Superdirective Beampattern Research Articles

Robust superdirective beamforming for arbitrary sensor arrays

This paper presents a method of designing robust superdirective beamformers for arbitrary sensor arrays. First, the optimal weighting vector is transformed into an equivalent form based on the eigen–decomposition of the noise correlation matrix in an isotropic uniform noise field, and its entries will be indexed using mode orders instead of element numbers. Second, a parameter is weighted at the maximum–order entry in the modified weighting vector. The beampattern, directivity factor, and error sensitivity function can be expressed as functions of the parameter. Adjusting the parameter will achieve arbitrary–order superdirective beampatterns with different directivity factors or error sensitivity functions. Third, the closed–form expression of the parameter can be derived given a desired directivity factor or error sensitivity function. The final robust broadband superdirective beampatterns can then be readily synthesized. The proposed method, which is an extension of that previously proposed for circular sensor arrays, provides flexible schemes to designing robust broadband superdirective beamformers for arbitrary arrays in simulations and experiments.

Arbitrary-order superdirectivity of circular sensor arrays.

Superdirectivity can yield large directivity using a small array aperture, but it suffers from high sensitivity to random errors. In a previously proposed model, robust Nth-order superdirectivity was obtained by combining low-order superdirectivity sub-solutions with orders from 0 to N. However, the orders N were exclusively positive integers, which may lead to an over-improved robustness with an excessively degraded directivity. This paper presents an arbitrary-order superdirectivity model of circular sensor arrays, which can achieve a more flexible compromise between directivity and robustness than the previous model. A coefficient η is weighted at the (N + 1)th-order entry of the optimal weighting vector and the new weighting vector will consist of the 0th- to Nth-order and the weighted (N + 1)th-order entries. The beampattern, the directivity factor, and the error sensitivity function can then be expressed as functions of the coefficient η. Moreover, broadband superdirective beampatterns with a desired directivity factor or error sensitivity function can be readily synthesized using the closed-form solutions derived from the arbitrary-order superdirectivity model. Simulations and experimental results show that the proposed model can easily synthesize superdirective beampatterns with arbitrary orders, and it is more flexible than the previous integer-order model. It is also demonstrated that the performance of the proposed method is similar to that of the optimization-based method, but it uses a simple closed-form weighting vector rather than a numerical algorithm.

Superdirectivity solutions of circular arrays with acoustic particle velocity sensors

This paper presents analytical superdirectivity solutions for circular arrays with acoustic particle velocity sensors (APVSs) based on the eigenbeam decomposition and synthesis theory. The main findings are summarized as follows. (1) The optimal beampattern, the global directivity factor, and the total error sensitivity function of circular APVS arrays, like those of circular arrays with pressure sensors, can also be expressed as the sums of eigenbeams, their directivity factors, and their error sensitivity functions, respectively. (2) The directivity factor increases but robustness decreases with an increase in the order number at low frequencies, and the reduced-rank technique can be used to synthesize robust superdirective beampatterns by truncating some high-order eigenbeams. (3) The limit theoretical maximum directivity factor and optimal beampattern are derived, which provide the upper bound of performance for circular APVS arrays and show that this type of arrays also has good potential for improving directivity. The reduced–rank technique is applied for circular APVS arrays to obtain robust high-order superdirectivity in simulations, which shows good performance for improving the robustness.

Robust Superdirective Frequency-Invariant Beamforming for Circular Sensor Arrays

This letter proposes a robust superdirective frequency-invariant beamforming method for circular sensor arrays. First, the beampattern is accurately transformed into an equivalent form using the property of circular arrays, and the expressions of directivity factor and white noise gain are correspondingly modified. Second, a closed-form weighting vector is accurately derived based on the minimum mean square error approximation to a desired superdirective beampattern that is expressed in a general form. When the number of elements is sufficiently large, the weighting vector can be further simplified, and the white noise gain is also modified to a concise form that is found to be an increasing function of the number of elements in the low-frequency range. Simulations and experimental results show that this method can provide a robust superdirective frequency-invariant beampattern over a wide frequency band given a suitable number of elements, and it can be readily applied to process broadband signals in practice using small circular sensor arrays.

Reduced-Order Robust Superdirective Beamforming With Uniform Linear Microphone Arrays

Sensor arrays for audio and speech signal acquisition are generally required to have frequency-invariant beampatterns to avoid adding spectral distortion to the broadband signals of interest. One way to obtain frequency-invariant beampatterns is via superdirective beamforming. However, traditional superdirective beamformers may cause significant white noise amplification (particularly at low frequencies), making them sensitive to uncorrelated white noise. To circumvent the problem of white noise amplification, a method was developed to find the superdirective beamforming filter with a constraint on the white noise gain (WNG), leading to the so-called WNG-constrained superdirective beamformer. But this method damages the frequency invariance of the beampattern. In this paper, we develop a flatness-constrained robust superdirective beamformer. We divide the overall beamformer into two subbeamformers, which are convolved together: one subbeamformer forms a lower order superdirective beampattern while the other attempts to improve the WNG. We show that this robust approach can improve the WNG while limiting the frequency dependency of the beampattern at the same time.

