Passive wideband cross correlation with differential Doppler compensation using the continuous wavelet transform

An estimate of the time difference for the signal emitted by a stationary source to arrive at two spatially separated sensors is given by the time displacement that maximizes the cross-correlation function. For a fast moving source, however, this estimate is found to be in error because the time scales of the received signals are different for the two sensors. The correct time delay can be extracted by evaluating the continuous wavelet transform, which has the same functional form as the wideband cross-ambiguity function. When the signal-to-noise ratio is high, the coordinates of the ambiguity surface’s global maximum provide reliable estimates of both the differential time of arrival (or time delay) and the ratio of the time scales of the signals received by the two sensors. The continuous wavelet transform is computed using the one-step chirp z-transform method, the cross-wavelet transform method, and the two-step methods where multirate sampled replicas of the sensor waveforms are cross correlated, or else the sensor waveforms are interpolated using the discrete Fourier transform prior to cross correlation. The latter method is applied to real acoustic data recorded from an orthogonal configuration of three microphones during the low-altitude transit of a jet aircraft. The resulting time delay estimates are used to calculate the variation with time of the azimuth and elevation angles of the aircraft during the transit.