A Fast Adaptive Filter Algorithm Using Eigenvalue Reciprocals as Stepsizes

It is shown that there exist finite optimum update positions in the gradient direction of the LMS algorithm. The optimum stepsizes to reach these positions are in a discrete set, i.e., the reciprocal eigenvalues of input signal autocorrelation matrix. On this basis a new adaptive filter (ADF) algorithm is proposed. The discrete cosine transform, a fairly good approximation of the Karhu-nen-Loeve transform for a large number of signal classes, is used to estimate the optimum stepsizes. A block-averaging operation is also used for smoothing the gradient estimate. Computer simulations show that the proposed ADF algorithm provides fast convergence rates when the input signal autocorrelation matrix has either large or small eigenvalue spread (ratio of the largest to the smallest eigenvalues). The number of multiplications required by the new ADF is about 11 log2 (N) + 12, which is comparable to the 10 log2 (N) + 8 required by the fast LMS algorithm.