An optimization method for frequency-invariant beamforming with arbitrary sensor arrays

This paper proposes a beampattern optimization method for frequency-invariant beamforming with arbitrary sensor arrays, which is an extension of the previously proposed method for circular sensor arrays. Based on the criterion of minimizing the mean square error between the synthesized beampatterns at each frequency and the desired beampattern, the relationship between the weighting vectors and the desired weighting vector can be accurately obtained. Then, the relevant performance parameters, such as the directivity factor, error sensitivity function and minimum mean square error, can all be expressed as the accurate closed-form functions of the desired weighting vector. With the above conclusions, a multi-constraint optimization problem is formulated to calculate the optimal desired weighting vector under different constraints. The optimal frequency-invariant beampatterns and their performance parameters can be readily achieved using the derived functions. Thus, the proposed method can not only obtain the optimal desired weighting vector, but also achieve the broadband frequency-invariant beampatterns with only one optimization calculation of the desired weighting vector. Simulations and experimental results show that the proposed method can provide a good trade-off among the directivity, robustness, and frequency invariance.