A new structure for adaptive AR spectral estimation based on multi-band decomposition of the linear prediction error is introduced and the mathematical background for the soulution of the related adaptive filtering problem is derived. The presented structure gives rise to AR spectral estimates that represent the true underlying spectrum with better fidelity than conventional LS methods by allowing an arbitrary trade-off between variance of spectral estimates and tracking ability of the estimator along the frequency spectrum. The linear prediction error is decomposed through a filter bank and components of each band are analyzed by different window lengths, allowing long windows to track slowly varying signals and short windows to observe fastly varying components. The correlation matrix of the input signal is shown to satisfy both time-update and order-update properties for rectangular windowing functions, and an RLS algorithm based on each property is presented. Adaptive forward and backward relations are used to derive a mathematical framework that serves as a basis for the design of fast RLS alogorithms. Also, computer experiments comparing the performance of conventional and the proposed multi-band methods are depicted and discussed.

A new adaptive AR spectral estimation method is proposed. While conventional least-squares methods use a single windowing function to analyze the linear prediction error, the proposed method uses a different window for each frequency band of the linear prediction error to define a cost function to be meinemized. With this approach, since time and frequency resolutions can be traded off throughout the frequency spectrum, an improvement on the precision of the estimates is achieved. In this paper, a wavelet-like time-frequency resolution grid is used so that low-frequency components of the linear prediction error are analyzed through long windows and high-frequency components are analyzed through short ones. To solve the optimization problem for the new cost function, special properties of the correlation matrix are used to derive an RLS algorithm on the order of M2, where M is the number of parameters of the AR model. Computer simulations comparing the performance of conventional RLS and the proposed methods are shown. In particular, it can be observed that the wavelet-based spectral estimation method gives fine frequency resolution at low frequencies and sharp time resolution at high frequencies, while with conventional methods it is possible to obtain only one of these characteristics.