Archives of Acoustics,

**34**, 4, pp. 491-506, 2009### 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 [ampersand]pi;/2, but also of a decaying non-oscillating

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.

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 [ampersand]pi;/2, but also of a decaying non-oscillating

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.

**Keywords**: Hilbert transform; analytic signal; irregularly shaped enclosures; reverberant room response; reverberation time; early and late decay times

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