Spectral coherence and hyperbolic solutions applied to time difference of arrival localisation

Time difference of arrival (TDOA) source localisation is based on intersecting hyperboloids from spatially diverse sensor pairs. TDOA is commonly applied to avian source localisation with field studies finding that sources located inside array boundaries are localised with higher accuracy than sources outside the array. We examine the constitutive hyperbolic equations to demonstrate that at locations interior to an array, hyperbolas intersect at large angles translating into higher localisation accuracy with insensitivity to noise and timing errors, and then use the equations to assess expected accuracy in response to noise and timing errors. We also introduce the use of cross-spectral coherence as a requisite for computation of time delays by cross-correlation in order to minimize the likelihood of interference from noise and reduce the need for manual preprocessing of spectrograms or correlation functions. A small-aperture array is evaluated with a field test finding results consistent with analytical solutions suggesting that a lack of sensor spatial diversity coupled with signal timing accuracy is the primary source of localisation error, but that bearing discrimination remains accurate even for a small-aperture and sources external to the array.