Response of the adaptive line enhancer to chirped sinusoids

The Adaptive Line Enhancer (ALE) was first described by Widrow et al, as a practical on-line technique for separating the coherent components from the incoherent components of an input signal. Subsequent work has shown this same adaptive filtering structure to be applicable to maximum entropy spectral estimation, predictive deconvolution, and narrowband interference rejection, as well as other applications which have historically used matrix inversion and Levinson's algorithm techniques. While an often cited advantage of adaptive filtering is its tolerance of slowly time-varying input statistics, the existing analyses of the ALE have concentrated on the stationary case. This paper extends these results, applying the theory to the case of inputs containing sinusoids whose frequencies slowly vary in time. This is approached by developing a time-varying eigenvalue-eigenvector description of the expected filter impulse response vector which holds for any slowly nonstationary input. These results are then used to predict the expected impulse response vector for the ALE input of stationary white noise plus a sinusoid with linearly swept frequency. The response of the ALE for this particular input signal provides useful benchmarks for dealing with more complex forms of frequency modulation.