Exploiting Symmetries in the Design and Implementation of Polynomial/Trigonometric Beamformers for Uniform Circular Arrays

The polynomial and trigonometric beamformers for uniform circular arrays (UCAs), which are respectively based on the polynomial and trigonometric interpolation, have been shown to enable dynamic and continuous beam steering over the entire 360° azimuth range through simple online parameter tuning. The ability of beam steering of the polynomial/trigonometric beamformers, however, comes at the cost of higher complexity in the beamformer design and implementation. This paper derives some symmetry properties of the coefficients of the polynomial/trigonometric beamformers for UCAs. By jointly exploiting the symmetries of the beamformer coefficients, a reduced-dimension design approach and a simplified implementation structure for the polynomial/trigonometric beamformers for UCAs are proposed. It is shown that the dimension of the beamformer design optimization problem can be reduced by as much as over 60% using the reduced-dimension design, enabling a fast design process. Moreover, the simplified implementation structure is highly cost-effective. Compared with the original implementation structure, a reduction rate of nearly 75% in the coefficient storage requirement can be achieved. And the number of the multipliers and adders can also be substantially reduced by around 50%.