Non-Wiener solutions of the adaptive noise canceller with a noisy reference

The performance of the adaptive noise canceller is typically analyzed in terms of Wiener filter theory. However, it has recently been shown by Glover that a narrow-band signal in the reference at a frequency for which there is no correlated signal in the primary gives rise to a notch in the output spectrum, a solution which lies outside the scope of the classical theory. In this paper an analytic expression for the output spectrum is derived for the more general case of a reference signal containing a sinusoid in white noise. It is found that although the notch depth is a decreasing function of the reference signal-to-noise ratio, the exact relationship also depends on the primary input; thus, with noise present, there is no longer a time-invariant relationship between the output and primary spectra. These results are shown to reduce to those of Glover and the classical Wiener filter in the appropriate contexts.