Smoothing polynomial

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. 

In order to keep the average signal amplitude unaffected a scaling factor NORM is introduced, which is the sum off all convolutes, here 35. The smoothing procedure is now 

The effect of a 5-point, 17-point and 25-point quadratic smoothing of a Gaussian peak with 0.3% noise is shown in Fig. 40.22b. Peaks are distorted as 

Convolutes for quadratic and cubic smoothing (adapted from Refs. [8,10]) 

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Bromba, M. U. A., and Ziegler, H., Efficient Computation of Polynomial Smoothing Digital Filters, Anal. Chem. 51, 1979, 1760-1762. 

Enke, C. G., and Nieman, T. A., Signal-to-Noise Ratio Enhancement by Least-Squares Polynomial Smoothing, Ana/. Chem. 48, 1976, 705A-712A. 

Fig. 40.22. Distortion (hJhn) of a Gaussian peak for various window sizes (indicated within parentheses). (a) Moving average, (b) Polynomial smoothing.

Fig. 40.23. Polynomial smoothing (noise = N(0,3%)) 5-point 17-point 25-point smoothing window and the noise left after smoothing.

Fig. 40.24. Polynomial smoothing window of 7 data points fitted with polynomials of degrees 0,1,2, 3 and 4.

Mathematically this operation can be described by the same equation (eq. (40.13)) as derived for polynomial smoothing, namely ... 

Hz). Another feature of polynomial smoothing is that smoothing and differentiation (first and second derivative) can be combined in single step, which is explained in Section 40.5.5. 

C.G. Enke and T.A. Nieman, Signal-to-noise ratio enhancement by least-squares polynomial smoothing. Anal. Chem., 48 (1976) 705A-712A. 

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

As shown in Figure 1 data from the viscometer detector and DRI are combined to yield the Intrinsic viscosity as a function of retention volume (la). This curve then was fit to a polynomial and a smoothed curved calculated. At this stage of data reduction the analyst can choose to continue to use the polynomial smoothed values of log [n](V) throughout, or continue to use the unsmoothed values for further data reduction. 

For this work, the spectrometer function s(x) was determined by the method outlined in Section II.G.3 of Chapter 2. In digitizing the data, a sample density was chosen to accommodate about 70 samples taken across the full width at half maximum of s(x). A 25-point cubic polynomial smoothing filter was used in the deconvolution procedure to control high-frequency noise. Instead of the convolution in Eq. (13), the point-successive modification described in Section III.C.2 of Chapter 3 was employed. In Eq. (24) of Chapter 3, we replaced k with the expression... 

FIGURE 4.5 Illustration of polynomial smoothing on near-infrared spectra of water-methanol mixtures, (a) Original spectra, (b) Smoothed spectra. 

Polynomial smoothing does not possess an ideal frequency-response function and can potentially introduce distortions and artifacts in smoothed signals [7], Other methods of smoothing do not possess these shortcomings. A detailed discussion of these important points is presented in Chapter 10. 

Marchand, P. and Marmet, L., Binomial smoothing filter a way to avoid some pitfalls of least squares polynomial smoothing, Rev. Sci. Instrum., 54, 1034—1041, 1983. 

The process of polynomial smoothing extends the principle of the moving average by modifying the weight vector, ea, such that the elements of describe a convex polynomial. The central value in each window, therefore, adds more to the averaging process than values at the extremes of the window. 

Consider five data points forming a part of a spectrum described by the data set X recorded at equal wavelength intervals. Polynomial smoothing seeks to replace the value of the point Xj by a value calculated from the least-squares polynomial fitted to Xj-2, Xj-u Xj, Xj+i, and Xj+2 recorded at wavelengths denoted by Xj-2, k/ i, j, Xj+1, and j+2-... 

Figure 4.4 Similar to the sliding polynomial smoothing (Savitzky Golay filter, the coefficients for 2nd order fit to a parabola) is the effect of Bromba Ziegler filters [Bromba and Ziegler, (1983c), coefficients fit to a triangle upper figure]. Both have bad low pass filter characteristics, as shown in the lower figure with the Fourier transforms of filters through 21 points each.

The relative merits of these different methods can be compared by differentiating a known mathematical function. The model we will use is y = x + x /2), X = 0. .. 4, at X = 2. Various levels of noise are imposed on the signal y (Table 3.2). The resulting derivatives are shown in Table 3.3. As the noise level reduces and tends to zero, the derivative results from applying the five-point polynomial (Equation 3.4) converge more quickly towards the correct noise-free value of 3 for the first derivative, and 1 for the second derivative (Equation 3.5). As with polynomial smoothing, the Savitzky-Golay differentiation technique is available with many commercial spectrometers. 

FIGURE 5-14 Least-squares polynomial smoothing convolution integers (a) quadratic five-point integers, (b) first-derivative cubic five-point integers, (c) second-derivalive quadralic five-point integers. 

Ilecause least-squares polynomial smoothing is so widely used to enhance the quality of analytical data, it is important to note the advaniages and limiiations of... 

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