Smoothing spectra

FIGURE 4.5 Illustration of polynomial smoothing on near-infrared spectra of water-methanol mixtures, (a) Original spectra, (b) Smoothed spectra. 

Without doubt, however, the greatest difference between the two is that the sophisticated electronics of the newer instrument enable it to be directly connected to an Infrared Data Station , a microcomputer system which can smooth spectra, produce an average spectrum from a number of scans, subtract one spectrum from another, or reformat a spectrum. In addition, it is possible to identify possible structural features of an unknown compound and to match the spectrum against a reference library stored on magnetic disks, to facilitate a rapid identification of unknowns3). It is interesting to note that the basic optical specifications of the two instruments are not all that dissimilar ... 

The rectangular window function is a simple truncating function which can be applied to transformed data. This function has zero values above some pre-selected cut-off frequency, /c, and unit values at lower frequencies. Using various cut-off frequencies for the truncating function and applying the inverse transform results in the smoothed spectra shown in Figure 13. 

An increasing number of commercially available spectrophotometers are controlled by a computer and allow the user to extract spectral data in twoCricket Graph HI, Deltagraph and many more can be used to smooth spectra, perform subtractions and additions of whole spectra, normalizations and calculations of derivatives. 

Figure 16-1 shows a typical output from a commercial IR spectrophotometer. Although the y-axis is shown as linear in transmittance, modern computer-based spectrophotometers can also produce spectra that are linear in absorbance. The abscissa in this spectrum is linear in wavenumbers with units of reciprocal centimeters. A wavelength scale is also shown at the top of the plot. Computer-based spectrophotometers can also produce a variety of other spectral formats such as linear in wavelength, baseline corrected, and derivative and smoothed spectra,... 

The expression under the radical gives the multiplying factor for the noise standard deviation for the computed derivative (or smoothed spectrum, but that is not our topic here, we will address only the question of the effect on derivatives), and can be computed solely from the convolution coefficients themselves, independently of the effect of the convolution on the signal part of the S/N ratio. 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

Because of the smooth spectrum of energies available at the synchrotron (Fig. 1) a single-crystal monochromator (Fig. 2) is used to select a small spread of energies (typically with AE/E 10 ). In other respects, most of the laboratory hardware required for diffraction experiments at the synchrotron is similar to that used in the laboratory and will not be dealt with in detail here. Several excellent texts and at least one comprehensive volume (Klug and Alexander 1974), detail the requirements for powder diffractometry. 

Figure 8 Smoothing with a 5-point moving average. Each new point in the smoothed spectrum is formed by averaging a span of 5 points from the original data...

Assuming all noise is removed then the result is the true spectrum. Conversely, from Equation (33), if the smoothed spectrum is subtracted from the original, raw data, then a noise spectrum is obtained. The distribution of this noise as a function of wavelength may provide information regarding the source of the noise in spectrometers. The procedure is analogous to the analysis of residuals in regression analysis and modelling. 

Fig. 6.7. Absorption cross-section spectrum of three silicon samples at 2K (the concentration at the metal-insulator transition is 3.5 x 1018cm-3). Note the evolution from sharp isolated lines to asymmetrically broadened ones due to close pairs and finally to a smooth spectrum dominated by the absorption of random clusters (after [253]). Copyright 1981 by the American Physical Society...

Ideally, a source is required which delivers the highest flux in a wavelength range at least between 0.6 A and 1.6 A and with a smooth spectrum. 

Fig. 14. ATR-FTIR spectrum of the oxidized form of 3-methylthiophene polymer on the surface of a reflection element in contact with an electrolyte solution containing tetrabutylammonium perchlorate. The smoothed spectrum originates from 200 added interferograms. The spectrum of the electrolyte solution was subtracted.

One of the difficulties in data smoothing is the choice of the correct value for n. Unfortunately, there are no strict criteria for the selection of n. The analyst should be guided mainly by experience and by the general effects of smoothing on the results. In particular, if the data represent energy spectra with many peaks, the smoothed spectrum may tend to flatten the peaks and fill the valleys. 

The results are illustrated by Fig. 15, where the green oxide furnishes a fairly smooth spectrum with a possible maximum at 5p, and a depression at 3-2p. 

Randomized phase, amplitude, and time domain relationships on successive passes. Ensemble average of amplitude approaches smooth spectrum. External source... 

It is not possible to use the STRAIGHT parametrization in the HMMs, since estimating statistically reliable acoustic models using high-dimensional observations is very difficult. To avoid this problem, some systems (e.g. [ ]) have used mel-cepstral coefficients converted from the smoothed spectrum with a recursive algorithm [ ]. For the same reason, the aperiodicity measurements must also be averaged, usually on five frequency sub-bands (0-1000, 1000-2000, 2000-4000, 4000-6000 and 6000-8000 Hz). 

Fig. 4 77K excitation spectra of low-light membranes, detection wavelength= 840 nm. Smooth-spectrum vertical exc and. detection, noisy spectmm=polaiization... 

Application of the Blackman window removes the side lobes and gives a smooth spectrum with a peak at 10/2tt Hz, but a lower intensity than obtained from the truncated data. Berens and Wilson [72] use a four-term Blackman-Harris window in computing the rotational spectra of CO and Ar from m.d. They note that by... 

FD giving the Fourier spectrum bl. This spectrum is then multiplied by an apodization function (a boxcar function is shown in b2). Finally, the Fourier transform of the truncated Fourier spectrum is computed, yielding the smoothed spectrum. 

FIGURE 6.9 Spectra of wool (a) Original spectrum with a noise spike at 1800 nm, MPA (25 pts) smoothed spectrum, and Fourier (50 pair) smoothed spectrum, (b) MPA smoothing of the noise spike, (c) Polynomial smoothing of the noise spike, (d) Fourier smoothing of the noise spike. 

This curve is used to represent the spectrum between the two center points in this region. Cubic equations are computed for every four-point region of the spectrum, so that the entire range of interest is covered. This method is used by some manufacturers of FT-IR spectrometers to produce a smooth spectrum on a screen or on... 

Be aware that smoothing lowers the resolution. The final resolution is a function of the initial resolutipn, amount of zero filling and interpolation, the degree and number of points in the polynomial. It would be simple for manufacturers to calculate and display the resolution of the smoothed spectrum. 

Infrared Spectroscopy. The infrared spectroscopy of adsorbates has been studied for many years, especially for chemisorbed species (see Section XVIII-2C). In the case of physisorption, where the molecule remains intact, one is interested in how the molecular symmetry is altered on adsorption. Perhaps the conceptually simplest case is that of H2 on NaCl(lOO). Being homo-polar, Ha by itself has no allowed vibrational absorption (except for some weak collision-induced transitions) but when adsorbed, the reduced symmetry allows a vibrational spectrum to be observed. Fig. XVII-16 shows the infrared spectrum at 30 K for various degrees of monolayer coverage [96] (the adsorption is Langmuirian with half-coverage at about 10 atm). The bands labeled sf are for transitions of H2 on a smooth face and are from the 7 = 0 and J = 1 rotational states Q /fR) is assigned as a combination band. The bands labeled... 

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