Fourier transforming and filtering

Figure 8.40 The k ySk) extended X-ray absorption fine structure (EXAFS) signal, Fourier transformed and then retransformed after application of the filter window indicated, in (a) osmium metal and (b) a 1% osmium catalyst supported on silica. (Reproduced, with permission, Ifom Winnick, FI. and Doniach, S. (Eds), Synchrotron Radiation Research, p. 413, Plenum, New York, 1980)...

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. 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

It should be mentioned that when a peak from a Fourier transform is filtered and back-transformed to k space, the envelope represents the backscattering amplitude for the near neighbor involved. 

In Section 3.3 we discussed a number of linear filter functions that can be used to enhance the quality of spectra and chromatograms. When performing Fourier transforms, it is possible to apply filters to the raw (time domain) data prior to Fourier transformation, and this is a common method in spectroscopy to enhance resolution or signal to noise ratio, as an alternative to applying filters directly to the spectral data. 

Figure 4.4 Similar to the sliding polynomial smoothing (Savitzky Golay filter, the coefficients for 2nd order fit to a parabola) is the effect of Bromba Ziegler filters [Bromba and Ziegler, (1983c), coefficients fit to a triangle upper figure]. Both have bad low pass filter characteristics, as shown in the lower figure with the Fourier transforms of filters through 21 points each.

For evaluation of radial NMR images Frir) of circular objects, processing of the FID in two steps by Fourier transformation and subsequent inverse Abel transformation is preferred over straight forward Hankel transformation, because established phase correction, baseline correction, and filter routines can be used in calculation of the projections P(jc) as intermediate results [Majl]. As an alternative to Hankel and Abel transformations, the back-projection technique (cf. Section 6.1) can be applied for radial evaluation of circular objects, using copies of just one projection for input. As opposed to the inverse Abel transformation, however, this provides the radial information with nonuniform spatial resolution. 

Fig. I (a) Experimental (h) Fourier transformed and (c) wandet-transformed IR. spectrum of benzoic acid. Spectra (c) and Id) were derived from (a with a Daubechies > (, wavelet filter at re.solutioti levels J - I and J - 2. respectively.

The data must be filtered in some way, in order that the procedure remains stable. The matrix of the observable intensities contains also a certain content of noise. In particular, the high-frequency part of the noise could make the procedure instable. It is possible to filter off the high-frequency noise, as the spectra are subjected to a Fourier transform and the high-frequency part is removed in the ttansformed spectra. This method is referred to as digital filtering. If too much of the high-fl-equency part is filtered off, then significant information will be lost. 

Figure 3.34. The use of a digital filter to observe a selected region of a spectrum. The desired frequency window profile is subject to an inverse Fourier Transformation and the resulting time-domain function convoluted with the raw FID. Transformation of the modified data produces a spectrum containing only a subset of all resonances as defined by the digital filter.

There are four basically different instrument designs, based on how incident energy is selected (McClure, 1994). These are grating instruments (monochromators), Fourier transform instruments, filter instruments and Diode Array-based instruments. Figure4 illustrates the basic design of a scanning NIR spectrophotometer. 

The philosophy discussed above for the Fourier transformation and the window roughness concept holds in the same way for the different filtering techniques. In the limit case (band-pass filter width- 0), both window values (FFT and COF) should be equal and define a correct window roughness curve vs. wavelength. The defined integral of this function between two chosen limits is equal to the roughness for the analysed profile determined between the boundaries of the lower limit (resolution) and the upper limit (waviness). 

Unfortunately, real data from electrochemical experiments are usually confused by noise and experimental artifacts. Although, signal processing techniques, such as Fourier transformation and digital filtering, can be used to improve the quality of the data, there is still a need for techniques that can extract useful information where there is inherent data uncertainty which cannot be removed. Pattern recognition techniques are powerful tools for this purpose (34,, W). They have been applied to classify electrochemical data on the basis of electrode mechanism (37, ) and chemical structure (i9,4Q). 

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