Adaptive enhancement of multiple sinusoids in uncorrelated noise

The steady-state behavior of the adaptive line enhancer (ALE), a new implementation of adaptive filtering that has application in detecting and tracking narrow-band signals in broad-band noise, is analyzed for a stationary input consisting of multiple sinusoids in white noise. It is shown that the steady-state performance of an L-weight ALE for this case can be modeled by the L × L Wiener-Hopf matrix equation and that this matrix equation can be transformed into a set of 2N coupled linear equations, where N is the number of sinusoids. It is also shown that the expected values of the ALE weights in steady state can be written as a sum of sinusoids and that the amplitude of each sinusoid is coupled to that of all other sinusoids by coefficients that approach zero as the number of ALE weights becomes large. The analytical results are compared to experimental results obtained with a hardware implementation of the ALE of variable length (up to 256 weights) and show good agreement. Theoretical expressions for linear predictive spectral estimates are also derived for multiple sinusoids in white noise. Comparisons are made between the magnitude of the discrete Fourier transform of the ALE weights and the linear predictive spectral estimate for two sinusoids in white noise.