Synchronous and asynchronous spectra

Keywords Covariance NMR, Synchronous and asynchronous spectra. Non-uniform sampling. Fast methods. Structure elucidation... 

As an apphcation of using synchronous and asynchronous spectra, Eads and Noda [50] described the covariance analysis of diffusion ordered... 

A correlation analysis is applied to the individual sine and cosine functions in the manner similar to the derivation for Equations (F28) and (F29), which in turn generates a number of synchronous and asynchronous spectra, that is, cospectra 4>s quad-spectra j/, for individual Fourier components with different frequencies s. 

Spectra. Examples of synchronous and asynchronous contour maps are presented in Fig. 5.4. 

The real and imaginary components, 3>(t i, V2) and F(vi, V2), of the cross-correlation function X(t) are referred to, respectively, as the synchronous and asynchronous correlation intensity. These quantities are related to the in-phase and quadrature spectra of dynamic dichroism by... 

Figure 3.12A and B, depicts schematic contour maps of synchronous and asynchronous 2D correlation spectra, respectively (the terms synchronous and asynchronous are always used even when the spectral variation is measured as a function of not time but another physical variable) (13). A one-dimensional refernce spectrum is provided at the top and left side of the contour map to show the basic feature of spectra of the system during experiment. A synchronous spectrum is symmetric with respect to a diagonal line corresponding to spectral coordinates = V2. Peaks... 

They are respectively referred to as the synchronous and asynchronous 2D infrared spectra. The synchronous spectrum characterizes the degree of coherence between the dynamic fluctuations of signals measured at two wavenumbers, and the correlation intensity becomes significant only if the reorientation rates of dipole transition moments are similar to each other. The asynchronous spectrum, however, characterizes the independent, uncoordinated out-of-phase fluctuations of the signals. Hence the asynchronous correlation intensity becomes non-vanishing only if the signals vary at difierent rates. 

In principle, there is no difference in the information content of a pair of synchronous and asynchronous correlation spectra and the corresponding power and phase spectra. Some spectroscopists find it easier to interpret the former pair of spectra and some the latter. However, the use of two-dimensional correlation maps to assist in the interpretation of spectral data is becoming extremely common. 

The basic concept and some examples of 2D correlation spectroscopy have been covered in this chapter. 2D correlation analysis is based on the simple mathematical treatment of a set of spectral data collected from a system under the influence of an applied perturbation during the measurement. This perturbation can take different forms of changes, including temperature, concentration, or pH, and the like. The set of spectra is then converted to the synchronous and asynchronous correlation spectra, respectively, representing the similarity and dissimilarity of perturbation-induced intensity variations between wavenumbers. 2D correlation peaks provide easier access to pertinent information, making it possible to determine the sequential order of the variations of spectral intensities, as well as relative directions. Highly overlapped peaks are often resolved more clearly. This technique can be a useful addition to the toolbox of experimental scientists. 

The synchronous and asynchronous spectrum, especially those expressed in terms of the amplitudes of cosine and sine function, clearly reveal the close resemblance of the functional forms to the ones given for the cospectrum and quad-spectrum in Equations (FIO) and (Fll). The amplitudes of cosine and sine component, respectively, of the dynamic spectrum with a single frequency 5 reflect the real and imaginary parts of the Fourier transform of the dynamic spectrum at the Fourier frequency of s = 5. Alternatively, the more general synchronous spectrum and asynchronous spectrum in Equation (F15) derived for the dynamic spectrum with arbitrary waveforms may be viewed as the collective sum totals of individual correlation spectra obtained for the corresponding Fourier components. 

Synchronous 2D correlation spectra represent coupled or related changes of spectral intensities, while asynchronous correlation spectra represent independent or separate variations [1007]. The 2D cross-correlation analysis enhances similarities and differences of the variations of individual spectral intensities, providing spectral information not readily accessible from ID spectra. 

Figure 3-32 Synchronous (A) and asynchronous (B) 2D FT-Raman spectra in the range 1620 1290 cm-1 constructed from the spectra of a set of blends containing PS and PPE polymers at the ratios of 100/0, 90/10 and 70/30. Black peaks indicate negative contours. (Reproduced with permission from Ref. 99.)...

Figure 10.6 2D-IR synchronous (left) and asynchronous (right) maps of LDL (bottom) and HDL (top) in the 1600-1700 cm region corresponding to the amide I region at the 20-40 °C interval. The spectra were taken in a D2O medium...

To increase interpretability, the dynamic IR spectra are snbjected to mathematical cross-correlation to prodnce two different types of 2D1R correlation spectra, or two-dimensional correlation maps. These maps, in which the. r- and y- axes are independent wavenumber axes (vi, V2), show the relative proportions of in-phase (synchronous) and ont-of-phase (asynchronons) response (Figs. 3.51 and 3.52). Initially, the mathematical formalism was based on the complex Fourier transformation of dynamic spectra [277]. To simplify the computational difficulties, the Hilbert pansform approach was developed [280], which produces two-dimensional correlation maps from a set of dynamic spectra as follows. First, the average spectrum y(v) is subtracted from each spectrum in the set, y(v, Pj) = y v, Pj) — y(v), where Pj is the dynamic parameter. Then, the synchronous spectrum, 5 (vi,V2), and the asynchronous spectrum, A(vi,V2), are calculated as... 

Figure 4.25 Synchronous (a) and asynchronous (b) 2D IR plots of a set of40/60 PLA/ Nodax blend spectra recorded sequentially at 40,60, 80,100,120,140, and 160°C...

Figure 3.12. Schematic contour maps of synchronous (A) and asynchronous (B) spectra. A one-dimensional reference is also provided at the top and side of the 2D map. [Reproduced from Ref. 13 with permission. Copyright (1993) Society of Applied Spectroscopy].

Figure 21.1 Schematic illustrations of (a) synchronous and (b) asynchronous 2D correlation spectra. White and gray areas in the contour maps represent positive and negative correlation intensities, respectively.

Figure 21.3 (a) Synchronous and (b) asynchronous correlation spectra directly calculated... 

The formal definition of synchronous spectrum and asynchronous correlation spectrum given in Equation (F3) is mathematically concise and rigorous. However, the requirement for obtaining the Fourier transforms of signals with respect to the variable t at every point of wavenumber v for a given dynamic spectrum makes the computation of correlation spectra rather cumbersome, even with the aid of the fast Fourier transform (FFT) algorithm. Fortunately, there is a simple way to circumvent the use of the Fourier transforms to efficiently compute the desired correlation spectra [2]. 

The synchronous spectrum and asynchronous correlation spectrum are now obtained as nxn correlation matrices, 0 and F. Their matrix elements correspond, respectively, to the value of correlation spectra 0 = (Vp, Vq) and = P(Vp, Vq). Equations (F30) and (F31) are now presented concisely by the matrix notation as... 

To extraa more information from the spearal data, 2D-COS can be employed. Basically, this analysis method ae-ates a pair of synchronous (vj,v2) and asynchronous F(vi,V2) 2D correlation spectra, where the spectral variables vi and V2 are wavenumbers. The synchronous 2D correlation intensity (vi,V2) represents the overall similarity or coincidental changes between two separate intensity variations measured at different spectral variables during variation of the external perturbation. The as3mchronous 2D correlation intensity 1 (vi,V2) may be regarded as a measure of dissimilarity or more strictly speaking, out-of-phase charaaer of the spectral intensity variations. 

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