Abstract
An exact analysis is presented for the LMS algorithm with tonal reference signals in the presence of frequency mismatch. First, the time-varying linear system describing the LMS algorithm is converted into a time-invariant linear system. Then, a necessary and sufficient condition about the step sizes for convergence of the algorithm is derived using the Lyapunov function method and a transient behavior is analyzed. Finally, the effects of observation noise and frequency mismatch are examined without any approximations. The validity of the obtained results is shown by simulations.
Highlights
The LMS algorithm is most frequently used in the field of adaptive filtering because of its simplicity in implementation and robustness of performance
The results based on independence assumption (IA) are sometimes justified by the averaging method [2]
The averaging method relies on the slow adaptation limit so that the results are valid for small values of the step size parameter
Summary
The LMS algorithm is most frequently used in the field of adaptive filtering because of its simplicity in implementation and robustness of performance. The independence assumption (IA) is often made for stochastic situations [1]. This assumption does not hold for most of the adaptive filtering problems. The averaging method relies on the slow adaptation limit so that the results are valid for small values of the step size parameter. In general deterministic situations an exact analysis using the Lyapunov function method is presented in [2]. We treat the general case of p-dimensional input vector whose elements are complex sinusoids. A necessary and sufficient condition about the step size parameters for convergence of the algorithm is derived using the Lyapunov function method. The validity of the obtained results is shown by simulations
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