Recent Advances in Overcoming the Independence Assumption in Adaptive Filtering Analyses

One important strength of adaptive filtering algorithms is their ability to learn from their interactions with the environment. Such a flexibility comes with a price, since their performance strongly depends on adjustable parameters. In order to provide to the designer guidelines and guaranties for both performance and stability, the open literature often relies on stochastic models that predict the learning capabilities of different algorithms. Unfortunately, usually some statistical hypotheses are assumed to make the mathematics tractable. The most critical of them is the independence assumption, which states that the adaptive coefficients are statistically independent from the excitation data. Such an assumption hampers the accuracy of the theoretical predictions in the case of large step sizes, which is a common choice in practice. Fortunately, a technique that avoids the independence assumption, respectively exact expectation analysis, has experienced some advances. In this paper, it is demonstrated that such recent advances can be used to address a complex configuration of the least mean squares algorithm, where the additive noise is colored and the ideal plant the adaptive scheme intends to estimate is nonlinear.