The spread function

Let us introduce a spread function as a measure of difference between the resolving kernel and the delta function  

We will prove that the resolving kernel given by (3.121) delivers us the minimum of 

We then c tlcul tt( the derivative of the. sjiread lunctioii witli resjx ct to h,(ri) )  

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The variance of the spreading function, i.e. the spreading factor of monodisperse polymer in a SEC column may be written as... 

In optics and spectroscopy, resolution is often limited by diffraction. To a good approximation, the spread function may appear as a single-slit diffraction pattern (Section II). If equal-intensity objects (spectral lines) are placed close to one another so that the first zero of one sine-squared diffraction pattern is superimposed on the peak of the adjacent pattern, they are said to be separated by the Rayleigh distance (Strong, 1958). This separation gives rise to a 19% dip between the peaks of the superimposed patterns. 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

For convenience and simplicity in this section, we assume that the spread function st has its peak value at l = 0 and that it has L nonvanishing values on either side of s0. The development based on these assumptions may easily be generalized. 

Now let us assume that the true object is zero for all subscript values less than 1. When we do the convolution, then, im must have a value of zero for m < 1 — L. When we perform the numerical convolution starting with m 1 — L, we compute zeros as we slide the spread function toward the region of finite ok. Finally, when the end of the spread function encounters the first nonzero ok, we obtain the first nonzero value of ik. We illustrate this particular sum of products with an elementary example in which L = 1 and m — 0 ... 

The next value of the image is given by sliding the spread function down and forming the appropriate sum of products ... 

When the spread function is normalized, we obtain oc = ic. The estimates of the object are thus easy to determine by starting at the ends. 

Under these conditions, the spread function is termed shift invariant and s is called a Toeplitz matrix. When Eq. (17) does not hold, the spread function is called shift variant, and the spreading can no longer be described by simple convolution. 

Kawata et al. (1979) introduced a concept of reblurring to guarantee that the transfer function does not have negative values. They recognized that the convolution of the spread function with a reversed version of itself yields, in Fourier space, a function that cannot have negative values ... 

Here o and i are the object and image vectors, respectively, and s is the spread-function matrix as defined in Chapter 3. Gold was able to show that proper convergence is assured if the following conditions hold ... 

We may write the spread-function contribution due to the x-ray source as the following Lorentzian doublet ... 

An analytical integration of an integrodifferential equation under a singular time boundary is always a complicated matter. The treatment of the method, based on a representation of the delta functional as a Fourier transform, and working in the complex plane, would be out of place in this report. It can be found in detail in Ref. 7) where also the solution obtained is discussed. It is shown that this solution is especially simple if the elution curves show a positive skewness, i.e. if they are tailed on the right-hand-side of their maximum (this is always true in PDC and GPC). A renormalization of the found concentration profile and a recalculation of the coordinates (z, t) to the elution volumina (V, V) then yield the spreading function of the considered column (Greschner 7))... 

There is however, a way conceivable to avoid these difficulties, namely the combination of GPC- and PDC-measurements performed with the same sample for which the resolution of the GPC-column is good at a possibly narrow MWD. Since the mathematical structure of the spreading functions of the GPC- and the PDC-column is the same, the parameters of Eq. (44a) (e.g. D(P), ctD(P), yD(P) and SD(P)) can then be fitted for GPC by comparing the MWDs calculated from GPC- and PDC-measurements on the same sample by the standard method shown below. Although inverted integral transforms would have to be included in such a non-linear fit, it should not be too hard to find a suitable mathematical algorithm for that iteration. However, so far no efforts have been made in this direction. 

A convolution of these values with the cochlear spreading function follows. Due to the non-normalized nature of the spreading function, the convolved versions of eb and cb should be renormalized. The convolved unpredictability, ch, is mapped to the tonality index, tb, using a log transform just as the unpredictability was mapped to the tonality index, c(t, CO), from equation (2.12). 

The solution of the above equation in order to obtain W(y) requires an appropriate form of the spreading function and the numerical values of its parameters. Furthermore, to convert W(y) into a size distribution requires a relationship between the mean retention volume y and the particle diameter D (i.e., a calibration curve). 

Estimation of the Spreading Function. When the injected sample is monodispersed, peak broadening occurs solely due to axial dispersion if y is the mean retention volume of the chromatogram, then the detector response is given by ... 

For many cases, especially if the spreading is small, the spreading function can be approximated by the normal Gaussian distribution function G in the form ... 

Fig.2. The spreading function. AFg = unit sample volume, AFg = unitelementvolume, AFs/AFg = constant. (Reprinted, with permission, from Methods in Enzymology, Vol. 137. San Diego, CA Academic, 1988.)...

In such cases, as stated by Meira and co-workers [10], the quality of results depends on computational refinements. It is necessary to add specific constraints related to the chromatographic problem rejection of negative values or unrealistic fluctuations in the weight distribution. The normal way is to invert the large matrix defining the spreading function for any position on the elution volume scale. With modem computational facilities, that becomes easy, but it is still not trivial to obtain stable results, and proper filtering processes are useful. 

A raw fractogram h V) is a convolution of the fractogram corrected for the zone broadening g(Y) and the spreading function G(V, Y) which is a detector response to a uniform species having the elution volume Y ... 

For correcting band broadening, the main difficulty is in obtaining precise mapping of the spreading function of the system. Normally, this needs very high quality standards and TGIC offers new possibilities in that area. Computational techniques are now sufficiently efficient... 

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