Why and How Savitzky–Golay Filters Should Be Replaced

Savitzky–Golay
(SG) filtering, based on local least-squares
fitting of the data by polynomials, is a popular method for smoothing
data and calculations of derivatives of noisy data. At frequencies
above the cutoff, SG filters have poor noise suppression; this unnecessarily
reduces the signal-to-noise ratio, especially when calculating derivatives
of the data. In addition, SG filtering near the boundaries of the
data range is prone to artifacts, which are especially strong when
using SG filters for calculating derivatives of the data. We show
how these disadvantages can be avoided while keeping the advantageous
properties of SG filters. We present two classes of finite impulse
response (FIR) filters with substantially improved frequency response:
(i) SG filters with fitting weights in the shape of a window function
and (ii) convolution kernels based on the sinc function with a Gaussian-like
window function and additional corrections for improving the frequency
response in the passband (modified sinc kernel). Compared with standard
SG filters, the only price to pay for the improvement is a moderate
increase in the kernel size. Smoothing at the boundaries of the data
can be improved with a non-FIR method, the Whittaker–Henderson
smoother, or by linear extrapolation of the data, followed by convolution
with a modified sinc kernel, and we show that the latter is preferable
in most cases. We provide computer programs and equations for the
smoothing parameters of these smoothers when used as plug-in replacements
for SG filters and describe how to choose smoothing parameters to
preserve peak heights in spectra.