Scaling Compressive Equivalent Source Method for Wide-Area Sound Field Reconstruction From Non-Synchronous Measurements

Abstract

Non-synchronous measurements (NSM), in which a single microphone array is sequentially moved, can reduce the number of simultaneously acquired channels required for wide-area sound field reconstruction. However, amplitude and phase relationships among measurements are unknown, and methods relying on spatial continuity or correlation between nearby array positions can lose accuracy when the positions are widely separated. This paper proposes the Scaling Compressive Equivalent Source Method (Scaling-CESM), which is formulated for a stationary sound field generated by a single coherent source whose position and directivity remain unchanged throughout the sequential measurements. At each frequency, complex scaling coefficients are assigned to the measurement positions and jointly estimated with a common sparse equivalent-source vector through a convex second-order cone program. Numerical simulations used a 25-channel <inline-formula> <tex-math notation="LaTeX">$5\times 5$ </tex-math></inline-formula> grid microphone array with 0.05 m spacing at nine measurement positions over 100–2000 Hz and <inline-formula> <tex-math notation="LaTeX">$N=200$ </tex-math></inline-formula> equivalent sources randomly distributed within a 7.5 cm radius around the source. Scaling-CESM was compared with single-measurement, CSM-based non-synchronous, and synchronous reconstruction methods. In coverage-expansion simulations, Scaling-CESM maintained an NMSE of approximately −17 dB at movement intervals of 0.6–0.7 m, outperformed the CSM-based NSM methods, and required only 2.23 s per frequency versus more than 90 s for the CSM-based methods. It maintained high accuracy under free-field and very weakly reverberant conditions (<inline-formula> <tex-math notation="LaTeX">$\mathrm {RT}_{60}\leq 0.02$ </tex-math></inline-formula> s) but degraded substantially for <inline-formula> <tex-math notation="LaTeX">$\mathrm {RT}_{60}\geq 0.05$ </tex-math></inline-formula> s. Semi-experimental results using measured room impulse responses showed the same tendency: accurate wide-area reconstruction for direct sound, but substantial degradation under the measured reverberation condition (<inline-formula> <tex-math notation="LaTeX">$\mathrm {RT}_{60}=0.38$ </tex-math></inline-formula> s).