Filtering digital

As a rule analog filters are connected on line with the signal source. Digital filtering takes place after AD conversion and storage of the data it is seldom possible to work up the data source immediately. They should thus be stored temporarily on the PC-integrated RAM or permanently on a discette or on tape. 

If only a single measurement is available or only a few scans can be made, smoothing operations must be carried out. In this case the shape of the signals becomes more or less distorted, and one must enter into a compromise between resolution and alteration of the shape. Fortunately, alteration is not important if derivatives are used for quantitative measurement and the conditions of differentiation are held constant, because only the amplitudes are necessary for evaluation rather than the true A position of the extrema. Moreover, the standard line, or generally stated the standard curve, corrects possible deviations from the linearity of the peak height to the quantity of the components. 

The most common digital filtering polynomial methods, according to Savitzky and Golay [22, 23], involve a shortened least-square computation using a sliding window with variable data points but other algorithms are also sometimes practicable. 

Smoothing by Fourier transform filtering is more arduous with respect to programming computation and time, but this algorithm is now being used time by time. 

The best solution of the problem of digital filtering may be true arithmetic time averaging of as many independent scans (or other signals) as possible. A minimum of eight accumulated scans are recommended. If the SNR is small, then 32, 64, or more scans are necessary. The optimal number can be determined by independent repetition of this procedure. If the results are congruent, the best solution is given. 

Positive values of a t improve the resolution at the expense of sensitivity. Another method of resolution enhancement without significant reduction of signal noise is referred to as Gauss multiplication [21 b]. This involves multiplication of the FID signal with an exponential of second order, e- - 2 where a 0 and b 0. The best value of a is the negative digital resolution (Section 2.5.6.2), while the optimum of b is related to the time after which the FID is practically zero. 

In the previous sections we focused on steps that must be taken to transform data faithfully from the time domain to the final displayed spectrum. However, if we are willing to accept some modification in the true Lorentzian line shape, we have an opportunity to process data both before and after Fourier transformation in such a way as to make the final spectrum more useful. Clearly, such processing must be undertaken carefully in order not to lose important information or to introduce features that could lead us to erroneous conclusions. 

Prior to Fourier transformation the time domain data may be modified by multiplication by a function q(t), a process commonly called digital filtering. By suitable choice of q(t), digital filtering may be used in NMR spectroscopy to enhance sensitivity, to improve spectral resolution, or to avoid truncation effects. Conceptually, there is very little difference between filtering of ID and 2D NMR spectra, so the treatment here may later be extended readily to the two-dimensional case. 

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The demodulation algorithm is very simple the DSP multiplies the received signal by each carrier, and then filters the result using a FIR filter. This kind of digital filter is phase linear, (constant group delay important for the EC combinations). Other filters may be programmed, other demodulation algorithms may be used. 

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. 

Consequently, the surface generation becomes a question of producing a matrix of random data that obey the defined height distribution and a prescribed ACF This can be carried out efficiently through the procedure of a 2-D digital filter, proposed by Hu and Tonder [47], as summarized as follows. 

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

Jones, R., High-Pass and Band-Pass Digital Filtering with Peak to Trough Measurement Applied to Quantitative Ultraviolet Spectrometry, Analyst 112, November 1987, 1495-1498. 

Bialkowski, S. E., Real-Time Digital Filters Finite Impulse Response Filters, Anal. Chem. 60, 1988, 355A-361A. 

S. Agaian et ai, Binary Polynomial Transfomns and Nonlinear Digital Filters (1995)... 

S.C. Rutan, Fast on-line digital filtering. Chemom. Intell. Lab. Syst., 6 (1989) 191-201. 

Fig. 7a-e Proton spectra of 1 a Dissolved in CDC13 b in D20 c in D20/H20 d with presaturation of the water signal e with presaturation using a digital filter. Signals marked with are due to an impurity (solvent from recrystallization of 1)... 

We can improve the appearance of the spectrum by applying a so-called digital filter the result is shown in spectrum (e). 

Sharaf MA, Illman DL, Kowalski BR (1986) Chemometrics. Wiley, New York Williams CS (1986) Designing digital filters. Prentice Hall, New York Wolf D (1999) Signaltheorie. Springer, Berlin Heidelberg New York... 

Phillips, S.C. Essex, J.W. Edge, C.M., Digitally filtered molecular dynamics the frequency specific control of molecular dynamics simulations, J. Chem. Phys. 2000,112, 2586-2597... 

This technique is of high accuracy and is meant to be used in precision measurement instrumentation, for it is inherently insensitive to the DC-offset and the AC-noise in the sinusoidal signal which can be substantially reduced by a great variety of electronic devices ranging from various electronic analogue filters, and digital filters to the most effective lock-in amplifiers. 

Example 18.11. Suppose we want to find a discrete approximation for a first-order lag. This would be called a first-order digital filter. Let x, be the output of the filter and m ) be the input. 

Example 18.12. Repeating Example 18.11 for a digital filter, using trapezoidal approximation gives... 

Rearranging gives the transfer function for the digital filter using trapezoidal approximation... 

This is the transfer function for a first-order digital filter. Thus the precompensator slows down the input to the controller so that it does not see a step change in the error signal (see (Fig. 20.3h). 

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