Abstract
This paper proposes a general model of superdirectivity to provide analytical and closed-form solutions for arbitrary sensor arrays. Based on the equivalence between the maximum directivity factor and the maximum array gain in the isotropic noise field, Gram-Schmidt orthogonalization is introduced and recursively transformed into a matrix form to conduct pre-whitening and matching operations that result in superdirectivity solutions. A Gram-Schmidt mode-beam decomposition and synthesis method is then presented to formally implement these solutions. Illustrative examples for different arrays are provided to demonstrate the feasibility of this method, and a reduced rank technique is used to deal with the practical array design for robust beamforming and acceptable high-order superdirectivity. Experimental results that are provided for a linear array consisting of nine hydrophones show the good performance of the technique. A superdirective beampattern with a beamwidth of 48.05° in the endfire direction is typically achieved when the inter-sensor spacing is only 0.09λ (λ is the wavelength), and the directivity index is up to 12 dB, which outperforms that of the conventional delay-and-sum counterpart by 6 dB.
Highlights
Sensor array signal processing has played a significant role in many diverse application areas, including sonar, radar, audio engineering, and wireless communication [1]
After the development of optimal sensor array theory, the maximum directivity factor (DF) of a sensor array is equivalent to the maximum array gain (AG) when the array is used in an isotropic noise field [25]
Because the noise correlation coefficients between sensors can be determined in the isotropic noise field, all the solutions of superdirectivity have been accurately expressed in full closed-form based on the GS orthogonalization scheme with the use of the frequency and array geometric parameters
Summary
Sensor array signal processing has played a significant role in many diverse application areas, including sonar, radar, audio engineering, and wireless communication [1]. Wang et al EURASIP Journal on Advances in Signal Processing (2015) 2015:68 computed using the probability density functions of the array errors These methods provide good insights into superdirectivity, but they require a priori knowledge of errors. A general model that provides an analytical and closed-form solution is still lacking to accurately calculate a superdirective beamformer and allow the decomposition of the beampattern into components with different error sensitivities for robust implementation. The eigen-beam decomposition becomes feasible because of the circulant property of the data covariance matrix of circular arrays This condition is not the case for sensor arrays with other geometries, and a new model that can be generally applicable to arbitrarily shaped arrays is required.
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