Improving the frequency response of Savitzky-Golay filters via colored-noise models

Abstract

Savitzky-Golay (SG) filters are finite impulse response (FIR) realizations of least-squares polynomial regression and they are widely used for filtering (e.g. smoothing, interpolating, predicting, differentiating) and processing (e.g. detecting and classifying) non-stationary signals in non-Gaussian noise. For such inputs, the Wiener filter is biased and the Kalman filter is sub-optimal. Sequentially-correlated (i.e. ‘colored’) noise models are an integral part of the Wiener filter and an optional addition to the Kalman filter; however, their use in SG-filters has been overlooked in recent times. It is shown here that colored (wide-band and narrow-band) noise models are readily incorporated into a standard SG-filter and that this also addresses the well-known deficiency of their poor frequency-selectivity/configurability. A wide-band noise model sets the main-lobe/side-lobe width/height and provides physical justification for band-limited design procedures described elsewhere. The proposed narrow-band noise model, with arbitrarily placed side-lobe nulls, has the potential to outperform other SG filters when sinusoidal interferers of known frequency are present. The utility of these ‘whitened’ SG-filters is illustrated in a hypothetical pulse/peak-detection application using a test statistic that is shaped by the noise model.