Fast Evaluation of the Room Transfer Function Using Multipole Expansion

Reverberation in rooms is often simulated with the image method due to Allen and Berkley (1979). This method has an asymptotic complexity that is cubic in terms of the simulated reverberation length. When employed in the frequency domain, it is relatively computationally expensive if there are many receivers in the room or if the source or receiver positions are changing with time. The computational complexity of the image method is due to the repeated summation of the fields generated by a large number of image sources. In this paper, a fast method to perform such summations is presented. The method is based on multipole expansion of the monopole source potential. For offline computation of the room transfer function for N image sources and M receiver points, use of the Allen-Berkley algorithm requires O(NM) operations, whereas use of the proposed method requires only O(N+M) operations, resulting in significantly faster computation of reverberant sound fields. The proposed method also has a considerable speed advantage in situations where the room transfer function must be rapidly updated online in response to source/receiver location changes. Simulation results are presented, and algorithm accuracy, speed, and implementation details are discussed. For problems that require frequency-domain computations, the algorithm is found to generate sound fields identical to the ones obtained with the frequency-domain version of the Allen-Berkley algorithm at a fraction of computational cost