Transform-domain adaptive filters: an analytical approach

Transform-domain adaptive filters refer to LMS filters whose inputs are preprocessed with a unitary data-independent transformation followed by a power normalization stage. The transformation is typically chosen to be the discrete Fourier transform (DFT), although other transformations, such as the cosine transform (DCT), the Hartley transform (DHT), or the Walsh-Hadamard transform, have also been proposed in the literature. The resulting algorithms are generally called DFT-LMS, DCT-LMS, etc. This preprocessing improves the eigenvalue distribution of the input autocorrelation matrix of the LMS filter and, as a consequence, ameliorates its convergence speed. In this paper, we start with a brief intuitive explanation of transform-domain algorithms. We then analyze the effects of the preprocessing performed in DFT-LMS and DCT-LMS for first-order Markov inputs. In particular, we show that for Markov-1 inputs of correlation parameter ρ∈[0,1], the eigenvalue spread after DFT and power normalization tends to (1+ρ)l(1-ρ) as the size of the filter gets large, whereas after DCT and power normalization, it reduces to (1+ρ). For comparison, the eigenvalue spread before transformation is asymptotically equal to (1+ρ)²/(1-ρ)². The analytical method used in the paper provides additional insight into how the algorithms work and is expected to extend to other input signal classes and other transformations