A least squares based optimization method for grating lobes suppression in parametric array loudspeakers

The Parametric Array Loudspeaker (PAL) is a highly directional sound source with a small aperture. By using phased array technology, PALs can achieve sound beam steering without mechanically rotating the source. However, a challenge arises due to the ultrasonic wavelength being typically smaller than the size of the transducers, which violates the spatial Nyquist criterion and leads to the formation of unwanted grating lobes in the radiation pattern. To address this issue, this paper employs a complex weighting scheme that incorporates both element amplitude and phase shifts, effectively suppressing the grating lobes for given element positions while allowing for a compact configuration. The matrix formulation of the PAL convolution model is provided, and a least squares (LS) optimization method is proposed to obtain the optimal complex weights. Simulation and experimental results demonstrate that the LS-based optimization effectively mitigates the grating lobes of the steerable PAL, resulting in significantly reduced computation time, enabling real-time calculation of the required element weights for steering. Compared to conventional simulated annealing algorithms, the proposed method achieves a 2300-fold acceleration in computational efficiency, enabling real-time beamforming in dynamic scenarios. Additionally, the paper analyzes the impact of the number of array elements, carrier frequency, and audio frequency on PAL grating lobes behavior. The analysis reveals the existence of beam steering blind spots where grating lobe suppression is difficult to achieve.