In acoustic array signal processing, spatial spectrum estimation and the corresponding directionofarrival estimation are sometimes affected by stationary nearfield interferences, presenting a considerable challenge for the target detection. To address the challenge, this paper proposes a beamspace spatial spectrum reconstruction algorithm. The proposed algorithm overcomes the limitations of common spatial spectrum estimation algorithms designed for nearfield interference scenarios, which require knowledge of the nearfield interference array manifold. The robustness and efficacy of the proposed algorithm under strong stationary nearfield interference are confirmed through the analysis of simulated and reallife experimental data.
1. Introduction
Stationary nearfield interferences are frequently encountered in many underwater systems. For instance, the array towed behind the ship inevitably receives the strong nearfield interferences caused by the loud vibration noise of the towing vessel, which can be regarded as stationary within a certain observation period. Unlike spatial white noise, nearfield interference tends to have directionality, high energy (often significantly greater than that of the targets due to its proximity to the array), and various other spatial and statistical properties. These characteristics pose great challenges for spatial spectrum estimation algorithms. Specifically, strong nearfield interference can create masking regions^{1} in the spatial spectrum for conventional beamforming (CBF), resulting in substantial estimation errors, and even failed detection of targets located within the masking region. Some highresolution spectrum searching algorithms, including minimumvariance distortionless response (MVDR)^{2} and multiple signal classification (MUSIC),^{3} can effectively alleviate these issues. However, they are sensitive to signal mismatch.^{4,5} Furthermore, they experience compromised estimation accuracy when the nearfield interference closely resembles farfield targets, such as when both are situated near the endfire of an array.
Recently, algorithms targeting nearfield interference environments have been proposed, such as matrix filter techniques,^{6} which eliminate interference prior to conventional spectrum estimation algorithms, and multisnapshot sparse Bayesian (MSBL) algorithms that incorporate the array manifold of nearfield interference into the dictionary matrix.^{7} However, these algorithms face the same challenge of requiring knowledge of the array manifold of nearfield interference, which is equivalent to knowing the position of nearfield interference if the nearfield interference follows spherical wave propagation. Although stationary nearfield interferences allow for preestimating the position of nearfield interference, the algorithms' performance still experiences a significant decline. This is due to the estimation results of the nearfield interference often being accompanied by significant errors, especially in scenarios involving multiple nearfield interferers.
To tackle the stationary nearfield interferences more effectively, we propose a systematic approach, called beamspace spatial spectrum reconstruction algorithm. Initially, we designed a sample matrix inversion (SMI) adaptive beamformer^{4,5} using precollected snapshots of interference plus noise. This beamformer serves as a spatial filter to eliminate strong nearfield interferences, deriving the beamspace signal. Subsequently, inspired by the remarkable performance of sparse Bayesian learning (SBL) framework in sparse reconstruction^{8} and the intrinsic nonnegative property of the spatial spectrum, this paper exploits the nonnegative fast sparse Bayesian learning (NNFSBL) algorithm to reconstruct the spatial spectrum from the beamspace signal. The salient features (e.g., superresolution and robustness) of the proposed algorithm are demonstrated by simulation and reallife experimental results.
2. Problem formulation
2.1 Array signal model
2.2 Adaptive beamformer design
Consider a measurement vector composed of N sequentially arranged beam outputs at different $\varphi n$ s, that is $y=[y(\varphi 1),\u2026,y(\varphi n)]T\u2208R+N$ ( $R+$ is the field of nonnegative real numbers). Based on Eq. (5), $y$ can be expressed as the linear combination of the beam power vectors for varying arriving angles, i.e., ${hk=[Bp(\varphi 1,\theta k),\u2026,Bp(\varphi N,\theta k)]T}k=1,\u2026,K$, with the signal power $\sigma k2$ being the associated coefficient. This interpretation motivates a sparsityaware problem formulation, which has witnessed great success in modeling elementspace measurements.^{10,11}
3. Nonnegative fast sparse Bayesian learning
To exploit the nonnegative characteristics of the spatial spectrum, we incorporate a nonnegativity constraint in the offtheshelf fast sparse Bayesian learning (FSBL)^{12} algorithm, leading to the development of the NNFSBL algorithm.
