Triple cross-correlation for enhancing noisy signals

Triple-correlation-based cancelers are derived for enhancing noisy signals. The filter impulse response is obtained by solving a set of linear equations involving second-order statistics of the available data. It is shown that the filter thus derived attempts to cancel the noise by essentially matching the cross-correlation between the reference signal and the unknown noise and that between the reference signal and the estimate of the noise. As an extension of this property a filter is derived that matches the triple cross-correlation between the reference signal and the unknown noise with the triple cross-correlation between the reference signal and the estimate of the noise. The impulse response of this filter is obtained by solving a set of linear equations involving third-order statistics of the observations. It is shown that if the noise and the reference signal are related by a linear time-invariant transformation then cancelers based on the true, second-order and higher-order statistics are equivalent and cancel the noise exactly. In other cases the second-order canceler cancels the noise in the mean-square sense. The triple-correlation-based noise canceler outperforms the classical design when the reference signal is corrupted by additive Gaussian noise of unknown covariance. Simulation results illustrating the performance of the second-order and higher-order canceler implemented in batch form are presented.