Spatial correlation and coherence in reverberant acoustic fields: Extension to microphones with arbitrary first-order directivity

Spatial correlation and coherence functions in reverberant sound fields are relevant to the acoustics of enclosed spaces and related areas. Theoretical expressions for the spatial correlation and coherence functions between signals representing the pressure and/or the components of the particle velocity vector in a reverberant sound field are established in the literature and most of these have also been corroborated with measurements F. Jacobsen, J. Acoust. Soc. Am. 108, 204-210 (2000)]. In the present paper, these expressions are generalized to microphones of first-order directivity, whereby the directivity can be expressed in terms of pressure and pressure gradient. It is shown that the resulting spatial correlation and coherence functions can be expressed in terms of the established spatial correlation and coherence functions. The derived theoretical expression for the spatial coherence function is validated with a modeled diffuse sound field. Further, it is compared with the experimental coherence obtained from the reverberant tails of room impulse responses measured with two common surround sound microphone setups in a concert and a lecture hall.