True spectrum

Slow scanning (i) of the mass spectrum over a GC peak for substance A gives spectrum (a), but rapid scanning (ii) gives spectrum (b), which is much closer to the true spectrum (c). 

Column bleed gives a mass spectrum (a) that is mixed with an eluting component to give a complex spectrum (b). By subtracting (a) from (b), the true spectrum (c) of the eluting component is obtained. 

The projection of yinto V is y , , which is usually is much closer to the true spectrum ytrue . than y , itself. A substantial amount of the noise is removed in the projection but not all. 

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

In principle we could deconvolute the experimental spectrum with the instrumental lineshape, if that were known, to recover the true spectrum. In our example we have some good experimental evidence as to the form of the instrumental lineshape since the acetone signal is (apart from small carbon-13 satellites) a singlet, its experimental shape is just the instrumental lineshape convoluted by a Lorentzian of width l/(7rr2 ), where is the spin-spin relaxation time of the acetone protons. How can we use this experimental evidence to correct the imperfect experimental spectrum The simplest way to deconvolute one function fi uj) by another f2 ( ) is to Fourier transform the ratio of their inverse Fourier transforms ... 

If we are observing a spectrum o(x ) with the aid of an instrument having a characteristic response function s(x — x ), then i represents the data acquired. If we have a perfectly resolving instrument, then s(x — x ) is a Dirac function, and our data i(x) directly represent the true spectrum, that is, o(x). In this case we have no need for deconvolution. 

When we are solving for an unknown spectrum, each data point contains some information about the component spectral lines. A data point far from the center of a given line would be expected to contain very little information about that line. A hypothetical model of the spectrum could incorporate widely varying estimates of the amplitude of that line without influencing the fit to the data point in question. Three data points moderately distributed near the line center—say, spanning the interval between the half-maximum points—affix the parameters of the line more reliably. Instead of taking equally spaced samples of a trial solution as independent unknowns (the deconvolution approach), we can express the sought-after true spectrum in terms of its spectral-line parameters—amplitudes, half-widths, positions, and so on—provided that we can assume such a model with some confidence. 

A ratio containing this quantity and the data may be used to construct a new estimate of the true spectrum,... 

With the present definition of r, however, an overcorrection that would normally disappear gradually through ensuing iterations results in a value of d(k)(x) that vanishes for all subsequent iterations. This behavior occurs because further corrections to that value are prohibited. To use the method, the investigator is compelled to take small values for r0. Even in this case, erroneously nonphysical values of o(k) that have been forced to zero are never allowed to return to the finite range that might better represent the true spectrum o(x). This form of the method therefore demands excessive computation and yields a solution that, although physically realizable, is not the best achievable estimate. 

Subsequent research by Herschel (1971), Kikuchi and Softer (1977), and Frieden (Chapter 8) has refined the concept in a way that provides an explicit and sensible accounting for noise contributions. This work also provides solutions that incorporate a type of prior knowledge not used before. In particular, the users may express their bias by proposing a prior spectrum or guess as to what the true spectrum o(x) might look like. Furthermore, they may express their relative confidence in the guess by specifying a probability of occurrence for each value that may be assumed by an element of the estimate o(x). Both the prior spectrum and its associated user-conviction probability function may be obtained from past experience by statistical analysis. In Chapter 8, Frieden examines the possibilities of maximum and minimum conviction in connection with the types of prior... 

The luminescence emission spectrum of a specimen is a plot of luminescence intensity, measured in relative numbers of quanta per unit frequency interval, against frequency. When the luminescence monochromator is scanned at constant slit width and constant amplifier sensitivity, the curve obtained is the apparent emission spectrum. To determine the true spectrum the apparent curve has to be corrected for changes of the sensitivity of the photomultiplier, the bandwidth of the monochromator, and the transmission of the monochromator with fre-... 

The scanning rate for the emission wavelength has to be slow enough to allow the instrument to respond to changes in intensity with wavelength. Too fast a scan, even with a number of accumulations, results in an apparent spectrum that can differ markedly from the true spectrum (Fig. B3.6.4). When, as is usually the case with proteins, the emission band is rather broad (Fig. B3.6.I), a scan rate setting in the region of 100 nm/min will usually be appropriate. 

The true spectrum of a compound can be recovered if its concentration window is not completely embedded inside the concentration window of a different compound. 

The true spectrum has a pair of doublets each split by an identical amount. Note that no line appears at the true chemical shift, but it is easy to measure the chemical shift by taking the midpoint of the doublet. 

The process of Fourier transformation converts the raw data (e.g. a time series) to two frequency domain spectra, one which is called a real spectrum and die other imaginary (diis terminology conies from complex numbers). The true spectrum is represented only by half the transformed data as indicated in Figure 3.18. Hence if there are 1000 datapoints in the original time series, 500 will correspond to the real transform and 500 to the imaginary transform. 

Prior knowledge is available before the experiment. There is almost always some information available about chemical data. An example is that a true spectrum will always be positive we can reject statistical solutions that result in negative intensities. Sometimes much more detailed information such as lineshapes or compound concentrations is known. 

Fig. 2 The VCD spectra of camphor with the anisotropy ratio on the order of magnitude of 10 5 are used as examples to illustrate the performance of several generations of FTIR-VCD spectrometers. Raw (bottom traces) and subtracted (top traces) VCD spectra of 0.6 M R-camphor in CC14 in a 0.10 mm CaF2 cell from 4,000 to 2,000 cm-1 are shown (a) SPM, single PEM only, for a pair of enantiomers. The subtracted spectrum is the true spectrum (b) SPM, single PEM, with only solvent baseline correction (c) SPM/RHWP, single PEM (SPM) with RHWP, with only solvent baseline correction (d) DPM, DPM only, with only solvent baseline correction (e) DPM/RHWP, with both DPM and RHWP, with only solvent baseline correction. Reproduced with permission from [41]. Copyright (2008) Springer...

FIGURE 3.6 (Continued) (b) Example of foldover of frequencies above the Nyquist frequency with two phase-sensitive detectors. The lower spectrum (spectral width 5000 Hz) faithfully reproduces the true spectrum, but the upper spectrum (spectral width 3600 Hz) shows that peaks near + 2000 Hz now display an aliased frequency near — 1800 Hz and appear near the right-hand end of the spectrum. Sample D-glucorono-6,3-lactone-l,2-acetonide in DMSO-<4 at 500 MHz. Spectra courtesy of Joseph J. Barchi (National Institutes of Health). 

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