Spatial correlation for directional sensors in arbitrary noise fields

Spatial correlation provides the basis for determining the covariance matrix used in calculating the optimum gain performance and Direction-of-Arrival (DOA) Cramer-Rao Lower Bound (CRLB) for arrays of directional sensors. A general formulation for the spatial correlation between two pressure sensors in a noise field of arbitrary directional distribution F was presented in [1]. The approach involves the expansion of the directional noise density F(¿, ¿, ¿) in cylindrical or spherical surface harmonics. This approach has been extended to the case of two directional sensor components l and k whose directional amplitude responses are g_l(¿, ¿, ¿) and g_k(¿, ¿, ¿). The spatial correlation between the directional sensors can be obtained using a similar harmonic expansion procedure by replacing F(¿, ¿, ¿) with H(¿, ¿, ¿) = g_l(¿, ¿, ¿)g_k(¿, ¿, ¿)F(¿, ¿, ¿).