Comparison of Expectation-Maximization to Direct Cosine Transforms for FT-Raman Data
FT-Raman spectroscopy is thought to be limited by the multiplex disadvantage. The Raman optical signal is optically filtered to reduce Rayleigh scattering. The scattering spectrum is subsequently multiplexed with a Michaelson interferometer. The Michaelson interferometer produces an interferogram signal as a function of mirror displacement. The interferogram signal is "transformed," producing the Raman scattering signal. The later is typically performed using a Fast Fourier Transform (FFT) algorithm. The FFT is limited by the Nyquist sampling frequency and intrinsic phase information.
Shown below is an one-sided interferogram produced from Nd:YAG excited sample. The sample and interferogram were supplied by Dr. Kirk Michaelian from the Canadian NRC in Alberta. The interferogram is relatively free of phase errors. The raw interferogram was cropped to produce the one-sided one illustrated below. In addition, the baseline was restored to the average value, as is necessarily true for real interferograms. Preliminary analysis showed that the sample interval corresponds to a mirror displacement of 1/2 the HeNe laser wavelength. This is typical of Michaelson interferometers. An internal HeNe laser serves as a wavelength marker and triggers the digital data sampling. In this case only the first 512 data points are shown. The interferogram was used to estimate the Stokes-shifted Raman spectrum with up to 4096 data points.
The direct cosine
transform was first used to reconstruct the Raman spectrum. The direct cosine transform is
a least-squares fit of the cosine-like functions that are produced (in theory) with the
Michaelson interferometer. Shown below are the spectrum estimatesobtained using
consecutively longer interferogram data sets. The dark blue spectrum is that obtained with
512 points, the red with 1025 points, the yeallow with 2048 points, and the light blue
with 3096 data points in the interferogram. Spectra amplitudes get consecutively
larger due to the fact that there is more signal in the interferogram with increased
number of points. Although it is difficult to see at this resolution, the spectrum
features get sharper with number of interferogram points. The 4096-point spectrum estimate
shows large amplitude features near around 9400 wavenumbers.
The spectra estimated
using the Expectation-Maximization (EM) algorithm are similar to those found using the
direct cosine transform. The main difference between the two methods of transformation is
the apparent lack of Gibbs phenomina in the EM estimates in the large features arounf 9400
wavenumber. The Gibbs phenomina arises due to a finite sampling (mirror displacement)
range. It is a natural consequence of transforms using transendental functions. The lack
of Gibbs phenomina may help interpretation of FT-Raman spectra when there are a number of
sharp features.
Link to the Word 97 spread sheet, EM-TRANS.XLS, here.