A general superdirectivity model for arbitrary sensor arrays

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.

Design of Robust Superdirective Arrays With a Tunable Tradeoff Between Directivity and Frequency-Invariance

Frequency-invariant beam patterns are often required by systems using an array of sensors to process broadband signals. If the spatial aperture is shorter than the involved wavelengths, using a superdirective beamforming is essential to get an efficient system. In this context, robustness of array imperfections is a crucial feature. In the literature, only a few approaches have been proposed to design a robust, superdirective, frequency-invariant beamformer based on filter-and-sum architecture; all of them achieve frequency-invariance by imposing a desired beam pattern. However, the choice of a suitable desired beam pattern is critical; an improper choice results in unsatisfactory performance. This paper proposes a new method of array synthesis that allows the design of a robust broadband beamformer with tunable tradeoff between frequency-invariance and directivity, without the need for imposing a desired beam pattern. The latter is defined as a set of variables that do not depend on frequency and are included in the vector of variables to be optimized. To this end, a suitable cost function has been devised whose minimum can be found in closed form. Therefore, the method is analytical and computationally inexpensive. In addition, a technique that allows obtaining a beam pattern with a linear phase over frequency is described. The results show the effectiveness of the proposed method in designing robust superdirective beam patterns for linear arrays receiving far-field signals, with special attention to microphone arrays of limited aperture.

Global synthesis of superdirective frequency‐invariant beam patterns

Frequency‐invariant beam patterns are often required by systems using an array of sensors to process broadband signals. If the spatial aperture is shorter than the involved wavelengths, the use of a superdirective beam pattern is essential to attain an efficient system. In this context, robustness to array imperfections is a crucial feature. In the literature, only a few approaches have been proposed to design a robust, superdirective, frequency‐invariant beamformer, based on a filter and sum architecture: in all of them, the frequency invariance is achieved imposing an a priori desired beam pattern. However the choice of a suitable desired beam pattern is not trivial and depends on the specific design case: an improper selection of the desired beam pattern can produce unsatisfactory performances. We propose a new method of global synthesis, computationally inexpensive, allowing to design a robust broadband beam pattern with an optimal trade‐off between the frequency invariance and the directivity, without the need of imposing a priori a desired beam pattern. The results show that the synthesized beam patterns have a directivity, a frequency‐invariance, and a robustness that are very similar to or better than those of the beam patterns obtained by the literature methods.

A Stochastic Approach to the Synthesis of a Robust Frequency-Invariant Filter-and-Sum Beamformer

Frequency-invariant beam patterns (FIBPs) are often required by systems using an array of sensors to process broadband signals. Although several methods have been proposed to design a broadband beamformer (typically characterized by a finite impulse response (FIR) filter for each sensor) with an FIBP, until now, the case in which the spatial aperture is shorter than the involved wavelengths has very rarely been considered. In such a case, the use of a superdirective beam pattern is essential to attaining an efficient system. In this context, robustness to array imperfections and random errors is a very crucial feature. In this paper, a method to design a robust broadband beamformer that produces an FIBP for a data-independent superdirective array is proposed and compared with other potential approaches. The method generates a far-field beam pattern that reproduces the desired profile over a very wide frequency band (also if the array is shorter than the wavelength) and is based on a stochastic approach to the direct synthesis of the FIR filters. The very simple implementation and the resulting robustness of the attained filter-and-sum beamformer to array imperfections increase the applicability of the system. This fact is particularly important in the context of a audio signal processing carried out by microphone arrays, which is the main application considered in this paper.

Designing superdirective microphone arrays with a frequency-invariant beam pattern

Frequency-invariant beam patterns are often required in systems using an array of sensors to process broadband signals. Although several methods have been proposed to design a broadband beamformer [typically realized with a finite-impulse-response (FIR) filter for each sensor] with a frequency-invariant beam pattern (FIBP), until now the case in which the spatial aperture is shorter than the involved wavelengths has very rarely been addressed. In such a case, the use of a superdirective beam pattern is essential for attaining an efficient system. In this paper, a novel method to design a broadband beamformer that produces an FIBP for a data-independent superdirective array is proposed and compared with other potential approaches. The method generates a far-field beam pattern that reproduces the desired profile over a very wide frequency band, also if the array is shorter than the wavelength. Two steps are necessary: 1) the generation of many apodizing windows at different frequency values by a stochastic method and 2) the synthesis of the FIR filters with the Parks-McClellan technique. At the end of the design chain, the very simple implementation and the robustness of the attained broadband beamformer to array imperfections increases the applicability of the system, for instance, in audio signal processing using microphone arrays