Given the values of these hyperparameters, Eq. (17) leads to a birthanddeath procedure in FSBL for determining the values of hyperparameters ${\alpha i}i$ along with adding/deleting one basis vector in each iteration (for more insight, see Ref. 12). Fortunately, this scheme of FSBL provides a viable way to enforce the nonnegativeness of $x\xaf$ after certain modifications, while still guaranteeing the increment of the objective function $L(\alpha ,\sigma 2)$.
Specifically, we calculate the potential $x\xaf$ and the increment of the marginal likelihood (i.e.,the objective function) of FSBL for each basis candidate in the dictionary matrix. Then, in each iteration, we select the candidate that ensures both the nonnegativity of $x\xaf$ and the largest increment in marginal likelihood for the update. This procedure provides a viable way to enforce the nonnegativeness of $x\xaf$ while preserving the algorithm's convergence. For further details, please refer to Ref. 13 where the NNFSBL algorithm was first proposed.
4. Spatial spectrum reconstruction using beamspace signal
After introducing the SMI adaptive beamformer design and the NNFSBL algorithm, we are now ready to present the developed beamspace spatial spectrum reconstruction method. Following the consecutive blocks in Fig. 1, the proposed method can be summarized in the following steps:

Step 1: Design an SMI adaptive beamformer using interferenceplusnoise snapshots.

Step 2: Let the received array signal mixed with target sources and nearfield interferences pass the adaptive beamformer to obtain the beamspace signal.

Step 3: Use the NNFSBL algorithm to reconstruct the spatial spectrum from beamspace signal.
It is worth noting that the adaptive beamformer design and sparse learning are not two independent modules that are simply combined together. The adaptive beamformer not only enables the suppression of nearfield interference but also leads to spectrum estimation in the beam space. Compared with previous approaches that directly perform sparse reconstruction algorithms on the array signals, the proposed algorithm offers a more concise solution by requiring much fewer unknowns to be determined. Specifically, in element space, the variables to be solved are complex signal amplitudes across multiple snapshots, with the total number of variables being the product of the number of grid points and the number of snapshots, i.e., $L\xd7T$. However, since our proposed algorithm operates in beam space, the variables to be solved are L nonnegative real numbers representing the signal power at grid points and the number of these variables is independent of the number of received snapshots, T.
Moreover, the dimension of the measurement vector $y\u2208R+N$ can be flexibly adjusted by artificially tuning the number of steering directions, $\varphi $. This strikes an excellent balance between reconstruction probability and computational burden, a feat difficult to achieve with classical sparse learning methods performed in the element space, where the dimension of the measurement vector is fixed to the number of array sensors, M. In scenarios where the number of sensors is large, which is common with towed arrays, this property is highly valuable, as the computation cost can be excessively high for certain sparse reconstruction algorithms such as the basis pursuit (BP) algorithm.
5. Simulation
In this section, we validate the effectiveness of the proposed algorithm using simulated data and compare it with classical and recent stateoftheart algorithms, including CBF, MVDR, and a variant of MSBL designed for nearfield interference scenarios^{7} (mentioned in Sec. 1, represented as MSBL1), to demonstrate its robustness against signal mismatch.
In simulation, we assume that the SNR of the target at $80\xb0$ is 15 dB, and the other targets are 10 dB, and 16element ULA with interelement spacing of 0.75 m is applied to receive signals at a frequency of 1000 Hz. Onethousand snapshots of interference plus noise are used to construct the SMI adaptive beamformer, while 200 snapshots containing the signals of interest are utilized for subsequent spatial spectrum estimation processing. To fairly compare the performances, CBF deals with the case assuming no nearfield interference, since it does not have mechanisms for interference suppression.
Comparing Figs. 2(a) and 2(b), the MVDR algorithm is sensitive to signal mismatches, leading to a significant performance degradation by underestimating the target strength by 15 dB. Meanwhile, MSBL1 encounters severe basis mismatch problems^{14} under nearfield interference, resulting in substantial estimation errors and numerous false targets.
