Signal smoothing

Various mathematical manipulation schemes are available to smooth spectral data here we shall concentrate on smoothing techniques that serve to average a section of the data. They are all simple to implement on personal computers. This ease of use has led to their widespread application, but their selection and tuning is somewhat empirical and depends on the application in-hand. 

The mathematical process of implementing the moving average technique is termed convolution. The resultant spectrum, x (as a vector), is said to be the result of convolution of the original spectrum vector, x, with a filter function, tv, i.e. 

The simple moving average technique can be represented mathematically by 

The process of polynomial smoothing extends the principle of the moving average by modifying the weight vector, w, such that the elements of m 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 and the shape of a spectral peak is better preserved. 

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 xy 2,xy i,x,xy+i, and Xy+2 recorded at wavelengths denoted by Ay 2, 2y i, Ay, Ay+i, and Ay+2. 

The mathematical process of implementing the moving average technique is 

---

A wavelet defined as above is called a first-order wavelet. From Eq. (21) we conclude that the extrema points of the first-order wavelet transform provide the position of the inflexion points of the scaled signal at any level of scale. Similarly, if i/ (f) = d it)/dt, then the zero crossings of the wavelet transform correspond to the inflexion points of the original signal smoothed (i.e., scaled) by the scaling function, tj/it) (Mallat, 1991). 

Any of several weighting functions designed to bring the time-domain signal smoothly to zero at both ends of its period is suitable. 

Accelerated termination of cGMP signal Smooth muscle Rybalkin et al. 2002... 

A very simple window function for resolution enhancement is the sine bell (Fig. 3.34), which is just the function sin(x) for x = 0 to 180°. This function grows for the first half of the FID and then brings the signal smoothly to zero during the second half. We saw examples of this window in Chapter 2 (Figs. 2.9 and 2.10). We will see that the sine-bell family of... 

In addition to the smoothing methods based on digital filters and on transformations, some further possibilities exist for signal smoothing. Among these are the following ... 

Goncalves, W., Lanffedi, A.J.C., Crespilho, F.N. Development of numerical methods for signal smoothing and noise modeling in single wire-based electrochemical biosensors. J. Phys. Chem. C 115 (32), 16172-16179 (2011)... 

Figure 3-49. Smoothing by Fourier analysis (according to [76]). a) Five overlapping peaks with background noise, 128 single data points b) signal smoothing with 4 Fourier coefficients c) with 10 Fourier coefficients d) with 20 Fourier coefficients, (c) shows the best compromise between smoothing and loss of information.

Figure 4.4 System setup for LA-ICP-MS equipped with a signal smoothing device. The internal volume of the stabilizer was 52 ml. With the stabilizer device, the signal intensity can be smoothed by both the buffering of the pulsed flow and the removal of the large-sized particles.

Tunheng, A. and Hirata, T. (2004) Development of signal smoothing device For precise elemental analysis using laser ablation-ICP-mass spectrometry. 

Fig. 5.18 Area measurements of Ptgg during a measurement up to 453 K, only the runs 10, 12, 14 show a signal (smoothed) and are labeled in the legend (a). Representative maximum areas, after measurement (up to 453 K), for different sizes along with larger particles and the thin film (b)...

The following classifications concerning signal smoothing and filtering can be made ... 

---