Popular Filters in Spectrometry Revisited

Filtering Using Gauss Filters

Gaussian filters are essentially moving average filters with filter weights whose time-dependent probability density function follows a Gaussian distribution. The sum of these weights must remain constant at one. Compared to a moving average with identical filter weights, the values of the weights are maximized at the center, at the peak of the Gaussian bell curve, and decrease symmetrically on both sides. This symmetry ensures that the position of the absorption bands remains unchanged.

Similar to running means, these filters also have other undesirable properties, as evident when analyzing their characteristics in the frequency domain (or wavenumber domain).

[h_sg,w] = freqz(b_sg,1,512);

f = fs*w/(2*pi);

figure('Position',[100 800 800 600],...
    'Color',[1 1 1])
axes('Position',[0.1 0.1 0.35 0.8],...
    'XGrid','On',...
    'YGrid','On')
line(f,abs(h_sg),...
    'LineWidth',1)
title('Magnitude Response')
axes('Position',[0.55 0.1 0.35 0.8],...
    'XGrid','On',...
    'YGrid','On')
line(f,angle(h_sg),...
    'LineWidth',1)
title('Phase Response')

Filtering Using Butterworth Lowpass Filters

Now, we use a frequency-selective filter, specifically a lowpass filter designed using the Butterworth approach. Our goal is to remove noise from xn while preserving all wavelengths of the three absorption bands, which range from approximately 200 nm to 10 nm. These wavelengths correspond to wavenumbers of 1/200 nm^(-1) = 0.0050 nm^(-1) and 1/10 nm^(-1) = 0.1 nm^(-1). Consequently, the filter should eliminate wavelengths shorter than about 5 nm, which corresponds to a wavenumber of 1/5 nm^(-1) = 0.2 nm^(-1). We employ a Butterworth filter of the order 10, resulting in a filter length of 11, similar to the other filters used previously. Additionally, we set a cutoff wavelength of 5 nm or a cutoff wavenumber of 1/5 nm^(-1) = 0.2 nm^(-1).

These filters do not have any of the undesirable effects of the previous filters. They do not alter the amplitudes of the absorption bands within the passband of the filter. In contrast, they completely eliminate the amplitudes in the filter’s stopband. As you can observe from the magnitude and phase characteristics, these filters do not share the undesirable effects of the previous filters. They do not have zeros and do not exhibit the amplitude flipping as described below, which is also mentioned on page 313 in Trauth et al., 2025, and in the blog post titled Better Avoid Running Means. The phase response is not perfectly linear, indicating some distortion—albeit minimal—along the wavelength axis. However, this distortion is corrected by using a filter.

Comparing the Frequency Response of the Filters

The frequency response of the filters, including the magnitude response (the absolute value of the complex-valued frequency response) and the phase response (the angle of the complex-valued frequency response), offers deeper insights into how the filters affect the signal.