Enhancement of sinusoids in colored noise and the whitening performance of exact least squares predictors

The extraction of sinusoids in white noise using least squares predictors has attracted a lot of attention in the past, mainly in the context of the adaptive line enhancer (ALE). However, very few results exist for the practical colored noise case or for the whitening performance of the predictors. We use a matrix formulation to derive the optimal least squares coefficients and frequency response of the D-step predictor or ALE for sinusoids (real or complex) in additive colored noise. Several cases are considered, and in particular, new formulas for the amplitude gain are obtained. In low-pass background noise, the amplitude gain of the sinusoids becomes essentially a monotonically increasing function of their frequency, and a decreasing function for high-pass noise. For the whitening application, signal-to-noise ratio (SNR) bounds of the output are derived when the input is a white signal plus a sinusoidal interference. We also give a state-space model and stochastic interpretations of our analysis of the D-step predictor, providing connections to other related areas. To enable filtering of nonstationary complex inputs, as well as multichannel and multiexperiment data, a complex vector version of the ladder algorithm is presented that can be used to implement the ALE, noise cancelling, and noise inversion for narrow-band interference rejection.