Simple Algorithms for Fast Adaptive Filtering

The LMS algorithm invented by Widrow and Hoo in 1959 is the simplest, most robust, and one of the most widely used algorithms for adaptive ltering. Unfortunately , its convergence rate is highly dependent upon the conditioning of the autocorrelation matrix of its inputs: the higher the input eigenvalue spread, the slower the convergence of the adaptive weights. This problem can be overcome by preprocessing the inputs to the LMS lter with a xed data-independent transformation that, at least partially, decorrelates the inputs. Typically, the preprocessing consists of a DFT or a DCT transformation followed by a power normalization stage. The resulting algorithms are called DFT-LMS and DCT-LMS. This technique is to be contrasted with more traditional approaches such as recursive least squares algorithms , where an estimate of the inverse input autocor-relation matrix is used to improve the lter convergence speed. After placing DFT-LMS and DCT-LMS into context, we propose three diierent approaches to explain the algorithms both intuitively and analytically. We discuss the convergence speed improvement brought by these algorithms over conventional LMS, and we make a short analysis of their computational cost.