Exponential filter

Now load the 64K "C FID of peracetylated glucose D NMRDATA GLLICOSE 1D C GC 001001.FID and apply forward LP to improve the signal-to-noise ratio of the corresponding spectrum. Carefully inspect the FID to define the First Point used for LP and Last Point used for LP and the Number of Coefficients. Follow the rules given before and your experience acquired in the last Check its, perform several calculations varying the LP parameters to optimize the spectral quality. Compare the results with the spectrum obtained with/without applying a matched exponential filter and without LP. 

The difficulty is that real spectra always contain noise. Figure 3.25 represents a noisy time series, together with the exponentially filtered data. The filtered time series amplifies noise substantially, which can interfere with signals. Although the peak width of the new transform has indeed decreased, the noise has increased. In addition to making peaks hard to identify, noise also reduces the ability to determine integrals and so concentrations and sometimes to accurately pinpoint peak positions. 

Because the signal decays as a function of time, while the noise is stationary, data points at the beginning of the FID contribute more to the resonance intensity than do points near the end of the FID. One common way to discriminate against later points with poor S/N without abruptly truncating the FID is to use an exponential filter, with... 

FIGURE 3.10 Use of an exponential filter to improve signal/noise ratio, (a) Acquisition time T = 4T2, with no filter. (6) T = 4T2, with a matched filter, showing better S/N but a broader line. (c) Relative S/N as a function of acquisition time and without a matched filter. 

A related method has been used to demonstrate that lifetimes as ort as 200 ps can be measured usii the mode noise in a free-running a on-ion laser to produce variations in the excited state population of a fluorophore . Meaairement of the rf power spectmm of the resilting fluctuations then reveals the excited state lifetime. Mode noise contains very high frequency fluctuations which the excited state population cannot follow because of its finite lifetime, and thus these hi frequency components are absent from the rf spectrum of the fluorescence fluctuations. The fluorescence process thus acts like a low pass exponential filter, and comparison of the fluorescence power spectmm with that of the source provides the decay time data, as demonstrated below. 

In quantum scattering calculations we are typically interested in the eigenstates in the low energy end of the spectrum. For this purpose tm exponential filter f Hs) = cxp[—— Hmin)] is useful as it will dilate the eigenvalues at low energies. The action of the filter on the vectors can be performed via the Chebyshev polynoinicds[63, 64],... 

Fig. 0 shows a typical curve of the GST for the exponential filter. It is clear that the approximate spectral transform well represents the profile of the exponential filter at low energies, where the curve is also smooth and monotonous. 

Exponential filtering, a fast procedure, is equivalent to RC-tjq>e analog filtering. It must be used with great care, because it involves a lag in the curve which can cause errors in the case of overlapping peaks. The general filtering formula is... 

In gas chromatography-computer S5retems exponential filtering 44,53) or a least-square fit 54,55) may be used alone, or bracketing and a linear moving average can be combined with a least-square curve fit .32,55), a method of exponential filtering combined with a nine-point least-square fit 26) has also been published. 

As can be seen in Fig.3 the exponential filter causes a decrease in the peak height (and increase in the peak width due to the constant peak area) even when the noise level is 0 % and should not be used when smoothing ac polarograms. The rectangular filter is found to give the best result. This is obviously due to the fact that the shape of the ac... 

Figure 4 shows the frequency domain spectra obtained with our pulsed Fourier transform spectrometer for c(13)h20 in natural abundance displayed on an oscilloscope. The formaldehyde pressure was approximately 1 mTorr. The spectra cover 25 MHz and each frequency point corresponds to 100 KHz. The displayed line corresponds to the 111 llO t otational transition of c(13)h20 at 4593.3 MHz. The carrier frequency, ff/ o, was kept at 4 MHz off-resonance in the upper spectrum and 21 MHz off-resonance in the lower spectrum. Both spectra were obtained after an averaging time of 15 sec and an optimum exponential filter was used in the digital conversion. The line has an absorption coefficient of 6 x 10 8 cm"l. The obtained signal-to-noise ratio (peak signal amplitude to rms noise amplitude) is approximately 50 1. 

This procedure is nothing more than application of a digital filter with an exponential time constant equal to twice the muon lifetime. Figure 7 shows the effect of this transformation on the data of Figure 6. Shorter time constants for the exponential filter may be appropriate for short-lived signals, in order to discriminate against noise at later times however, broadening of the peaks in the Fourier spectrum is an unavoidable and undesirable consequence. 

Table 2 Muon precession frequencies in 4.4M 2,3-dimethyl-2-butene in cyclohexane. 1024 bins of 2.61 ns width were Fourier analyzed using an exponential filter with an 0.5 time constant.

Figure 3.- The proton decoupled 9.12 MHz NMR spectra at 27 C of five ISig-enriched intact gram positive bacterial cells. Spectra of 1.5 cc packed cells ( 300 mg. dry wt.) were obtained on a Brucker UH-90 spectrometer using 10,000 accumulations, 90° pulse angle, 4 K data points, 2000 Hz spectral width, 2 Hz exponential filter, with quadrature and alternating phase detection.

The coefficient of a one-sided exponential filter leads to a geometric progression. In this case the number of data points to be considered should be infinite and, therefore, the exponential filter can only be approximately realized by an iterative filter . A more common method of realization is based on the recursive function [69, 70]. 

To estimate the dynamic response to fast temperature changes, the pressure sensor is first immersed in a warm water bath. After it has adjusted, the sensor is quickly placed in a cold water bath at the same pressure. Hie sensor response together with the temperature is recorded. If the response is still significant after all the manufacturer s comjiensations have been applied, the method of Miiller et al. (1995) may be used to reduce the error. It assumes that the dynamic pressure correction Pcd is proportional to the temperature difference between the outer and the inner parts of the pressure sensor. If not measured, the temperatures at the outer and the inner parts of the pressure sensor are estimated from the measured main temperature of the CTD using a recursive exponential filter. 

DCS have generally standardised on the first order exponential filter. This introduces an engineer-configurable lag (with time constant xfi on the PV. It is implemented as... 

Higher orders of filter are possible. For example a second order exponential filter (with both lags set to Tf) can be developed by connecting two first order filters in series. The first would be as Equation (5.23) generating an intermediate output Y that becomes the input to a second filter, i.e. 

It is possible to modify the first order exponential filter, described by Equation (5.23), to make it nonlinear. Instead of the parameter P being set by the engineer it is changed automatically according to the formula... 

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