Study Of Superdirectivity Principles And Methods For Sensor Arrays

Sensor array signal processing has been widely used in sonar, radar, audio engineering, and communication, etc., whose main goal is spatial resolution, interference suppression, and signal-to-noise ratio enhancement using beamforming techniques. The contradiction between array size and performance for traditional arrays is becoming more and more prominent, owing to the limitations from different application scenarios. Consequently, determining how to improve spatial directivity of a sensor array in constraint of its shape, geometrical size, element number, and element characteristics is an important practical problem in sensor array signal processing. The superdirectivity theory is significantly attractive in solving this problem, because it can provide high directivity, good angular discrimination, excellent wideband frequency-invariant performance, and remarkable reduction of array aperture. However, achieving good superdirectivity is still a challenge beacause of its high sensitivity to errors. This dissertation presents an in-depth study of superdirectivity, and proposes and completes a series of superdirective beamforming methods for different arrays. The performance of these methods are fully demonstrated with simulations and experiments, which can be readily used to design small superdirective sensor arrays. The main contributions are as follows.1. A general beamforming model based on the concept of phase modes for circular sensor arrays is proposed, which integrates the delay-and-sum beamforming, minimum variance distortionless response beamforming, and direct robust superdirective beamforming methods in mode space, making the existing modal beamforming theory more complete. A robust modal beamformer is then straightforwardly derived for circular sensor arrays mounted on acoustically transparent baffles, rigid spheres, infinite and finite rigid cylinders using sound scattering theory, which can provide a suitable trade-off among multiple conflicting performance measures, such as directivity factor, robustness, and sidelobe level.2. The sensitivity function(SF) used as a robustness measurement for circular sensor arrays in the eigen-beam decomposition and synthesis superdirectivity(EBDS) model is accurately decomposed into a finite series of closed-form SFs of eigen-beams, so that some general rules presenting the relationship among frequency, eigen-beam order, and SF are revealed. Based on these results, the reduced-rank technique for achieving robust high-order superdirectivity is further studied in depth. These achievements largely complete the EBDS model and make it more practical. The EBDS model is also extended to uniform circular arrays mounted on rigid spheres and infinite rigid cylinders with an arbitrary number of elements, showing that the baffles can improve the robustness. The limit expressions of the maximum DF and optimal beampattern are then derived using the EBDS model, which show that both baffled and unbaffled circular arrays possess good potential for directivity improvement. Moreover, a robust superdirective beamformer with sidelobe constraints is designed, in which the robustness constraint parameter is determined by the estimated maximum order eigen-beam. This beamformer can give more flexible superdirective beampatterns for circular sensor arrays.3. A general Gram-Schmidt mode-beam decomposition and synthesis(GSMDS) superdirectivity model is proposed to provide analytical and closed-form solutions for arbitrary sensor arrays. Because the noise correlation coefficients between sensors can be easily determined in the isotropic noise field, all the solutions of superdirectivity are accurately expressed in full closed-form based on the GS orthogonalization scheme with the use of the frequency and array geometric parameters. The optimal beampattern and its corresponding DF can be expressed as the sum of GS mode-beams and sum of their associated DFs, respectively. Design examples for three different arrays show that the robustness of mode-beams decreases with an increase in the order number. Therefore, robust superdirective beampatterns can be synthesized with the use of the reduced-rank treatment.4. A superdirective beamforming method based on pattern synthesis for circular sensor arrays is proposed by minimizing the mean square error(MSE) between the desired and synthesized beampatterns. The accurate solutions are derived by utilizing the properties of circulant matrix. Both the array weighting vector and the synthesized beampattern can be expressed in closed-form when the desired beampattern is properly formulated. The MSE is also modified to a more concise form. Broadband beampatterns with optimal superdirectivity and frequency-invariance are achieved for circular arrays mounted on acoustically transparent baffles, rigid spheres, infinite rigid and elastic cylinders based on these results.5. Since the manifolds and noise cross-spectral matrices of the practical sensor arrays cannot be analytically expressed, the pattern synthesis method is also applied to study the performance of superdirective beamformers for these arrays, in which in-situ measurements and numerical methods are indispensable. The superdirective method based on desired beampattern fitting is applied for a practical circular array mounted on a prism. Specifically, the phase modes are extracted from real data using the least-square principle, which are then multiplied and summed to synthesize the final superdirective beampattern, in which the weighting vector is obtained from the desired beampattern. Moreover, the superdirective method based on direct optimization is applied for a practical conformal vector array. In this case, a superdirective beamformer with specific spatial responses is designed using an optimization method given a priori knowledge of ambient noise directivity, which provides good signal-to-noise ratio enhancement.