Algorithms, filtering

Because the pseudo-inverse filter is chosen from the class of additive filters, the regularization can be done without taking into account the noise, (n). At the end of this procedure the noise is transformed to the output of the pseudo-inverse filter (long dashed lines on Fig. 1). The regularization criteria F(a,a) has to fulfill the next conditions (i) leading to an additive filter algorithm, (ii) having the asymptotic property a, —> a, for K,M... 

For an industrial application it is necessary to separate the response of a real crack from artifacts, and to derive information about the geometry and the location of the crack. For this purpose we have developed a filter which is sensitive to the characteristic features of a signal caused by a crack and amplifies it, whereas signals without these typical features are suppressed. In Fig. 5.1 first results obtained with such an iterative filter algorithm are shown. 

Furthermore, it was possible to suppress liftoff and offset effects by using efficient digital filter algorithms. These measures provide the tester with a nearly unbiased picture of the wheel that is to be tested and he can thus conduct a reliable evaluation. 

Kalman filter algorithm equations for time-invariant system states... 

The procedure of applying Eqns (3)-(7) recursively to each measurement point of a spectrum performs the Kalman filter algorithm. To start the first iteration, i.e for k=l, some initial guesses for X(0) and P(0) are required, which will allow the predicted estimates X(1 0) and P 1 0) to be obtained prior to filtering the first measurement. 

The computational approach described here, based on the combination of the Kalman filter algorithm and iterative optimization by the simulated annealing method, was able to find the optimal alignment of the pure component peaks with respect to the shifted components in the overlapped spectra, and hence, to correctly estimate the contributions of each component in the mixture. The simulated annealing demonstrated superior ability over the other optimization methods, simplex and steepest descent, in yielding more reliable convergences at the expense of not much more computer time, at least for resolving ternary shifted overlapped spectra. 

One alternative to the direct online measurement of polymer properties is to use a process model in conjunction with optimal state estimation techniques to predict the polymer properties. Indeed, several online state estimation techniques such as Kalman filters, nonlinear extended Kalman filters (EKF), and observers have been developed and applied to polymerization process systems. ° In implementing the online state estimator, several issues arise. For example, the standard filtering algorithm needs to be modified to accommodate time-delayed offline measurements (e.g., MWD, composition, conversion). The estimation update frequency needs to be optimally selected to compensate for the model inaccuracy. Table 5 shows the extended Kalman filter algorithm with delayed offline measurements. Fig. 2 illustrates the use of online state estimator... 

Table 5 Extended Kalman filter algorithm with online and offline measurements...

Numerous software data treatments authorize the elucidation of mixture composition from spectra. One of the best-known methods is the Kalman s least squares filter algorithm, which operates through successive approximations based upon calculations using weighted coefficients (additivity law of absorbances) of the individual spectra of each components contained in the spectral library. Other software for determining the concentration of two or more components within a mixture uses vector quantification mathematics. These are automated methods better known by their initials PLS (partial least square), PCR (principal component regression), or MLS (multiple least squares) (Figure 9.26). 

Provided we know the scaling coefficients at some resolution level j, the remaining scaling coefficients and wavelet coefficients can be found by the pyramidal filtering algorithm without even having to construct a wavelet or scaling function. We need only work with the filter coefficients Ik and hk. 

Another, more challenging, application of micro times would be to include them into the correlation to obtain information about the homogeneity of the lifetime. A simple, yet not very efficient way to use the lifetime information is to correlate photons in different micro time windows. [44] uses a filter algorithm which separates the FCS curves of molecules of different lifetime [65]. An ex-... 

Zf. Furthermore, it is assumed to be statistically independent to F. The essential steps of the Kalman filter algorithm are to predict and filter at each time step with the data set P// = yi, yz, , yw - When the measurements up to the nth time step T> = yi, y2,..., yn are available, the predicting procedure is applied to estimate y +i by using the conditional PDF p(X +i T>n, C), which is multi-variate Gaussian for linear systems. By using Equation (2.181), the predicted state vector at the (n + l)th time step can be estimated from the filtered state at the nth time step ... 

Zhu P, Tong W, Alton K, Chowdhury S. An accurate-mass-based spectral-averaging isotope-pattern-filtering algorithm for extraction of drug metabolites possessing a distinct isotope pattern from LC-MS data. Anal Chem 2009b 81 5910—5917. 

Determination of a singie species in a mixture Simuitaneous kinetic-based determinations Classical differential kinetic methods Logarithmic-extrapolation method Proportional-equation method Multipoint methods Curve-fitting methods Kalman filter algorithm Artificial neural networks Multivariate calibration methods... 

X-i-F Fluoro complexes Fe lll)-Zr(IV) Use of a fluoride-selective electrode. Kalman filter algorithm... 

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