This paper aims at providing a tutorial introduction to adaptive filters
operating in the subband or transform domain. In doing so we start off
by reviewing the classical time-domain algorithms such as the {\em least
mean square} algorithm (LMS) and its block oriented relative (BLMS).
Following this we introduce subband
domain algorithms through the block oriented discrete Fourier transform
(DFT) adaptive algorithms.
It is pointed out that these algorithms are essentially
computationally efficient implementations of the BLMS algorithms with some
possibilities for improved convergence speed. By virtue of the block
oriented DFT being a particular instantiation of a critically sampled
filter bank, we argue that this family of algorithms belong to the class
of subband adaptive filtering algorithms. Subsequently we introduce
true subband adaptive filters based on more general subband decompositions.
Reasons why intuitively appealing structures based on critically sampled
filter banks does not work well are given. After indicating some possible
remedies for this, we present structures using orthogonal transforms for
the *non-subsampled* signal decomposition. These algorithms lead to
improved convergence when compared to the LMS algorithm, but at the expense
of some additional computations. All the subband adaptive filters
presented in this paper make
use of the LMS algorithm in some way or another. The objective of introducing
the subband decomposition is to either reduce computational complexity or
to improve the speed of convergence through signal conditioning.