Adaptive Filters

Notation

NOTATION

The objective of this book is not limited to providing a treatment of adaptive filters, but also to bring forth connections between adaptive filtering and other filtering theories. To do so, it becomes necessary to adopt a notation that captures with relative ease the similarities and connections that exist among different filtering theories. One main reason behind our choice of notation is that in our treatment of adaptive filters we need to distinguish between four types of variables:

random, scalar, vector, and matrix variables

While in many treatments of adaptive filters, no notational distinction is usually made between ran dom quantities and their realizations, it is nevertheless important in our treatment of the subject to distinguish between the stochastic and nonstochastic domains. This distinction allows us, among other results, to describe with more transparency the connections that exist between filters that are derived in the stochastic domain (e.g., Kalman filters) and filters that are motivated by working with signal realizations (e.g., least-squares filters).

Once the reader becomes familiar with our convention, it will be straightforward to deduce the nature of a variable appearing in an equation by simply recalling the rules listed below. Whenever exceptions to these rules are used in the text, they will be obvious from the context. In some rare instances, rather than insist on following a strict convention, we may opt to relax our notation ...