$\mathbf{R}\left(n\right)\triangleq \lambda \mathbf{R}(n-1)+\mathit{x}\left(n\right){\mathit{x}}^H\left(n\right)+{\displaystyle \frac{1-{\lambda }^M}{1-\lambda }}{\mathit{l}}_{k\left(n\right)}{\mathit{l}}_{k\left(n\right)}^H,$$

(26)

in which ${\mathit{l}}_{k\left(n\right)}$ denotes the $k\left(n\right)$-th column of $\mathbf{L}$. The second RLS update stage is then modified to:

$$\mathbf{P}\left(n\right)={\mathbf{P}}_x\left(n\right)-{\displaystyle \frac{{\mathbf{P}}_{\mathbf{x}}\left(n\right)\mathbf{v}\left(n\right){\mathbf{v}}^H\left(n\right){\mathbf{P}}_x\left(n\right)}{1+{\mathbf{v}}^H\left(n\right){\mathbf{P}}_x\left(n\right)\mathbf{v}\left(n\right)}},$$

(27)

where $\mathit{v}\left(n\right)=\sqrt{{\displaystyle \frac{1-{\lambda }^M}{1-\lambda }}}{\mathit{l}}_{k\left(n\right)}$. Notably, the vectors ${\left\{\mathit{v}\left(k\right)\right\}}_{k=1}^M$ may be pre-computed at initialization to further reduce the real-time computational load.

To verify the correctness of the algorithm, we prove the following theorem:

Theorem 1

(Convergence of Dual-Sequential Structured Loading). Let $\left\{\mathit{x}\right(k\left)\right\}$ be a wide-sense stationary (WSS) stochastic process with a constant covariance matrix ${\mathbf{R}}_{\mathit{x}}=\mathbb{E}\left[\mathit{x}\left(k\right){\mathit{x}}^H\left(k\right)\right]$. Given the recursive update as (26), then the expectation of the covariance matrix in steady-state converges as follows:

$$\underset{n\to \infty ,\lambda \to 1}{lim}\mathbb{E}\left[\mathbf{R}\left(n\right)\right]={\displaystyle \frac{1}{1-\lambda }}({\mathbf{R}}_{\mathit{x}}+\alpha {\mathbf{I}}_M+\beta \mathbf{S}).$$

(28)

Proof. 

Assuming the initial state $\mathbf{R}\left(0\right)=0$, the instantaneous matrix $\mathbf{R}\left(n\right)$ can be expanded as:

$$\mathbf{R}\left(n\right)={\displaystyle \sum _{k=1}^n}{\lambda }^{n-k}\mathit{x}\left(k\right){\mathit{x}}^H\left(k\right)+{\displaystyle \sum _{k=1}^n}{\lambda }^{n-k}\mathit{v}\left(k\right){\mathit{v}}^H\left(k\right).$$

(29)

Taking the expectation on both sides and utilizing the wide-sense stationarity of $\mathit{x}\left(k\right)$, the first term on the R.H.S. becomes:

$$\mathbb{E}\left[{\displaystyle \sum _{k=1}^n}{\lambda }^{n-k}\mathit{x}\left(k\right){\mathit{x}}^H\left(k\right)\right]={\displaystyle \sum _{k=1}^n}{\lambda }^{n-k}\mathbb{E}\left[\mathit{x}\left(k\right){\mathit{x}}^H\left(k\right)\right]={\displaystyle \frac{1-{\lambda }^n}{1-\lambda }}{\mathbf{R}}_{\mathit{x}},$$

(30)

as $n\to \infty$, this term converges to the deterministic expectation ${\displaystyle \frac{1}{1-\lambda }}{\mathbf{R}}_{\mathit{x}}$.

The second term, defined as the loading matrix $\mathbf{D}\left(n\right)$, is deterministic and can be partitioned by grouping the contributions of each basis vector ${\mathit{l}}_j$ ($j=1,\dots ,M$):

$$\mathbf{D}\left(n\right)={\displaystyle \frac{1-{\lambda }^M}{1-\lambda }}{\displaystyle \sum _{j=1}^M}{\mathit{l}}_j{\mathit{l}}_j^H\left({\displaystyle \sum _{i=0}^{\lfloor {\displaystyle \frac{n-j}{M}}\rfloor }}{\lambda }^{d_j\left(n\right)+iM}\right),$$

(31)

where $d_j\left(n\right)=(n-j)modM$ represents the time lag, satisfying $0\le d_j\left(n\right)\le M-1$.

