Application of Hilbert Transform-Based Methodology to Computer Modelling of Reverberant Sound Decay in Irregularly Shaped Rooms

A study into the application of the Hilbert transform in numerical simulations of reverberant sound decay in irregularly shaped enclosures is presented. It is shown that there are some limitations in the use of this integral transform to exponentially decaying harmonic signals because the integration result consists not only of a signal, which differs from the original one by a phase shift of π/2, but also of a decaying nonoscillating signal occurring due to the fact that a spectrum of exponential function is unbounded. An initial amplitude of this signal is directly proportional to the ratio between a damping coefficient and a mode frequency, thus for lightly damped rooms, where this ratio is much smaller than unity, it can be neglected. Results of numerical simulation carried out for L-shaped enclosure indicate that the Hilbert transform is a useful tool in calculating instantaneous properties of reverberant sound, especially an envelope of the sound pressure level. It is of special importance in the case of irregularly shaped rooms, where a deviation from the exponential sound decay often occurs because of differences between reverberant responses for particular modes.