Window optimization issues in recursive least-squares adaptive filtering and tracking

• Main Authors: Sadiki, T., Triki, M., Slock, D.T.M.
• Format: Conference Proceeding
• Language: English
• Subjects: Adaptive filters; Applied sciences; Circuit properties; Convergence; Costs; Covariance matrix; Detection, estimation, filtering, equalization, prediction; Eigenvalues and eigenfunctions; Electric, optical and optoelectronic circuits; Electronic circuits; Electronics; Exact sciences and technology; Filtering algorithms; Frequency filters; IIR filters; Information, signal and communications theory; Noise measurement; Resonance light scattering; Shape; Signal and communications theory; Signal, noise; Telecommunications and information theory
• DOI: 10.1109/ACSSC.2004.1399277

In this paper we consider tracking of an optimal filter modeled as a stationary vector process. The filtering operation is interpreted by the RLS algorithm, which depends on the window used in the least-squares criterion. To arrive at a recursive LS algorithm requires that the window impulse response can be expressed recursively (output of an IIR filter). In practice, only two popular window choices exist (with each one tuning parameter): the exponential weighting (W-RLS) and the rectangular window (SWC-RLS). However, the rectangular window can be generalized at a small cost for the resulting RLS algorithm to a window with three parameters (GSW-RLS) instead of just one, encompassing both SWC and W-RLS as special cases. Since the complexity of SWC-RLS essentially doubles with respect to W-RLS, it is generally believed that this increase in complexity allows for some improvement in tracking performance. We show that, with equal estimation noise, W-RLS generally outperforms SWC-RLS in causal tracking, with GSW-RLS still performing better, whereas for noncausal tracking SWC-RLS is by far the best (with GSW-RLS not being able to improve). When the window parameters are optimized for causal tracking MSE, GSW-RLS outperforms W-RLS, which outperforms SWC-RLS. We also derive the optimal window shapes for causal and noncausal tracking of arbitrary variation spectra. It turns outs that W-RLS is optimal for causal tracking of AR (1) parameter variations whereas SWC-RLS if optimal for noncausal tracking of integrated white jumping parameters, all optimal filter parameters having proportional variation spectra in both cases.