As $n\to \infty$, the summation of the geometric series within the parentheses converges and is given by:

$$\underset{n\to \infty }{lim}{\displaystyle \sum _{i=0}^{\lfloor {\displaystyle \frac{n-j}{M}}\rfloor }}{\lambda }^{d_j\left(n\right)+iM}={\displaystyle \frac{{\lambda }^{d_j\left(n\right)}}{1-{\lambda }^M}}.$$

(32)

Substituting this into the expression for the steady-state loading matrix $\mathbf{D}(\infty )$:

$$\mathbf{D}(\infty )={\displaystyle \frac{1-{\lambda }^M}{1-\lambda }}{\displaystyle \sum _{j=1}^M}{\mathit{l}}_j{\mathit{l}}_j^H{\displaystyle \frac{{\lambda }^{d_j\left(n\right)}}{1-{\lambda }^M}}={\displaystyle \frac{1}{1-\lambda }}{\displaystyle \sum _{j=1}^M}{\lambda }^{d_j\left(n\right)}{\mathit{l}}_j{\mathit{l}}_j^H.$$

(33)

Under the condition that $\lambda \to 1$ and noting that the lag $d_j\left(n\right)$ is bounded by M, the factor ${\lambda }^{d_j\left(n\right)}$ approaches unity. Consequently:

$$\underset{\lambda \to 1}{lim}\mathbf{D}(\infty )={\displaystyle \frac{1}{1-\lambda }}{\displaystyle \sum _{j=1}^M}{\mathit{l}}_j{\mathit{l}}_j^H={\displaystyle \frac{1}{1-\lambda }}(\alpha {\mathbf{I}}_M+\beta \mathbf{S}).$$

(34)

Combining the limits of both terms, the proof is complete. □

Remark 1.

In the special case where $\beta =0$, the structured loading term disappears, and the result in (28) reduces to the conventional diagonal loading (DL) form. This indicates that our proposed method can be viewed as a generalized framework that encompasses the standard DL as a special case.

2.4. Ripple Analysis and Steady-State Error Bounds

Since the basis vectors ${\mathit{l}}_j$ are injected sequentially rather than simultaneously, each vector experiences a distinct exponential decay determined by its specific lag $d_j\left(n\right)$. This non-uniform decay induces a subtle periodic fluctuation in the loading matrix $\mathbf{D}\left(n\right)$, referred to as the steady-state ripple.

We define the absolute ripple matrix, $\Delta \mathbf{D}\left(n\right)$, as the perturbation of the instantaneous loading matrix from its ideal expectation:

$$\Delta \mathbf{D}\left(n\right)\triangleq \mathbf{D}(\infty )-{\displaystyle \frac{1}{1-\lambda }}\mathbf{G}={\displaystyle \frac{1}{1-\lambda }}{\displaystyle \sum _{j=1}^M}\left({\lambda }^{d_j\left(n\right)}-1\right){\mathit{l}}_j{\mathit{l}}_j^H.$$

(35)

By introducing $ϵ=1-\lambda$ (typically $ϵ\in [{10}^{-3},{10}^{-2}]$) and applying a first-order Taylor expansion for $ϵ\ll 1$, we have:

$${\lambda }^{d_j\left(n\right)}={(1-ϵ)}^{d_j\left(n\right)}\approx 1-d_j\left(n\right)ϵ,$$

(36)

substituting this approximation into the expression for $\Delta \mathbf{D}\left(n\right)$ yields:

$$\Delta \mathbf{D}\left(n\right)\approx {\displaystyle \frac{1}{ϵ}}{\displaystyle \sum _{j=1}^M}(-d_j\left(n\right)ϵ){\mathit{l}}_j{\mathit{l}}_j^H=-{\displaystyle \sum _{j=1}^M}{d}_j\left(n\right){\mathit{l}}_j{\mathit{l}}_j^H.$$

(37)

