An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties

The LMS algorithm and learning identification, which presently are typical adaptive algorithms, have a problem in that the speed of convergence may decrease greatly depending on the property of the input signal. To avoid this problem, this paper presents a geometrical discussion as to the origin of that defect, and proposes a new adaptive algorithm based on the result of the investigation. Comparing the convergence speeds of the proposed algorithm and the learning identification by numerical experiment by computer, great improvement was verified. The algorithm is extended to a group of algorithms which includes the original algorithm and the learning identification, which are called APA (affine projection algorithm). It is shown that APA has some desirable properties, such as, the coefficient vector approaches the true value monotonically and the convergence speed is independent of the amplitude of the input signal. Clear conclusions are also obtained for the problem as to what noise is included in the output signal when an external disturbance is impressed or the degree of the adaptive filter is not sufficient.