An error bound for a noise canceller

The degradation in performance of the least-mean-square estimation (LMSE) is evaluated after replacing the covariance matrices with estimates. Taking advantage of the statistical properties of the complex Wishart matrix and its inverse, the decrease in performance stemming from this substitution is investigated. The performance criterion is based on the computation of the final output error, which includes covariance deviations. This is carried out for a certain class of linear observation models, involving an underlying problem of noise cancelling using noise alone references (NARs), namely, array processing techniques using a known array manifold. Performance limits of the adaptive noise canceller are investigated for the case of a spectral filter to illustrate the ideas. It is shown that its performance depends on the quadratic coherency between the corrupted signal and the NAR. It is preferable to run no processing when this coherency is not strong enough. A rule is proposed in order to decide when to switch to the no-processing scheme. Contrary to some previous results, the results given are available without any approximations, provided the statistical properties are satisfied. In the case of a special noise canceller, these properties are asymptotically satisfied. In the case of a transversal noise canceller acting in the time domain, these results are valid for white signals and noises