Wavelet transform domain adaptive FIR filtering

This paper presents and studies two new wavelet transform domain least mean square (LMS) algorithms. The algorithms exploit the special sparse structure of the wavelet transform of wide classes of correlation matrices and their Cholesky factors in order to compute a whitening transformation of the input data in the wavelet domain and minimize computational complexity. The procedures differ in the exact estimates they use and in the way they identify the data dependent whitening transformation. The first approach explicitly computes a sparse estimate of the wavelet domain correlation matrix of the input process. It then computes the Cholesky factor of that matrix and uses its inverse to whiten the input. The complexity of this first approach is O[N log(2) (N)]. In contrast, the second approach computes a sparse estimate of the Cholesky factor of the wavelet domain correlation matrix of the input process directly. This second approach has a computational complexity of O[N log (N)] floating-point operations. However, it requires a more complex bookkeeping procedure. Both algorithms have a convergence rate that is faster than that of time-domain LMS and discrete Fourier transform (DFT) or discrete cosine transform (DCT)-based LMS procedures. The paper compares the two procedures and analyzes their mean and mean square performance.