Eigenbeam-Space Transformation for Steerable Frequency-Invariant Differential Beamforming in Linear Arrays

Beamforming is a fundamental technique in signal processing commonly used to enhance the signal-to-noise ratio when dealing with corrupted signals by noise and reverberations.  This is achieved by combining signals from multiple sensors in an array with appropriate delays and weights to constructively or destructively interfere with the signals. The technique enhances the desired signal while suppressing noise and interference. Traditional beamformers, such as the delay-and-sum beamformer and the superdirective beamformer have been used to achieve maximum white noise gain and directivity index, respectively. However, the synthesis of steerable, frequency-invariant beampatterns using linear arrays remains a challenging area of research. Recently, differential beamformers, which use the spatial derivatives of the acoustic pressure field, have shown promise due to their ability to achieve high directivity with small apertures. To this end, new study published in Signal Processing by Professor Gaokun Yu from the College of Information Science and Engineering at Ocean University of China proposed an eigenbeam-space transformation-based steerable differential beamforming approach for linear arrays, to address key limitations in existing methods.

The core objective of differential beamforming is to design a beamformer that maximizes the directivity factor while maintaining a distortionless constraint in the desired direction. This involves solving an optimization problem in the eigenbeam-space domain, ensuring the beamformer is robust against noise and interference. The finite difference method is typically used to approximate the spatial derivatives, but this introduces errors that degrade directivity performance at higher frequencies. To mitigate these issues, robust differential beamformers are designed using more sensors than the order of the beamformer, employing the angular spectrum method to obtain the spatial derivatives. The signal model considers a source located in the far field, radiating sound waves received by a linear array of 𝑀 omnidirectional sensors with an interelement spacing 𝛿. The phase vector d(f,θ), is defined, capturing the response of the array to the incident wave from direction 𝜃. By introducing a weighting vector (𝑓), the beamformer output can be expressed, and constraints are imposed to ensure the desired beampattern and interference suppression.

The eigenbeam-space transformation decomposes the sound field acquired by sensors into eigenbeams, leading to a frequency-independent beampattern. This transformation is crucial for designing robust differential beamformers. The approach leverages the angular spectrum method and singular value decomposition to achieve accurate eigenbeam representations, even at high frequencies. The transformation matrix B(f,m_t​) converts the array response into the virtual array response q_N​(f,θ), which approximates the ideal dipole beampatterns. This enables the synthesis of steerable differential beamformers in the eigenbeam-space domain, with parameters m_t​ (truncation parameter) and 𝑁 (highest order of the eigenbeam) playing critical roles in controlling robustness and accuracy. According to the author, the weighted least-squares method was employed to synthesize steerable differential beamformers, minimizing the error between the synthesized and desired beampatterns. The optimization problem is formulated, incorporating constraints to ensure the robustness of the beamformer. The optimal weighting vector a_N​(m_t​) is determined through a closed-form solution, and the resulting beampattern is evaluated for performance.

Professor Gaokun Yu validated the simulations and experimental results of the proposed method. Various scenarios demonstrate the effectiveness of the hybrid least-squares-error solution in achieving frequency-invariant beampatterns with high directivity and robustness. When the author compared the results with traditional beamforming methods it highlighted the advantages of the eigenbeam-space transformation approach, specially in terms of performance at higher frequencies and ease of parameter determination. The significance of Professor Gaokun Yu’s study is mainly in the innovative approach that addresses the challenges in differential beamforming for linear arrays. The new eigenbeam-space transformation, is indeed a robust method for achieving steerable, frequency-invariant beampatterns, which are essential for processing broadband signals in various applications. For instance, the new method can significantly improve the directivity and robustness of differential beamformers, particularly in environments with strong interference and noise. This advancement is critical for applications requiring precise directional signal processing, such as sonar, radar, and microphone arrays. Moreover, the successful achievement of frequency-invariant beampatterns ensures consistent performance across a wide range of frequencies. This is essential for broadband signal processing applications, where maintaining a stable beampattern is important for accurate signal detection and localization. Furthermore, Professor Gaokun Yu provided an analytical closed-form solution for the optimization problem, which simplify the design process and make it more accessible for practical implementations. The use of eigenbeam-space transformation allows for a systematic approach to controlling beamformer performance through experimentally determined parameters.

In conclusion, the eigenbeam-space transformation-based steerable differential beamforming method found a solution for key challenges in the design of differential beamformers. Professor Gaokun Yu’s optimization of the eigenbeam-space domain, achieved robust, frequency-invariant beampatterns. His experimental validation confirmed the effectiveness of the method making it a significant advancement in the field of beamforming for linear arrays.