Remarkably, the scaling factor ${(1-\lambda )}^{-1}$ is canceled by the first-order infinitesimal term from the Taylor expansion. To evaluate the impact of this ripple on system performance, we define the relative ripple bound based on the Frobenius norm:

$${\rho }_{\mathrm{ripple}}\le {\displaystyle \frac{{\parallel \Delta \mathbf{D}\left(n\right)\parallel }_F}{\parallel \frac{1}{1-\lambda }{\mathbf{G}\parallel }_F}}.$$

(38)

Given that $0\le d_j\left(n\right)\le M-1$, the Frobenius norm of the absolute ripple is bounded by:

$${\parallel \Delta \mathbf{D}\left(n\right)\parallel }_F\le (M-1){∥{\displaystyle \sum _{j=1}^M}{\mathit{l}}_j{\mathit{l}}_j^H∥}_F=(M-1){\parallel \mathbf{G}\parallel }_F.$$

(39)

Consequently, the relative ripple bound simplifies to:

$${\rho }_{\mathrm{ripple}}\le {\displaystyle \frac{{(M-1)\parallel \mathbf{G}\parallel }_F}{\frac{1}{1-\lambda }{\parallel \mathbf{G}\parallel }_F}}=(1-\lambda )(M-1).$$

(40)

The result indicates that the relative steady-state ripple is strictly bounded and directly proportional to $(1-\lambda )M$. In practical scenarios, as long as the array dimension M is small relative to the integration time constant ${(1-\lambda )}^{-1}$ (e.g., $M=16$ and $\lambda =0.995$ yields ${\rho }_{\mathrm{ripple}}\approx 0.075$), the structural perturbation to the covariance matrix is minimal, ensuring that the impact on adaptive beamforming performance, such as mainlobe distortion and nulling depth, is negligible.

2.5. Scope and Applicability

It is important to note that the proposed Recursive MFN-RLS is formulated within an online sequential processing framework. While conventional batch-mode spatial filters often struggle in low-snapshot regimes due to the singularity of the sample covariance matrix, the recursive nature of the RLS algorithm, coupled with the inherent regularization term $\alpha {\mathbf{I}}_M$, ensures numerical stability and solution validity from the first iteration. Thus, the algorithm is primarily optimized for dynamic environments requiring continuous adaptation rather than static scenarios with strictly limited snapshots.

2.6. Hyperparameter Selection Strategy

The proposed algorithm involves four hyperparameters, $M_r,\alpha ,\beta$, and $\lambda$, which critically affect spatial filtering performance. While exhaustive grid search is conceptually straightforward, its computational cost becomes prohibitive due to the high-dimensional search space. Therefore, a structured and computationally efficient selection strategy is adopted.

First, $M_r$ determines the rank of the subspace projection and can be selected by analyzing the spatial attenuation curves corresponding to candidate values, allowing for a deterministic choice without search.

Second, $\alpha$ governs the shrinkage behavior of the filter response. From (17), $\mathbf{B}$ can be expressed as a function of $\alpha$:

$$\mathbf{B}\left(\alpha \right)=\alpha {({\mathbf{R}}_{\mathit{x}}+\alpha {\mathbf{I}}_M)}^{-1}{\mathbf{B}}_q,$$

(41)

by imposing the practical constraint

$${∥\mathbf{B}\left(\alpha \right)-{\mathbf{B}}_q∥}_F^2=ϵ,$$

(42)

where $ϵ>0$ is a small predefined tolerance, $\alpha$ can be determined efficiently via a one-dimensional root-finding algorithm (e.g., Newton’s method), eliminating the need for grid search entirely.

Third, $\beta$ is selected via a lightweight grid search over a narrow, empirically motivated range, as its impact is less analytically tractable but computationally inexpensive to evaluate.

Finally, the choice of $\lambda$ controls the trade-off between RLS convergence speed and steady-state stability [22]. A moderate value is typically sufficient to balance tracking agility and estimation accuracy.

This hierarchical and partially analytic hyperparameter selection strategy significantly reduces computational overhead, enhancing the practicality of the proposed algorithm.

