Identification of impulse response by use of smooth input signals

Among the many identification methods of impulse response of linear discrete system, the recursive least squares method is well known because of its ease of handling and excellent convergency. This paper analyzes why the least squares estimates of an impulse response sequence degrade in convergency when one utilizes smooth input signals for identification and the number of data is finite. It also presents an effective identification method even under such a condition. The magnitude of the eigenvalue of the correlation matrix of input signal is adopted as a quantitative index of input signal smoothness and it is demonstrated that small valued eigenvalues increase the mean squares error of least squares estimate. Using the idea of eigenvalue expansion, we also present a new online identification method which truncates small eigenvalues associated with the input autocorrelation matrix. It is shown that the optimal number of eigenvalues can be theoretically determined in consideration of the tradeoff between the parameter error due to observation noise and the parameter error due to the truncation of small eigenvalues. Moreover, by recursively estimating the eigenvalues and the eigenvectors using the power method, the proposed method is generalized to a recursive estimation method to evaluate the estimate at each time instance. By also recursively updating the optimal number of eigenvalues, the proposed method is shown to have an excellent convergency through numerical examples compared to the conventional recursive least squares method.