Filtering, Fourier transforms

Arnold, M. A, Small, G. W. (1990) Determination of physiological levels of glucose in an aqueous matrix with digitally filtered Fourier-transform near-infrared spectra. Ana/. Chem. 62,1457-1464. 

Mattu, M. J. Small, G. W, (1995) Quantitative analysis of bandpass-filtered Fourier-transform infrared interferograms. Ana/yfica/ Chemistry 67,2269-2278. 

Shaffer, R. E., Small, G. W., Combs, R. J., Knapp, R. B. Kroutil, R. T. (1995) Experimental-Design Protocol For the Pattern-Recognition Analysis of Bandpass Filtered Fourier-Transform Infrared Interferograms. Chemometrics and Intelligent Laboratory Systems 29,89-108. 

Computer-Aided Experiments Computers have become common tools for the analysis of NMR data as Fourier transform spectrometers have replaced CW instruments. These computers perform the basic processing of data, including digital filtering, Fourier transformation, and phase correction. 

Ion Cyclotron Resonance Mass Filter Like Magnetic Sector mass filters, Fourier Transform Ion Cyclotron Resonance (FT-ICR) mass filters use magnetic fields to determine the m/g ratio of ions. Although capable of extreme mass resolution (m/Am > 100,000), these have only been recently introduced in the field of SIMS albeit in highly specialized research environments (Pahnblad et al. 2000 Todd et al. 2002 Smith et al. 2011). These are also sometimes referred to as Fourier Transform Mass Spectrometers (FT-MS). 

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

In FT-Raman spectroscopy the radiation emerging from the sample contains not only the Raman scattering but also the extremely intense laser radiation used to produce it. If this were allowed to contribute to the interferogram, before Fourier transformation, the corresponding cosine wave would overwhelm those due to the Raman scattering. To avoid this, a sharp cut-off (interference) filter is inserted after the sample cell to remove 1064 nm (and lower wavelength) radiation. 

Figure 8.39 Fourier transformed Fe extended X-ray absorption fine structure (EXAFS) and retransformation, after applying a 0.9-3.5 A filter window, of (a) a rubredoxin, (b) a plant ferredoxin and (c) a bacterial ferredoxin, whose structures are also shown. (Reproduced, with permission, Ifom Teo, B. K. and Joy, D. C. (Eds), EXAFS Spectroscopy, p. 15, Plenum, New York, 1981)...

Figures Fourier transform (soiid curve), Osir ) versus r (A, without phase-shift correction), of the Mo K-edge EXAFS of Figure 5 for moiybdenum metal foii. The Fourier filtering window (dashed curve) is applied over the region -1.5-4.0 A to isolate the two nearest Mo-Mo peaks.

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

The Fourier transform H(f) of the impulse response h(t) is called the system function. The system function relates the Fourier transforms of the input and output time functions by means of the extremely simple Eq. (3-298), which states that the action of the filter is to modify that part of the input consisting of a complex exponential at frequency / by multiplying its amplitude (magnitude) by i7(/)j and adding arg [ (/)] to its phase angle (argument). 

Figure 5.49. (a) STM image (unfiltered) of the initially sodium-contaminated Pt(l 1 l)-(2x2)-0 adlattice (b) corresponding Fourier transform spectrum (c) Fourier-filtered STM image of the overlapping Pt(l 1 l)-(2x2)-0 and Pt(111)-(12x12)-Na adlayers (bias Ut = 80 mV, tunelling current I, = 10 nA, total scan size 319 A).78 Reprinted with permission from Elsevier Science. 

The solution is given by applying a linear filter f to the data and the Fourier transform of the solution writes ... 

Fig. 5 XANES region, -weighted Fourier transformed of the raw EXAFS functions and the corresponding first shell filtered, Fourier back transform (a, b and c, respectively) of TS-1 activated at 400 °C (full lines), after interaction with water (wet sample, dashed lines) and after interaction with NH3 (Pnh3 = 50 Torr, dotted lines). Adapted from [64] with permission. Copyright (2002) by the ACS...

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