The Use of Orthogonal Transforms for Improving Performance of Adaptive Filters

The least mean square (LMS) adaptive algorithm is probably the best known and the most widely used real time adaptive filtering algorithm due to its simple computational requirements. However, as VLSI digital processors become cheaper and more readily available, the question arises as to whether more effective real time algorithms can be found that take advantage of increased computational resources as they become available. It has been shown in the literature that a real time decomposition of the incoming signal into a set of partially uncorrelated components via an orthogonal transform, and a subsequent adaptation on these individual components, leads to faster convergence rates. In this paper, transform domain processing is characterized by the effect of the transform on the shape of the mean-square error surface. It is shown that the effect of an ideal transform is to convert equal error contours, that are initially hyperellipses in the parameter space, into hyperspheres. Five specific real-valued orthogonal transforms are then compared in terms of learning characteristics and computational complexity. Since the Karhunen-Loeéve transform (KLT) is the ideal transform for this application, and since the KLT is defined in terms of the statistics of the input signal, it is certain that no fixed-parameter transform will deliver optimal learning characteristics for all input signals. However, the simulations suggest that, with a little trial and error, transforms can be found which give much improved performance in a given situation.