Power spectrum method

Several algorithms have been used to determine surface fractal dimension from SPM images. The most popular ones are the power-spectrum method,1 2 68"71 the triangulation method,10 37 ... 

The methods for calculating profile fractal dimension mainly include the yardstick, the box counting, the variation, the structure function and the power spectrum method (Sahoo, 2005). 

To measure the strength of the forces exerted on particles, various analytical techniques have been developed [6, 7]. Unfortunately, since most of these techniques are based on hydrodynamics, assumption of the potential profiles is required and the viscosities of the fiuid and the particle sizes must be precisely determined in separate experiments, for example, using the viscous flow technique [8,9] and power spectrum analysis of position fluctuation [10]. Furthermore, these methods provide information on ensemble averages for a mass of many particles. The sizes, shapes, and physical and chemical properties of individual particles may be different from each other, which will result in a variety of force strengths. Thus, single-particle... 

EXAFS analysis is a powerful spectroscopic method for structural analysis which has been extensively applied to the problem of structure determination in nanoparticles, and especially bimetallic nanoparticles [170-172]. The X-ray absorption spectrum of an element contains absorption edges corresponding to the excitation of electrons from various electronic states at energies characteristic of that element, i.e., K edges arise from the excitation of electrons from Is states, and LI, II, III edges from excitations from 2s, 2p 1/2, and 2p3/2 states. When the X-ray energy is increased above an edge, oscillations (fine... 

While I am no longer working in this field, and cannot easily do simulations, I think that a 2 factor PCR or PLS model would fully model the simulated spectra. At any wavelength in your simulation, a second degree power series applies, which is linear in coefficients, and the coefficients of a 2 factor PCR or PLS model will be a linear function of the coefficients of the power series. (This assumes an adequate number of calibration spectra, that is, at least as many spectra as factors and a sufficient number of wavelength, which the full spectrum method assures.) The PCR or PLS regression should find the linear combination of these PCR/PLS coefficients that is linear in concentration. 

Finlayson and Hordley (2001 a,b) have developed a method for color constancy at a pixel. They make two assumptions. One that the camera s sensors are sufficiently narrow band and that the illuminant can be approximated by a black-body radiator. If the sensor s response functions are not sufficiently narrow band, a sharpening technique can be used (Barnard et al. 2001 Finlayson and Funt 1996 Finlayson et al. 1994a,b). The power spectrum of a black-body radiator was described in Section 3.5. The radiance of a black-body radiator at a temperature T, measured in Kelvin, at wavelength /. is given by (Haken and Wolf 1990 ... 

The basic mathematical method for power spectrum analysis is the Fourier transformation. By the way. transient fluctuation can be expressed as the sum of the number of simple harmonic waves, which is helpful for understanding fluctuation. A frequency spectrum analysis for pressure signals can yield a profile of the frequencies and that of the amplitude along the frequencies. The basic equation of Fourier transformation can be expressed as... 

The third noteworthy feature of near-infrared spectra presented in Figure 13.3 is the uniqueness of the spectral patterns for each analyte. Although the spectral features are highly overlapping, the spectrum for glucose is notably unique relative to the others. The uniqueness of each spectrum provides the selectivity that is required for sound analytical measurements. However, the extensive overlap dictates that an analysis of the full spectrum is needed to extract the unique spectral signature for the targeted analyte relative to the sample matrix. Powerful multivariate methods are available for this purpose, as described in Chapter 12. 

One of the most important phenomenon, chemically induced dynamic nuclear polarization (CIDNP), deserves more detailed consideration, since it forms the basis of one of the most powerful modem methods for the investigation of the structure and reactivity of short-lived (from nano- to microseconds) paramagnetic precursors of the reaction products. CIDNP manifests itself in the form of unusual line intensities and/or phases of NMR signals observed when the radical reaction takes place directly in the probe of the spectrometer. These anomalous NMR signals—enhanced absorption or emission — are observed within the time of nuclear relaxation of the diamagnetic molecule (from several seconds to several minutes). Later on, the NMR spectrum re-acquires its equilibrium form. 

Figure 7.5 Pb (A) and U (B) background/baseline maps compiled using spatial and spectral analysis (S—A method). The values have been obtained by decomposition of the Pb and U geochemical maps (Fig. 7.1A and B, respectively) into separate components using the fractal filters selected on the S—A plot. Pb background/baseline map is generated only with signals with power spectrum (E) >15,713 (selected cutoff in Fig. 7.6A). U background/baseline map is generated only with signals with power spectrum (E)>706 (selected cutoff in Fig. 7.6B). For both maps pixel values have been reclassified by fractal C—A plot (see text).

Burg135 derived a means of estimating a power spectrum from the limited number of autocorrelation coefficients of a short time series, without adding extra information. It can be shown that this approach leads to equations identical to an autoregressive LP method, with the power spectrum P(f) given by... 

An alternative method of calculating the power spectrum is available in the fast Fourier transform technique [23], because ... 

Laser ablation can be carried out on any material without special sample preparation. The laser beam can be directed onto a defined spot of the sample or moved to different parts to analyse over a defined area. It can be moved in an XYZ plane using a stepper motor and driven in translational motions on which the cell is mounted and with more expensive models can be turned for analysis in other parts of the sample. Lasers can operate in UV, visible, and IR regions of the spectrum and a recent development in laser technology uses neodymium yttrium aluminium garnet (Nd YAG) which gives high repetition rate at a comparatively low power. This method of analysis is suited to bulk analysis of solid materials and the amount of volatility varies from sample to sample. The size of the laser spot can vary from 10 to 250 pm and little or no sample preparation is required. Errors are greatly reduced because of the simple sample preparation, and the fact that no solvents are required reduces interferences. 

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