Analysis of a first order complex recursive LMS adaptive predictor

The statistical behaviour of a first order complex LMS adaptive predictor for recovering or cancelling a complex narrowband sine wave in additive white noise is studied. The properties of the optimum complex scalar predictor weights are investigated. A coupled set of dynamical nonlinear difference equations is derived for the mean values of the two complex weights, one pole and one zero, the MSE and the correlation between the error and the desired signal for the recursive LMS adaptive predictor. The derivation is based on the assumption of `slow adaptation´ and uses weight-data expectation splitting arguments. The nonlinear difference equations are numerically evaluated and converge to the optimal weights for a wide range of parameter values. Monte Carlo simulations and the theory are shown to be in good agreement, supporting the assumptions used to derive the mathematical model. To determine the increase in steady-state MSE caused by the adaptation process, the misadjustment error, the weight covariance matrix is studied in the neighbourhood of convergence. A matrix difference equation for the weight covariance matrix is derived and solved in steady-state. The misadjustment error is evaluated to be (1/2)ζ₀{μ₁σ²+μ₂ ζ₀} where ζ₀ is the Wiener MSE, σ² is the input power and μ₁ and μ₂ are the feed-forward and feed-back algorithm step-sizes, respectively