Representation of the spatial impulse response of a room

Abstract

Microphone arrays allow for the measurement of the so-called spatial impulse response (SIR) of a room or of a concert hall. The SIR provides a local description of the reverberant field of that environment as a function of both time and space. It is shown that, under given assumptions, the SIR can be described by means of an integral operator, the so-called Herglotz wave function, which represents an infinite superposition of plane waves arriving from all possible directions. The kernel of this operator (the Herglotz kernel) contains all the information on the SIR. In practical cases only a limited amount of information is available to compute the Herglotz kernel, typically because a finite number of sensors is used for the measurement. In that respect, several alternatives are discussed to represent the Herglotz density as a sum of a finite number of basis functions. Some results for numerical simulations are then presented, which show the Herglotz kernel for simple examples. Finally, some limitations of this representation are discussed, especially those imposed by the use of real microphone arrays.