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.