Functions convolutions

Noticing the fact that the formula for determining surface deformation takes the form of convolution, the fast Fourier transform (FFT) technique has been applied in recent years to the calculations of deformation [35,36]. The FFT-based approach would give exact results if the convolution functions, i.e., pressure and surface topography take periodic form. For the concentrated contact problems, however. 

By way of illustration the spectrometry example is worked out. Two functions are involved in the process, the signal/(X,) and the convolution function h(k). Both functions should be measured in the same domain and should be digitized with the same interval and at the same r-values (in spectrometry X-values). Let us furthermore assume that the spectrum/(A,) and convolution function h(k) have a simple triangular shape but with a different half-height width. 

When tuning the spectrometer at another wavelength, the centre of the convolution function is moved to that wavelength. If we encode the convolution function relative to the set point h 0), then we obtain the following discrete values (normalized to a sum = 1) ... 

The convolution or smoothing function, h f), used in moving averaging is a simple block function. However, one could try and derive somewhat more complex convolution functions giving a better signal-to-noise ratio with less deformation of the underlying deterministic signal. 

We can multiply two polynomials together easily with the convolution function conv (). 

The mathematical symbol is used for the convolution function which means in detail... 

Through the use of these formulas, Savitzky-Golay convolution coefficients could be computed for a convolution function using any odd number of data points for the convolution. 

Equation 56-27 contains scaled coefficients for the zeroth through third derivative convolution functions, using a third degree polynomial fitting function. The first row of equation 56-27 contains the coefficients for smoothing, the second row contains the coefficients for the first derivative, and so forth. 

Therefore we now turn to the noise part of the S/N ratio. As we saw just above, the two-point derivative approximation can be put into the framework of the S-G convolution functions, and we will therefore not treat them as separate methods. 

The nature of the convolution function matters, however, and so do the details of the way it is computed. To see this, let us begin by considering the two-point derivative we have been dealing with in most of this sub-series of chapters. For our first examination of the effect, let us consider that we are computing the derivative from adjacent data points spaced 1 nm apart (such as in our initial discussion of derivatives [1]). 

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,... 

We turn now to the effect of using the Savitzky-Golay convolution functions. Table 57-1 presents a small subset of the convolutions from the tables. Since the tables were fairly extensive, the entries were scaled so that all of the coefficients could be presented as integers we have previously seen this. The nature of the values involved caused the entries to be difficult to compare directly, therefore we recomputed them to eliminate the normalization factors and using the actual direct coefficients, making the coefficients more easily comparable we present these in Table 57-2. For Table 57-2 we also computed the sums of the squares of the coefficients and present them in the last row. 

There are several directions that the convolutions can be varied one is the increase the amount of data used, by using longer convolution functions as we demonstrated above. Another is to increase the degree of the fitting polynomial, and the third is to compute higher-order derivatives. In Table 57-3, we present a very small selection of the effect of potential variations. 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

Very popular is the Savitzky-Golay filter As the method is used in almost any chromatographic data processing software package, the basic principles will be outlined hereafter. A least squares fit with a polynomial of the required order is performed over a window length. This is achieved by using a fixed convolution function. The shape of this function depends on the order of the chosen polynomial and the window length. The coefficients b of the convolution function are calculated from ... 

Figure 1.7 The convolution function (r— r ) accomplishes coarse graining of an object,...

In one dimension, an exemplary choice for a convolution function is (x — aq) = exp[(x — Xi)2/B2, where B is the characteristic coarse-grained length. With this choice, the coarse-grained one-dimensional concentration is... 

Twenty-two atoms were placed randomly on a discrete lattice at positions Xi, 0 < Xi < 100. The concentration curves are continuous and have areas that are approximately equal to the number of atoms in the random sample. Each atom contributes a unit area to the coarsegrained c(x). Broader convolution functions (higher values of B) produce greater degrees of coarse graining. 

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