3. Simulations

We demonstrate the proposed algorithm using a Uniform Linear Array (ULA), though the framework is readily extendable to other array manifolds. The ULA consists of $M=16$ omnidirectional sensors with half-wavelength spacing. For the simulations, the beamspace dimension $M_r$ is set to 4 and the number of snapshots N is 6400. We would like to clarify that the use of a large snapshot number (e.g., 6400) is intentional and crucial for fair benchmarking. Since our comparison includes recursive adaptive methods, this specific snapshot count was selected to ensure that all algorithms—especially the iterative ones—have reached their steady-state performance. This setup guarantees that the comparison evaluates the ultimate spatial filtering capabilities of the methods rather than their transient convergence behaviors, thereby ensuring a rigorous and equitable evaluation. Our methods are evaluated against conventional beamspace design techniques [1], as well as the robust approaches proposed in [11,12]. All baseline SOCP-based methods were re-implemented using the SeDuMi toolbox.

We assume two strong interference sources located at ${\theta }_{i_1}=-{30}^{\circ }$ and ${\theta }_{i_2}=-{11}^{\circ }$, each with an interference-to-noise ratio (INR) of 50 dB. The signals of interest (SOI) arrive from ${\theta }_{s_1}={15}^{\circ }$ and ${\theta }_{s_2}={17}^{\circ }$ with an SNR of 5 dB. The sector-of-interest is defined as ${\mathrm{\Theta }}_p=[5^{\circ },{30}^{\circ }]$, while the out-of-sector regions are ${\mathrm{\Theta }}_s=[-{90}^{\circ },0^{\circ }]\cup [{35}^{\circ },{90}^{°}]$. A uniform grid of $s=100$ points is utilized to define the stopband ${\left\{{\theta }_k\right\}}_{k=1}^s$.

We compare the following algorithms to evaluate the performance of our proposed methods:

• Element Space: The baseline processing method without spatial filtering.
• Spheroidal Sequences-Based Algorithm: A fixed (non-adaptive) spatial filter as proposed in [1].
• Adaptive SOCP Beamspace Algorithm 1: The adaptive filter from [10], formulated as an SOCP that jointly optimizes passband flatness and stopband attenuation.
• Adaptive SOCP Beamspace Algorithm 2: An SOCP-based adaptive filter proposed in [12], similar to Algorithm 1 but utilizing the formulation in Equation (13).
• Proposed MFN Algorithm 1: Our proposed Matrix Filter with Nulling (MFN) as derived in (10).
• Proposed MFN Algorithm 2: An extension of MFN based on (18), which explicitly incorporates stopband attenuation constraints.
• Proposed MFN-RLS Algorithm 1: The dual-sequential recursive update implementation of the MFN framework in (10), utilizing the dual-sequential update for enhanced computational efficiency.
• Proposed MFN-RLS Algorithm 2: The recursive version of (18), incorporating an explicit stopband attenuation constraint.

The concise specification comparison of the data adaptive methods are in

Table 1. Specifications of the adaptive methods.

Method	Optimization Algorithm	Computational Complexity	Real-Time Applicability
SOCP Beamspace algorithms	SOCP	$\mathcal{O}\left(M^{4.5}\right)$	no
Proposed MFN algorithms	Closed-form equation	$\mathcal{O}\left(M^3\right)$	yes
Proposed MFN-RLS algorithms	RLS	$\mathcal{O}\left(M^2\right)$	yes

.

The hyperparameters are configured as $\alpha$ = 37,365 and $\beta =0.3$ for the closed-form algorithms in (10) and (18). For their recursive RLS counterparts, these values are adjusted to $\alpha$ = 11,210 and $\beta =0.09$. For the benchmark methods, we set the parameters to $ϵ=1.0$ and $\gamma =0.166$ for [10], $ϵ=0.5$ and $\delta =0.01$ for [12].

3.1. Comparison of Spatial Filtering and Adaptive Nulling Performance

The beamspace attenuation is defined as:

$$g\left(\theta \right)\triangleq {\displaystyle \frac{|{\mathbf{B}}^H{\mathbf{a}\left(\theta \right)|}_2^2}{{\left|\mathbf{a}\left(\theta \right)\right|}_2^2}}.$$

(43)

Figure 1 illustrates the beamspace attenuation for different methods. As shown, the proposed adpative beamspace Matrix Filter with Nulling (MFN) algorithms—incorporating explicit passband constraints—achieves a significantly flatter response compared to the approach in [11]. Furthermore, the proposed methods provides substantial stopband attenuation, effectively suppressing strong interference.

