Fast evaluation of the room transfer function using the multipole method

Reverberation in rooms is often simulated with the image method due to Allen and Berkley (1979). This method has asymptotic complexity that is cubic in terms of the simulated reverberation length. When employed in the frequency domain, it is relatively computationally expensive to use for many receivers in the room or in a dynamically changing configuration 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 the 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 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 considerable speed advantage in situations where the room transfer function must be rapidly updated online in response to the 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 that are identical to the ones obtained with the frequency-domain version of the Allen-Berkley algorithm at a fraction of computational