In contrast, the proposed algorithm achieves satisfactory spatial spectrum reconstruction results both without and with signal mismatch, respectively. It can be seen from Fig. 2(b) that the proposed algorithm can achieve much more accurate strength estimation in the presence of signal mismatch, depending on the available prior information about the nearfield interference. It is noteworthy that the snapshots of interference plus noise required by the proposed algorithm are often milder conditions and more easily achievable compared to the specific prior knowledge of the location of nearfield interference required by MSBL1. Figures 2(c)–2(f) further prove that our proposed algorithm achieves better robustness in power estimation compared to the MVDR and MSBL1 algorithms. Note that, in addition to benefiting from the robustness of the SBL framework,^{11} the superior robustness against signal mismatch, comparable to that of the CBF algorithm, primarily originates from the beamforming operation within the algorithm. This highlights a significant advantage of processing signals in the beam space over the element space.
6. Experimental data analysis
In this section, the effectiveness and superior performance of the proposed method are validated using experimental data.
6.1 Laker experimental data
The experimental setting is illustrated in Fig. 3(a). In the experiment, the same ULA used in the previously noted simulation is deployed horizontally, positioned 10 m underwater. Two farfield target sources are positioned approximately 40 m below the surface and at a horizontal distance of 100 m from the array, at angles of $\theta =90\xb0$ and $\theta =120\xb0$ relative to the array endfire direction, respectively. With the array center as the origin of the coordinates, the nearfield interference is suspended from the experiment platform at (r, $\theta $) = (5 m, $90\xb0$).
Before receiving signals from two farfield targets, we recorded a 30s signal containing only nearfield interference and noise. Subsequently, two target sources started transmitting a singlefrequency signal at 1000 Hz and the signal mixed with both farfield target sources and nearfield interference was recorded for 25 s. A 50% overlapping FFT is performed on the received two segments of signals with a sliding time window of 0.2 s, resulting in two datasets containing 299 and 249 snapshots respectively. The 299 snapshots of interference plus noise are used to construct the SMI adaptive beamformer, while the 249 snapshots, containing the signals of interest, are utilized for subsequent spatial spectrum estimation processing. The angular space from $0\xb0$ to $180\xb0$ is uniformly discretized at $0.1\xb0$ in the data processing.
6.2 Results
Figure 3(b) shows the normalized spatial spectrum estimation results of CBF, MVDR, MSBL1, and the proposed method. It should be noted that spatial smoothing and forwardbackward averaging techniques^{5} are employed in the CBF, MVDR, and the proposed algorithm to mitigate the correlation of the sources, making them approximately satisfy the assumption of uncorrelated sources.
As observed in Fig. 3(b), nearfield interference creates a masking region from approximately $30\xb0$ to $90\xb0$ for CBF, making it challenging to observe the target at $90\xb0$. Since the other farfield target is located outside the masking region at $120\xb0$, a peak can be observed in this direction. However, next to this peak, there is an obvious false peak with comparable energy (reduced by less than 2 dB) due to the energy leakage caused by the high sidelobes of CBF. While MVDR can mitigate this deficiency, Fig. 3(b) shows that many false targets are still noticeable. MSBL1 reduces the number of false targets, however, its estimation errors for the farfield target at $90\xb0$, near the direction of the nearfield interferer, are significantly increased, especially in terms of target strength. This can be attributed to mismatches between the dictionary atoms and the actual array manifold, particularly with the manifold of the nearfield interference.
Compared with the algorithms mentioned previously, the proposed algorithm demonstrates superior estimation accuracy, both in terms of the DOA and strength of targets, see Fig. 3(b). Additionally, the proposed algorithm exhibits the sharpest peaks and cleanest backgrounds, showcasing the superresolution capability and excellent denoising performance, owing to the sparsity and robustness provided by the NNFSBL algorithm.
7. Conclusions
In this paper, to address the problem of farfield spatial spectrum reconstruction under strong stationary nearfield interferers, we design a systematic solution for scenarios where training data containing only nearfield interference and noise is available in advance. The proposed algorithm demonstrates superior robustness against signal mismatch resulting from changes in the gain and phase of the array sensors, particularly in terms of power estimation. Additionally, it also inherits the advantages of SBL, including superresolution and denoising capability, thereby producing sharp peaks and a clean background. In future research, we will explore extending the proposed algorithm to scenarios where nearfield interference moves rapidly relative to the receiving array.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No. 61831020 and the National Key Research and Development Program of China Grant No. 2016YFC1400100.
Author Declarations
Conflict of Interest
The authors confirm that there are no conflicts of interest associated with the publication of this research article. Any potential conflicts arising from funding sources or affiliations have been disclosed accordingly.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.