Adaptive Filters

CHAPTER 27 Efficient Block Convolution

The transform-domain adaptive filters of the previous chapter were motivated by the desire to improve the convergence performance of LMS by exploiting the de-correlation properties of unitary transforms such as the DFT and the DCT. It turns out that these same transforms, and other similar ones, are also useful in reducing the computational cost per iteration of LMS below the O(M) figure. This cost reduction can be achieved by processing the data on a block-by-block basis rather than on a sample-by-sample basis.

27.1 MOTIVATION

As motivation, consider the setting of Fig. 27.1, which shows an FIR channel of length M; assumed long. The channel is excited by a zero-mean random sequence {u(i)} and its output is another zero-mean random sequence {y(i)}. At any particular time instant i, the state of the channel is captured by the regression vector

and its output is measured in the presence of noise,

(27.1)

where the column vector g represents the channel impulse response, and v{i) is a zero-mean noise sequence uncorrelated with u_i Let G(z) denote the transfer function associated with g, i.e.,

(27.2)

where the {g(k)} are the individual samples ...