Adaptive Filters

CHAPTER 1 Scalar-Valued Data

In this first part of the book we focus on the basic, yet fundamental, problem of estimating an unobservable quantity from a collection of measurements in the least-mean-squares sense. The estimation task is made more or less difficult depending on how much information the measured data convey about the unobservable quantity. We shall study this estimation problem with increasing degrees of complexity, starting from a simple scenario and building up to more sophisticated cases.

The material is developed initially at a slow pace. This is done deliberately in order to familiarize readers (and especially students) with the basic concepts of estimation theory for both real- and complex-valued random variables, as well as for scalar- and vector-valued random variables. We hope that, by the end of our exposition, the reader will be convinced that these different scenarios (of real vs. complex and scalar vs. vector) can be masked by adopting a uniform vector and complex-conjugation notation. The notation is introduced gradually in the two initial chapters and will be used throughout the book thereafter.

Before plunging into a study of least-mean-squares estimation theory, and the reasons for its widespread use, the reader is advised to consult the review material in Sees. A. 1-A.4. These sections provide an intuitive explanation for what the variance of a random variable means. The sections also introduce several useful concepts such as complex- and vector-valued ...