Frequency domain tracking characteristics of adaptive algorithms

The problem of tracking time-varying linear systems is discussed. The focus is on the model quality in terms of the mean square error (MSE) between the true (momentary) transfer function and the estimated one. This MSE is thus a function of frequency. The exact expression for the MSE is complicated, but simple expressions that are asymptotic in the model order are developed for model structures of finite impulse response (FIR) character. Simulations verify that these simple expressions are quite reliable and insightful even for moderate model orders. Expressions are developed for three basic adaptation algorithms (recursive identification algorithms), viz. the least-mean-squares algorithm, the recursive least-squares algorithm with exponential forgetting, and a tracking algorithm based on the Kalman filter. The results apply both to slowly time-varying systems and to the model recovery after an abrupt change in the system dynamics