Quantitative comparisons of key performance metrics are summarized in

Table 2. Performance comparison of the adaptive methods. The upward arrow (↑) indicates that a higher value is better, while the downward arrow (↓) indicates that a lower value is better. Bold font denotes the optimal value.

Method	PRV (↓)	Max Attenu. (↑)	SMS $(\uparrow$)
Spheroidal Sequences-Based Algorithm [1]	0.0018	11.7121	9.1553
Adaptive SOCP Beamspace Algorithm (19) [10]	0.3339	26.1789	14.0716
Adaptive SOCP Beamspace Algorithm (13) [12]	0.0074	25.5760	5.6021
Proposed MFN Algorithm (10)	0.0025	16.6576	9.3191
Proposed MFN Algorithm (18)	0.0025	16.6575	9.3194
Proposed MFN-RLS Algorithm (10)	0.0269	26.3024	9.4811
Proposed MFN-RLS Algorithm (18)	0.0289	26.3219	9.4831

.

The passband response variance (PRV) measures the fluctuation of the filter response within the passband. Maximum attenuation denotes the ratio (in dB) of the peak passband response to the minimum stopband response. Stopband mean suppression (SMS) quantifies the ratio (in dB) between the maximum passband response and the average stopband response.

The results indicate that the proposed MFN algorithms achieve the best passband flatness, while the MFN-RLS variants provide the strongest interference suppression.

3.2. Comparison of Second-Stage DOA Estimation Performance

In this example, we examine a scenario where the INR of the interference sources is varied. We evaluate the performance of the proposed algorithm in conjunction with MUSIC and DAS. The signals of interest (SOI) arrive from ${\theta }_{s_1}={10}^{\circ }$ and ${\theta }_{s_2}={20}^{\circ }$, with the SNR fixed at 0 dB.

We evaluate the Root Mean Square Error (RMSE) of the DOA estimation after applying the beamspace filtering matrix. The RMSE is defined as:

$$\mathrm{RMSE}(\widehat{\mathit{\theta }},\mathit{\theta })\triangleq \sqrt{{\displaystyle \frac{1}{J}}\sum _{j=1}^J{({\widehat{\theta }}_j-{\theta }_j)}^2},$$

(44)

where J denotes the number of Monte Carlo trials. The vectors $\widehat{\mathit{\theta }}$ and $\mathit{\theta }$ represent the estimated and ground-truth DOAs, respectively, with ${\widehat{\theta }}_j$ and ${\theta }_j$ being their corresponding elements in the j-th trial.

As the proposed algorithm is readily applicable to arrays with arbitrary geometries, we employ spectral MUSIC rather than root-MUSIC for DOA estimation. We assume the number of sources is known a priori, and each numerical result is averaged over 100 independent Monte Carlo trials. As illustrated in Figure 2, our proposed closed-form methods remain robust across all scenarios, while the recursive versions exhibit a marginal performance degradation within the INR range of 3–27 dB. Notably, the performance of the method in [12] degrades significantly; this is because their beamspace filtering matrix introduces numerous spurious peaks in the MUSIC spatial spectrum, leading to estimation failures.

As illustrated in Figure 3, the proposed closed-form methods maintain superior performance for DAS-based DOA estimation. In contrast, the recursive implementations exhibit a slight performance drop when the INR ranges from −6 dB to 21 dB. The approach in [10] exhibits degraded performance even at considerably low INRs. This originates from the fact that these designs cannot preserve the desired passband flatness. Consequently, the resulting large passband ripples lead to the failure of the DAS algorithm, which fundamentally relies on a uniform gain across the passband for accurate estimation.

3.3. Comparison of Passband Signal Power Estimation Performance

In this example, the interference INR is fixed at 20 dB, while the SNR is varied to evaluate the accuracy of the signal power estimation. We consider a single signal of interest $s\left(t\right)$ arriving from $\theta ={15}^{\circ }$.

