Basis Expansion Adaptive Filters for Time-Varying System Identification

Abstract

ABSTRACTblock adaptive (OBA) and the OBA shifting (OBAS) algorithms [9]. However, the tracking performance of all these block algorithms We show by simulations that our proposed BE-LMS is able to track ple stochastic gradient type of algorithms, lead by the least mean the system time variation better than the NLMS, while maintaining squares (LMS) family, to more elaborated algorithms like those begood convergence properties in case of overlapping blocks. To even longing to the recursive least squares (RLS) class [1,2,3].further improve the tracking capabilities, we also propose another A common feature of most of these algorithms is that they have BEM-based adaptive algorithm that has a nice connection with the been initially derived assuming a time-invariant system model. In matrix generalization of the momentum LMS (MLMS) algorithm case of a time-varying model, adaptive algorithms like LMS try to [14]. track the time variation of the system impulse response, as analyzed in [4,5,6]. However, this tracking is done on a sample-by-sample 2. SYSTEM MODEL basis, by using limited information about the past. This past information can be better taken into account by using a block of data ratherThe system under consideration is a linear time-varying system with than one single sample. As a consequence, to track time-varying additive noise, whose input-output relation is expressed by systems, a block-by-block algorithm could potentially be more ben-L-1 eficial than a sample-by-sample approach, especially if the blocks L-1 are overlapping. 31n = h0,1xn l + en,(1) Traditionally, block adaptive filtering techniques have been de-1-0 veloped to reduce complexity in the time-invariant case, by exploitwhere yn is the output, hn,1 is the linear time-varying system iming fast convolution methods. For example, the block LMS (BLMS) pulse response, assumed finite with L taps, Xn is the input, and en algorithm [3,7] uses an average gradient over a block of samples is the additive noise. Given Xn and yn, our goal is to find a linear to estimate the impulse response. In case of non-overlapping blocks, time-varying filter hn,1 such that BLMS has approximately the same steady-state performance of LMS [7,8], with computational savings due to the shared operations. How-L-1 ever, if the blocks are overlapping, as in the sliding-window LMS Yn =E hn,lXn-1 (SW-LMS), the convergence and the steady-state performance of l=0 block adaptive algorithms improve, at the expense of an increased complexity caused by the repeated processing of the same data [3].is as close as possible to yn. The filter h,l can then be consideredThe convergence of both BLMS and SW-LMS can further be imas an estimate of the system impulse response hn,1.provd byallwinga tme-vryig stp sze, eadng t th optmum To describe the proposed adaptive algorithms, we will first reprovd b alowig a imevaringstepsiz, ladig totheoptmum shape the system input-output relation, and then we will make use This research was supported in part by NWO-STW under the VIDI proof the limited time-varyin...

Recursive inverse adaptive filtering algorithm

“…Adaptive filtering techniques are very well-known topics in signal processing [1], [2], [3]. The RLS algorithm is one of the most popular techniques [4], [5], [6].…”