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 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 ALE or the D-step predictor, for sinusoids (real or complex) in additive colored noise. Several cases are considered; 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. This performance is achievable using (possibly complex) exact least-squares recursions, such as our ladder form implementations.