Fast recursive estimation using the lattice structure

This paper presents the theory for a rapidly converging adaptive linear digital filter. The filter weights are updated for every new input sample. This way the filter is optimal (in the minimum mean square error sense) for all past data up to the present, at all instants of time. This adaptive filter has thus the fastest possible rate of convergence. Such an adaptive filter, which is highly desirable for use in dynamical systems, e.g., digital equalizers, used to require on the order of N
²
multiplications for an N-tap filter at each instant of time. Recent “fast” algorithms have reduced this number to like 10 N. One of these algorithms has the lattice form, and is shown here to have some interesting properties: It decorrelates the input data to a new set of orthogonal components using an adaptive, Gram-Schmidt like, transformation. Unlike other fast algorithms of the Kalman form, the filter length can be changed at any time with no need to restart or modify previous results. It is conjectured that these properties will make it less sensitive to digital quantization errors in finite word-length implementation.