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Stochastic models that predict adaptive filtering algorithms performance usually employ several assumptions in order to simplify the analysis. Although these simplifications facilitate the recursive update of the statistical quantities of interest, they by themselves may hamper the modeling accuracy. This paper simultaneously avoids for the first time the employment of two ubiquitous assumptions often adopted in the analysis of the least mean squares algorithm.The first of them is the so-called independence assumption, which presumes statistical independence between adaptive coefficients and input data. The second one assumes a sufficient-order configuration, in which the lengths of the unknown plant and the adaptive filter are equal. State equations that characterize both the mean and mean square performance of the deficient-length configuration without using the independence assumption are provided. The devised analysis, encompassing both transient and steady-state regimes, is not restricted neither to white nor to Gaussian input signals and is able to provide a proper step size upper bound that guarantees stability.

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In general, theoretical analyses of adaptive filtering algorithms employ statistical approximations in order to render the derivations tractable. Among such hypotheses, the statistical independence between the current adaptive coefficients and past input vectors is a very popular one. Unfortunately, this simplification gives rise to discrepancies with respect to empirical results, especially for large values of the step-size parameter. In this Letter, this issue is overcome by the usage of an exact expectation analysis (i.e. a stochastic model that does not employ the above-mentioned independence assumption) of the least-mean-squares adaptive algorithm. The authors analysis is also generalised in order to address the common case of coloured additive noise, an issue that is so far missing from the literature. The accuracy of the advanced model is verified through simulations.