Fast Fourier transform and singular value decomposition formulations for patch nearfield acoustical holography

Abstract

Nearfield acoustical holography (NAH) requires the measurement of the pressure field over a complete surface in order to recover the normal velocity on a nearby concentric surface, the latter generally coincident with a vibrator. Patch NAH provides a major simplification by eliminating the need for complete surface pressure scans-only a small area needs to be scanned to determine the normal velocity on the corresponding (small area) concentric patch on the vibrator. The theory of patch NAH is based on (1) an analytic continuation of the patch pressure which provides a spatially tapered aperture extension of the field and (2) a decomposition of the transfer function (pressure to velocity and/or pressure to pressure) between the two surfaces using the singular value decomposition (SVD) for general shapes and the fast Fourier transform (FFT) for planar surfaces. Inversion of the transfer function is stabilized using Tikhonov regularization and the Morozov discrepancy principle. Experimental results show that root mean square errors of the normal velocity reconstruction for a point-driven vibrator over 200-2700 Hz average less than 20% for two small, concentric patch surfaces 0.4 cm apart. Reconstruction of the active normal acoustic intensity was also successful, with less than 30% error over the frequency band.

Acoustic source localization exploring theory and practice

“…If the vibrations at the source extend beyond the field grid, the errors are larger and the differences are small. Contrary to the results found by Williams [96], who found a large improvement in accuracy for Patch holography, the results of the current study indicate that all methods except windowing and zero padding can be the most accurate depending on the source vector and the frequency. The differences between the methods in the area below the sensors tend to be less than 5%.…”

“…To achieve this, an algorithm named patch nearfield acoustic holography has been proposed by Williams and co-workers [95,96]. The algorithm uses the following steps.…”

Analytic solutions to the acoustic source reconstruction problem

“…The first approach concerns Fourier-based methods, such as the planar or the cylindrical NAH, which rely on the relationship in the spatial wavenumber domain between a given source distribution and its radiated field that, in principle, should be recorded over a complete hologram surface (Williams 1999). This constraint might, however, be alleviated if one uses procedures such as patch NAH for the analytic continuation of the sound field outside the measurement aperture (Williams et al 2003) or the statistically optimized NAH (SONAH) that makes use of a surface-to-surface projection scheme based on the plane wave expansion of the sound field (Steiner & Hald 2001). The SONAH process avoids the use of a spatial Fourier transform and its inherent truncation effects.…”

“…The SVD is a powerful tool that has been used in conjunction with the regularization techniques to solve a number of NAH problems, for instance to provide IBEM source reconstruction results robust to the presence of noise in the measured field data (Schuhmacher et al 2003), extract dominant acoustic modes in the HELS method (Zhao & Wu 2005) or extend patch NAH to complex source geometries while avoiding the replication problem of the measurement window (Williams et al 2003).…”

Patch near field acoustic holography based on particle velocity measurements

“…The key problem of PNAH is a numerical tangential extension of the measurement aperture. Saijyou and Yoshikawa 14 proposed a patch approach based on an iterative procedure, Williams and co-workers [15][16][17] extended it by using singular value decomposition and improved its accuracy by regularization, Sarkissian 18,19 developed a PNAH method using the superposition method, and Lee and Bolton 20 investigated PNAH in a cylindrical geometry. Yet another PNAH technique has been proposed by Steiner and Hald.…”