Optimum short-time polynomial regression for signal analysis

Abstract

We propose a short-time polynomial regression (STPR) for time-varying signal analysis. The advantage of using polynomials is that the notion of a spectrum is not needed and the signals can be analyzed in the time domain over short durations. In the presence of noise, such modeling becomes important, because the polynomial approximation performs smoothing leading to noise suppression. The problem of optimal smoothing depends on the duration over which a fixed-order polynomial regression is performed. Considering the STPR of a noisy signal, we derive the optimal smoothing window by minimizing the mean-square error (MSE). For a fixed polynomial order, the smoothing window duration depends on the rate of signal variation, which, in turn, depends on its derivatives. Since the derivatives are not available a priori, exact optimization is not feasible. However, approximate optimization can be achieved using only the variance expressions and the intersection-of-confidence-intervals (ICI) technique. The ICI technique is based on a consistency measure across confidence intervals corresponding to different window lengths. An approximate asymptotic analysis to determine the optimal confidence interval width shows that the asymptotic expressions are the same irrespective of whether one starts with a uniform sampling grid or a nonuniform one. Simulation results on sinusoids, chirps, and electrocardiogram (ECG) signals, and comparisons with standard wavelet denoising techniques, show that the proposed method is robust particularly in the low signal-to-noise ratio regime.