The estimated power is calculated as:

$$\widehat{P}={\left({\mathbf{B}}^H\mathit{a}\left(\theta \right)\right)}^H{\mathbf{R}}_{\mathit{z}}\left({\mathbf{B}}^H\mathit{a}\left(\theta \right)\right)=\mathit{a}{\left(\theta \right)}^H\mathbf{B}{\mathbf{B}}^H{\mathbf{R}}_{\mathit{x}}\mathbf{B}{\mathbf{B}}^H\mathit{a}\left(\theta \right),$$

(45)

where the oracle (true) power is defined as $P_o={\mathbb{E}\left\{\right|s\left(t\right)|}^2\}$.

As shown in Figure 4, the proposed closed-form methods achieve the highest accuracy across almost the entire SNR range. Meanwhile, the recursive versions perform similarly to the fixed spatial filter, demonstrating superior estimation performance in high-SNR scenarios. In contrast, the approach in [10] remains inaccurate even at high SNRs. This discrepancy arises because [10] lacks explicit constraints on passband flatness; consequently, when the signal of interest is strong, data adaptation distorts the beamspace passband gain, leading to a significant underestimation of the signal power.

3.4. Computational Complexity Evaluation

All methods were implemented in MATLAB R2020a on Windows, running on a PC with an Intel^® Core™ i5–10210U CPU @ 1.60 GHz and 16.0 GB RAM, without GPU acceleration.

The array size M was varied from 4 to 8192, with $M_r=min(4,M/4)$. Each result in Figure 5 is averaged over 30 Monte Carlo trials.

The runtime results in Figure 5 show that the methods in [10,12] are significantly slower than the proposed MFN-based approaches. Among all tested methods, MFN–RLS achieves the lowest execution time. The measured runtime scaling of the MFN method is approximately $\mathcal{O}\left(M^3\right)$, compared with near-quadratic scaling ($\mathcal{O}\left(M^2\right)$) for MFN–RLS.

The computational burden of the reference methods grows rapidly with M, leading to excessive runtime (>100 s) or out-of-memory failures, which renders them impractical for large-scale or real-time deployment.

4. Experimental Validation

To demonstrate the robustness of the proposed algorithms against array element position errors and non-Gaussian noise, as well as their applicability to arbitrary array geometries, a 64-element planar random microphone array was designed and fabricated.

The array employs Micro-Electro-Mechanical System (MEMS) microphones, and the element coordinates are listed in

Table 3. Array element coordinates (Unit: m).

No.	X Coord.	Y Coord.	No.	X Coord.	Y Coord.	No.	X Coord.	Y Coord.
1	−0.0080	0.0329	2	−0.0146	−0.0136	3	0.0275	−0.0123
4	−0.0025	0.0290	5	−0.0235	−0.0136	6	0.0340	−0.0101
7	−0.0073	0.0237	8	−0.032	−0.0169	9	0.0325	−0.0020
10	−0.0135	0.0281	11	−0.0262	−0.0212	12	0.0248	−0.0061
13	−0.0199	0.0296	14	$-0$.0185	−0.0227	15	0.0166	0.0003
16	−0.0242	0.0238	17	−0.0109	−0.0213	18	0.0113	0.0046
19	−0.0165	0.0214	20	−0.0034	−0.0268	21	0.0148	0.0109
22	−0.0332	0.0153	23	−0.0105	−0.0301	24	0.0214	0.0086
25	−0.0253	0.0159	26	−0.0187	−0.0307	27	0.0279	0.0044
28	−0.0162	0.0131	29	0.0004	−0.0341	30	0.0334	0.0102
31	−0.0082	0.0132	32	0.0128	−0.0338	33	0.0269	0.0140
34	−0.0101	0.0049	35	0.0066	−0.0290	36	0.0200	0.0192
37	−0.0170	0.0023	38	0.0175	−0.0270	39	0.0118	0.0172
40	−0.0234	0.0084	41	0.0104	−0.0223	42	0.0061	0.0115
43	−0.0319	0.0071	44	0.0017	−0.0210	45	−0.0013	0.0110
46	−0.0293	0.0002	47	0.0194	−0.0192	48	0.0004	0.0235
49	−0.0340	−0.0044	50	0.0276	−0.0205	51	0.0071	0.0240
52	−0.0294	−0.0103	53	0.0107	−0.0143	54	0.0145	0.0268
55	−0.0222	−0.0061	56	0.0036	−0.0114	57	0.0225	0.0258
58	−0.0133	−0.0058	59	0.0131	−0.0048	60	0.0121	0.0330
61	−0.0070	−0.0094	62	0.0185	−0.0117	63	0.0039	0.0321
64	−0.0045	−0.0164

.

