In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

Infrared spectroscopy of proteins.

Fourier transform ir (FTIR) spectra of 21 globular proteins have been obtained at 2 cm−1 resolution from 1600 to 1700 cm−1 in deuterium oxide solution. Fourier self-deconvolution was applied to all spectra, revealing that the amide I band of each protein except casein consists of six to nine components. The components are observed at 11 well-defined frequencies, although all proteins do not exhibit components at every characteristic frequency. The root mean square (RMS) deviation of 124 individual values from the 11 average characteristic frequencies is 1.9 cm−1. The observed components are assigned to helical segments, extended beta-segments, unordered segments, and turns. Segments with similar structures do not necessarily exhibit band components with identical frequencies. For instance, the lower frequency beta-structure band can vary within a range of approximately 15 cm−1. The relative areas of the individual components of the deconvolved spectra were determined by a Gauss–Newton, iterative curve-fitting procedure that assumed Gaussian band envelopes for the deconvolved components. The measured areas were used to estimate the percentage of helix and beta-structure for each of 21 globular proteins. The results are in good general agreement with values derived from x-ray data by Levitt and Greer. The RMS deviation between 22 values (alpha- and beta-content of 11 beta-rich proteins measured by both techniques) is 2.5 percentage points; the maximum absolute deviation is 4 percentage points.

Examination of the secondary structure of proteins by deconvolved FTIR spectra.

Infrared spectroscopy is one of the oldest and well established experimental techniques for the analysis of secondary structure of polypeptides and proteins. It is convenient, non-destructive, requires less sample preparation, and can be used under a wide variety of conditions. This review introduces the recent developments in Fourier transform infrared (FTIR) spectroscopy technique and its applications to protein structural studies. The experimental skills, data analysis, and correlations between the FTIR spectroscopic bands and protein secondary structure components are discussed. The applications of FTIR to the secondary structure analysis, conformational changes, structural dynamics and stability studies of proteins are also discussed.

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Fourier Transform Infrared Spectroscopic Analysis of Protein Secondary Structures

The theory and instrumentation for Fourier transform infrared spectrometry are discussed. These instruments measure infrared spectra of the same quality as spectra measured on grating spectrometers in about one thousandth of the time. Their sensitivity advantage for spectra measured in equal times is between a factor of 10 and 100. Commercial spectrometers are now available from nine vendors in North America. Important areas of chemistry include atmospheric monitoring, surface chemistry, and on-line identification of chromatographically separated materials. Many new biochemical and biomedical applications are also becoming apparent, including investigations of phase transitions in lipids and studies of the biocompatibility of implant polymers.

https://science.sciencemag.org/content/sci/222/4621/297.full.pdf

Fourier Transform Infrared Spectrometry

The general theory of Fourier self-deconvolution, i.e., spectral deconvolution using Fourier transforms and the intrinsic line-shape, is developed. The method provides a way of computationally resolving overlapped lines that can not be instrumentally resolved due to their intrinsic linewidth. Examples of the application of the technique to synthetic and experimental infrared spectra are presented, and potential applications are discussed. It is shown that lines in spectra having moderate signal/noise ratios (∼1000) can readily be reduced in width by a factor of 3. The method is applicable to a variety of spectroscopic techniques.

Fourier Self-Deconvolution: A Method for Resolving Intrinsically Overlapped Bands

A general formula for computing changes in the signal-to-noise ratio of a spectrum resulting from the Fourier self-deconvolution procedure is derived. Self-deconvolution reduces the intrinsic halfwidths of lines by a factor K, which is in practice limited by the noise in the spectrum. With the help of the derived formula, the rate of decrease in the SNR as a function of K for eight different smoothing (apodization) functions is studied. With high K values there are significant differences in the SNR as a result of the use of different smoothing functions. With K = 4 difference of more than 1 order of magnitude between two extreme cases is demonstrated, and with K = 5 a difference of almost 2 orders of magnitude in the SNR is predicted.

Noise in Fourier self-deconvolution.

Algorithms for the computation of bandwidths, and center of gravity and least-squares frequencies are developed. The sources of error are discussed, and it is shown that the above parameters can be determined with maximum uncertainties of hundredths or thousandths of a wave number.

Precision in Condensed Phase Vibrational Spectroscopy

We propose a new methodology to reduce the widths of spectral lines. This procedure, which is performed entirely in the time domain, combines elements of the methods of Fourier Self-Deconvolution (FSD), Maximum Entropy (MEM), and Linear Prediction (LP) in such a way as to take advantage of the efficacies of each of these procedures. The procedure is illustrated with examples.

A New Line-Narrowing Procedure Based on Fourier Self-Deconvolution, Maximum Entropy, and Linear Prediction

Several intrinsic characteristics of the Line Shape Optimized Maximum Entropy Linear Prediction (LOMEP) procedure as a line-narrowing method are discussed, and examples are presented that demonstrate its application to gas-phase IR absorption spectra, as well as to NMR spectra. An algorithm is introduced which determines the reliability of the Linear Prediction (LP) procedure on a given set of data and, hence, the reliability of the LOMEP output data.

Jyrki K. Kauppinen

Papers

Fourier Self-Deconvolution: A Method for Resolving Intrinsically Overlapped Bands

Noise in Fourier self-deconvolution.

Precision in Condensed Phase Vibrational Spectroscopy

A New Line-Narrowing Procedure Based on Fourier Self-Deconvolution, Maximum Entropy, and Linear Prediction

Characteristics of the LOMEP line-narrowing method