Spectral convolution

Steady-State Spectral Convolution. The steady state absorption and emission spectra of dilute dye samples can be measured using standard spectroscopic techniques. Once the extinction coefficient, e( ), and the normalized luminescence spectrum, f(v), are known for a particular dye, the self—absorption probability r over a pathlength L in the sample containing the dye at a concentration C is given by... 

The simplest way of dealing with the removal of the instrument slit width before transformation is to perform a Fourier transform on the appropriate slit profile (which may be Gaussian or triangular or which may be measured directly in Raman spectroscopy by observation of a scattering profile from spheres). The method is illustrated by equation 2.12, and it should be emphasised that the procedure works because the observed band is, in effect, the result of a spectral convolution of the true band profile with the slit profile. 

One way to improve the signal-to-noise ratio is through convolution of the spectrum with an appropriate function such as a boxcar, Lorentzian, or Gaussian function. The operation of spectral convolution has been presented in Section 2.3 Such operations tend to distort the spectrum, as the lineshape function is altered. The broader the convolution function, the greater is the distortion of the spectrum. The most common such convolution is the Savitzky-Golay smoothing algorithm [13]. 

Finally, instmmental broadening results from resolution limitations of the equipment. Resolution is often expressed as resolving power, v/Av, where Av is the probe linewidth or instmmental bandpass at frequency V. Unless Av is significantly smaller than the spectral width of the transition, the observed line is broadened, and its shape is the convolution of the instrumental line shape (apparatus function) and the tme transition profile. 

In SXAPS the X-ray photons emitted by the sample are detected, normally by letting them strike a photosensitive surface from which photoelectrons are collected, but also - with the advent of X-ray detectors of increased sensitivity - by direct detection. Above the X-ray emission threshold from a particular core level the excitation probability is a function of the densities of unoccupied electronic states. Because two electrons are involved, incident and the excited, the shape of the spectral structure is proportional to the self convolution of the unoccupied state densities. 

The conversion of radiated power (P in watts) to luminous flux (F in lumens) is achieved by considering the variation with wavelength of the human eye s photopie response. Then the spectral power from the source (PA in, lor example, W/nnt) is convoluted with the relative spectral response of the eye (V tabulated by the CIE) according to ... 

Therefore, if A represents the spectrum, the various a represent convolution coefficients and Var(A) represents a noise source that gives a constant noise level to the spectral values, then equation 57-36 gives the noise variance expected to be found on the computed resultant value, whether that is a smoothed spectral value, or any order derivative computed from a Savitzky-Golay convolution. For a more realistic computation, an interested (and energetic) reader may wish to compute and use the actual noise that will occur on a spectrum, from the information determined in the previous chapters [6-7] instead of using a constant-noise model. But for our current purposes we will retain the constant-noise model then equation 57-36 can be simplified slightly ... 

As we mentioned, the two-point first derivative is equivalent to using the convolution function -1, 1. We also treated this in our previous chapter, but it is worth repeating here. Therefore the multiplying factor of the spectral noise variance is — l2 + l2 = 2,... 

The degree of activation of the sample is measured by post-irradiation spectroscopy, usually performed with high-purity semiconductors. The time-resolved intensity measurements of one of the several spectral lines enables to get the half-life of the radioactive element and the total number of nuclear reactions occurred. In fact, the intensity of a given spectral line associated with the decay of the radioactive elements decreases with time as Aft) = Aoexp[—t/r], where Aq indicates the initial number of nuclei (at t = 0) and r is the decay time constant related to the element half-life (r = In2/ /2), which can be measured. Integrating this relation from t = 0 to the total acquisition time, and weighting it with the detector efficiency and natural abundance lines, the total number of reactions N can be derived. Then, if one compares this number with the value obtained from the convolution of... 

Alternatively, the right-hand side can be written in terms of a projection tensor and a convolution (Pope 2000). The form given here is used in the pseudo-spectral method. 

The difference in temporal dispersion in fibers, associated with the Stokes shift and broad spectral content of fluorescence as compared with the excitation, still has to be fully investigated with respect to the effect on convolution analysis. 

Due to the convolution with the instrumental resolution, an accurate determination of the spectral shape is not possible. Therefore, the value of has to be determined by other methods or from coherent scattering measurements. 

In kinetic analysis of complex reactions, 210, 382 fluorescence decay rate distributions, 210, 357 implementation in Laplace de-convolution noniterative method, 210, 293 in multiexponential decays, 210, 296 partial global analysis by simulated annealing methods, 210, 365 spectral resolution, 210, 299. 

If b and g are peaked functions (such as in a spectral line), the area under their convolution product is the product of their individual areas. Thus, if b represents instrumental spreading, the area under the spectral line is preserved through the convolution operation. In spectroscopy, we know this phenomenon as the invariance of the equivalent width of a spectral line when it is subjected to instrumental distortion. This property is again referred to in Section II.F of Chapter 2 and used in our discussion of a method to determine the instrument response function (Chapter 2, Section II.G). 

Even if perfectly narrow spectral lines are not available, we may take a clue from this approach and measure a spectral line of known shape. De-convolution should then yield the instrument function. This technique does indeed work (Chapter 2, Section II.G.3). Some of the information needed to generate the known line shape can even be obtained directly from the observed data. 

Z OL collision frequency fractional increase in instrument response-function breadth due to convolution with narrow spectral line... 

We have shown that the radiant flux spectrum, as recorded by the spectrometer, is given by the convolution of the true radiant flux spectrum (as it would be recorded by a perfect instrument) with the spectrometer response function. In absorption spectroscopy, absorption lines typically appear superimposed upon a spectral background that is determined by the emission spectrum of the source, the spectral response of the detector, and other effects. Because we are interested in the properties of the absorbing molecules, it is necessary to correct for this background, or baseline as it is sometimes called. Furthermore, we shall see that the valuable physical-realizability constraints presented in Chapter 4 are easiest to apply when the data have this form. 

It is now possible to see that the matrix formulation has the potential for describing the more-general Fredholm integral equation. This equation corresponds in spectroscopy to the situation where the functional form of s(x) varies across the spectral range of interest. In these circumstances, s is expressed as a function of two independent variables. Although we proceed with the present treatment formulated in terms of convolutions, the reader should bear the generalization in mind. 

The resolution of overlapping spectral peaks depends on their separations, intensities, and widths. Whereas separation and intensity are predominantly functions of the sample, peak width is strongly influenced by the instrument s design. The observed line is a convolution of the natural line, a function characteristic of inelastically scattered electrons that produces a skewed base line, and the instrument function. The instrument function is, in turn, the convolution of the x-ray excitation line shape, the broadening inherent in the electron energy analyzer, and the effect of electrical filtering. This description is summarized in Table I. 

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