The physical layout of the planar array is illustrated in Figure 6.

The fabricated printed circuit board (PCB) is shown in Figure 7.

The microphone array was connected to an FPGA-based acquisition platform for multichannel signal collection and preprocessing. The acquired time-domain signals were sampled at 150 kHz and segmented into frames of 512 samples with 50% overlap. The hyperparameters were set to $M_r=6,\alpha =98790$ and $\beta =0.5$, while the passband and stopband regions were defined as $\theta <{20}^{\circ }$ and $\theta \ge {25}^{\circ }$, respectively.

A spherical coordinate system was adopted, as shown in Figure 8, where $\theta$ denotes the zenith angle and $\varphi$ the azimuth angle. The array lies in the X–Y plane, with the positive X-axis corresponding to ${[\theta ,\varphi ]}^{\mathrm{T}}={[{90}^{\circ },0^{\circ }]}^{\mathrm{T}}$. The positive Z-axis aligns with $\theta =0^{\circ }$; consequently, the azimuth angle $\varphi$ is undefined at this pole.

Acoustic measurements were carried out in an anechoic chamber. Two continuous 10 kHz tonal sources were placed around the array. A weak target source was positioned at $[\theta ,\varphi ]=[{15}^{\circ },{95}^{\circ }]$, while a strong interference source was placed at $[\theta ,\varphi ]=[{60}^{\circ },{45}^{\circ }]$.

The fixed 2D matrix filter nulling attenuation is shown in Figure 9.

The spatial attenuation patterns of the fixed and adaptive matrix filters are shown in Figure 9a. The fixed filter exhibits a broad stopband, whereas the adaptive filter steers deep nulls precisely toward the interference direction at $[\theta ,\varphi ]=[{60}^{\circ },{45}^{\circ }]$, as illustrated in Figure 9b.

The MFN-RLS algorithm yielded a nearly identical attenuation characteristic and is therefore omitted for conciseness. Beamforming outputs after matrix spatial filtering remained sharply focused with no observable distortion, confirming the effectiveness of the proposed approach.

5. Conclusions and Future Work

This paper proposed a robust and computationally efficient beamspace matrix filtering framework with integrated nulling capabilities. Unlike conventional approaches relying on computationally intensive second-order cone programming (SOCP), the proposed method derives a closed-form solution for adaptive beamspace matrix calculation, reducing the complexity to $\mathcal{O}\left(M^3\right)$. Furthermore, the introduction of a dual-sequential update mechanism lowers the complexity to $\mathcal{O}\left(M^2\right)$, whereas existing SOCP-based benchmarks [9,10,11,12] typically scale at $\mathcal{O}\left(M^{4.5}\right)$. Extensive simulations and experimental results demonstrated that the proposed MFN and MFN-RLS algorithms achieve superior DOA estimation accuracy and power stability in high-INR environments while being well-suited for resource-constrained embedded platforms.

Despite these advantages, certain limitations warrant attention. First, the recursive update mechanism may face challenges in extremely non-stationary scenarios where interference fluctuates faster than the convergence rate. Second, the spatial degrees of freedom (DoFs) impose a fundamental limit on suppressing complex sound fields, making it difficult to suppress distributed or large-area interference when the interference rank exceeds the system’s DoF. Third, practical physical constraints—such as finite microphone size and inter-element spacing—inevitably lead to deviations from the ideal array manifold, causing spatial aliasing and gain distortion at high frequencies.

Future research will focus on three directions. First, we aim to enhance nulling depth and passband robustness under non-Gaussian noise conditions. Second, nonlinear mapping architectures, such as deep neural networks, will be explored to replace linear beamspace transformations, potentially increasing DoFs and improving dynamic range in non-stationary environments. Finally, a systematic hyperparameter sensitivity analysis will be conducted, and data-driven selection strategies (e.g., Bayesian optimization) will be investigated to replace manual tuning and further improve generalization.