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

Abstract

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.Index Terms-Beamforming robustness, broadband beamforming, filter-and-sum beamforming, frequency-invariant beam patterns, microphone arrays, superdirective arrays.

Enhancing resolution properties of array antennas via field extrapolation: application to MIMO systems

“…In the foregoing example, the transformation matrix (17) takes on the following values− whose rows can be regarded as beamforming weighing windows with nominal looking directions π/2, ± arc cos(1/2.1), and ± arccos(2/2.1), respectively. Inspection of the values of the entries in (18), in particular of those with alternating signs in the first row (associated to the broadside direction), indicates that the array is being operated in super-directive conditions, with a degree of super-directivity depending on the choice of parameters involved in the extrapolation procedure.…”

“…An interesting aspect that is worth pursuing relates to the properties of the beam patterns that are implicitly generated by the rows of the transformation matrix in (16) or (17). Actually, it can be shown that through a proper choice of the parameters M, Δ, and N, the number Q of these beams and their directivity can easily exceed those achievable through uniform-weighing beamforming applied to critically spaced arrays of equal length.…”

“…Actually, the index G n (also referred to in the literature with different names, e.g., white noise gain in [13, p. 1366], [17] and [21]) can be regarded as a margin offered by the array to counteract its own noise, that can be exploited to reduce complexity and cost of the front-end low-noise amplifiers. Now, while for a uniform-weighing beamformer the parameter G n is indeed greater than unity (and equal to M for the specific case of uniform-weighing vector), thus representing a true gain, instead for a super-directive array G n may occur to be less than one, and even far smaller than one as the array super-directive properties get more and more pronounced.…”

“…Similar concepts are also found in the theory underlying the Capon beamformer and variants thereof [13,Chap. 6] and in the immense literature dealing with super-directivity issues (examples of recent papers are in [14], e.g., [15] and [16], or [17,18], just to cite different contexts).…”

Design of minimax robust broadband beamformers with optimized microphone positions

“…Furthermore, as evident from earlier designs for superdirective narrowband arrays [12]- [16], broadband beamformers designed for physically-compact applications can likewise become very sensitive to errors in array imperfections and therefore robustness constraints need to be incorporated in the design [3,17]- [23]. In [17]- [21], the statistics of microphone characteristics are taken into account to derive broadband beamformers that are robust to microphone mismatches, while in [3,22,23] the white noise gain (WNG) is incorporated in the design to ensure that the beamformer is robust to spatial white noise and array imperfections. The use of the WNG constraint is not new and has been used in earlier beamformer designs to ensure robustness in superdirective beamformers [12]- [14].…”