Skew-circulant matrix formulation for transient near-field acoustical holography

Near-field acoustical holography is a widely used technique to visualize sound in three dimensions. However, the monitoring of short transients with such method is inaccurate because of the generation of wraparound. This is due to solving discrete convolutions with the fast Fourier transform algorithm. Yet, visualization of transient signals leads to useful applications, such as investigation of impact noises in industrial environments, which is a significant cause of hearing impairment. Typical solution to this problem involves adding zero padding to avoid wraparound; however, calculations are thus increased substantially. In the present paper, a solution is proposed based on the use of circulant and skew-circulant matrices. The method allows resolution of the discrete convolution with the fast Fourier transform algorithm without generating wraparound error. The resulting convolution is linear and does not require the use of zero padding. Current development of the method allows its application over the time domain, and standard convolution is applied to the spatial domains. When fully developed in three dimensions, computation cost should be similar to that of the classical formulation without zero padding. In this paper, the theoretical development of the circulant matrix holography method in 1D is presented, followed by an application to simulated data. The resulting sound field is then compared to the field obtained with standard near-field acoustical holography. It is shown that the proposed method is an efficient solution to accurately monitor transient sounds without generating wraparound error.