Critical band

This reaction occurs thermally in toluene at 30°C with an equilibrium constant (K) equal to 1.5 (13). Both bis(phosphine) and (carbonyl)phosphine Soret bands are present in the active catalyst solutions (see Decarbonylation Procedure), together with the unassigned, and likely critical, band at 420 nm. This could be due to some species giving rise to, or resulting from, a RuII + RCO reaction this is equivalent, of course, to a (Ru H-COR) acyl or a RuIII(C0)R (carbonyl)alkyl species, and the final elimination reaction after loss of CO could be written as ... 

Research on the hearing process carried out by many people (see [Scharf, 1970]) led to a frequency analysis model of the human auditory system. The scale that the ear appears to use is called the critical band scale. The critical bands can be defined in various ways that lead to subdivisions of the frequency domain similar to the one shown in table 2.1. A critical band corresponds to both a constant distance on the cochlea and the bandwidth within which signal intensities are added to decide whether the combined signal exceeds a masked threshold or not. The frequency scale that is derived by mapping frequencies to critical band numbers is called the Bark scale. The critical band model is most useful for steady-state tones and noise. 

QMF filter banks. Quadrature mirror filters (QMF, see [Esteban and Galand, 1977]) have often been proposed for audio coding. The most common configuration is the tree of filters with a two-way split. In one of the early examples [Theile et al., 1987] the 64d filter design from [Johnston, 1980] has been used. The decomposition tree is set up so that the filter bands resemble critical bands. The QMF halfband filters are non-perfect reconstruction, but with perfect alias cancellation by design. The reconstruction error of the analysis/synthesis pair can be held at small amplitudes by increasing the filter length. 

The magnitude values of the frequency domain representation are converted to a 1/3-critical band energy representation. This is done by adding the magnitude values within a threshold calculation partition. 

Scharf, 1970] Scharf, B. (1970). Critical Bands. In Tobias, J., editor, Foundations of Modem Auditory Theory, pages 159-202. Academic. 

The masking behavior of stereo signals is improved if a two-channel signal can be switched between left/right and sum/difference representation. Both broadband and critical band-wise switching has been proposed [Johnston, 1989a],... 

Long system delay The overall system delay can reach 250 ms, if the technique is used to design a filter bank with a frequency partitioning similar to critical bands. 

The key to the vibrational analyses can be understood from the five critical bands illustrated as stick spectra in Figure 2.12. These stick spectra portray the band structure that would be anticipated for a vibronically allowed, but electronically forbidden, transition, where the planar ground electronic state is... 

Colin et al. [SSS] have described a different method to construct a diagram that allows the prediction of optimum conditions. Their approach is based on the calculation of so-called critical bands. If the retention surface of a solute j is known, then a forbidden zone may be defined below the capacity factor kj. If the preceding solute i has a capacity factor kp which falls in this critical band, then the resolution between i and j is insufficient. Eqn.(1.20) relates the resolution to the capacity factors of the individual solutes ... 

On a column with a given (average) number of plates, the critical bands can be calculated for any value of the desired resolution Rs. If the retention line (in the case of a one parameter optimization problem) for solute j is straight, then eqn.(5.16) describes another straight line. Two applications of this approach are shown in the figures 5.18 and 5.19. 

Figure 5.18 (a) Figure showing the retention surfaces for some aromatic solutes in RPLC. Critical bands have beat constructed according to eqn.(5.16) below each solute. The dashed line indicates the optimum ternary mobile phase composition, (b) Chromatogram obtained at the predicted optimum composition. Figures taken from ref. [555], Reprinted with permission. 

The solute of interest can be separated from all the other solutes at compositions at which no other retention lines fall within the critical band. This is illustrated in figure 5.19b. 

Eqn.(5.17) can also be used to allow for the occurance of a solvent peak in the chromatogram. For example, a large resolution may be demanded between an imaginary peak at k= 0 and the first peak in the chromatogram. In that case all solutes may be assigned a critical band above (eqn.1.17) rather than below (eq.1.16) their retention lines. 

We may summarize the critical band method of Colin et al. as follows ... 

Sometimes 30 bands are used on a logarithmic scale to cover the audible range in 1/3 octave steps. These steps correspond roughly to an important resolution attribute of the human auditory system called the Critical Band. Sound intensity within a critical band can be manipulated in fairly extreme ways without changing the perceptual result, as shown in Figure 7.2. 

Figure 7.2. A critical band is roughly 1/3 octave. Equal noise intensity within a critical band soiuids the same.

Critical band method The same development is adopted to determine the retention models for all solutes. Below every retention surface it is possible to determine a forbidden zone (critical band) any solute whose retention would fall into a critical band would interfere with the compound considered with a resolution value (Rs) lower than a threshold value. The separation with Rs value higher than the threshold value is achieved at a composition for which none of the critical bands overlap. This is a graphical method for the optimization of a single parametei such as the composition of a binary mobile phase or of a ternary mobile phase constituted of mixtures of two isoeluotropic binary mobile phases. 

Whenever possible, select a column of the least polarity that provides acceptable separation of the most critical band pair in general, the more polar columns have a lower upper temperature limit and are more susceptible to water and oxygen-induced degredation. 

Selectivity, also known as the band-spacing factor, a, depends on column temperature and the nature of the stationary phase. Numerous examples of the effect of selectivity on a separation were presented earlier. Consider two separations (1) where the a value of a critical band pair is 1.1, or (a - 1)/q = 0.09 and (2) where a of the same critical band pair is 1.4, ox a - 1)/a = 0.29. A change of 0.3 in a produces greater than a threefold enhancement in resolution. Prudence suggests that manipulation of column temperature should be the first approach investigated in improving resolution, then changing to a column of different polarity should be considered. Also note that resolution approaches zero as selectivity approaches unity (i.e., coelution). 

Cossart-Magos C, Cossart D, Leach S, Maier JP, Misev L (1983) High-resolution gas phase emission and laser induced fluorescence excitation spectra of 1,3,5-C6fsh and l,3,5-c f3d critical bands in the Jahn-Teller effect analysis. J Chem Phys 78 3673... 

After windowing, Fourier analysis is performed on each frame, resulting in short-time Discrete Fourier Transform (DFT). Then derived values are then grouped together in critical bands and weighted by a triangular filter bank called mel-spaced filter banks. 

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