Asynchronous spectrum

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 5.4 2D IR correlation spectra based on the spectral changes induced by the static compression of linear low-density polyethylene (LLDPE) at 608 °C. Synchronous spectrum (left) Negative intensity peaks indicating anti-correlation are shaded. Asynchronous spectrum (right) Shaded peaks indicate that intensity changes at i/, occur at higher pressure than changes at i 2- Reprinted from Noda et al. [49], Copyright 1999, with permission from Elsevier.

It can be recognized from Fig. 5.4 that diagonal peaks are absent in the asynchronous spectrum. 

The lack of synchronous cross peaks between polystyrene and polyethylene bands indicates these polymers are reorienting independently of each other. Cross peaks appearing in the asynchronous spectrum (Figure 1-19) also verify the above conclusion. For an immiscible blend of polyethylene and polystyrene, where molecular-level interactions between the phase-separated components are absent, the time-dependent behavior of IR intensity fluctuations of one component of the sample... 

There is a strong correlation in the synchronous spectrum and no appreciable asynchronicity in the asynchronous spectrum between the mesogen band (2230 cm" ) and the (CH ) bands (2926 and 2863 cm" ). The main part of the intensity changes in the region of 2850-2950 cm" is due to the spacer thus, on the basis of the 2D results we may draw the conclusion that the spacer and the mesogen reorient simultaneously. 

The real and imaginary components of the complex 2D correlation intensities, 0(vi, V2) and (vi, V2), are referred to, respectively, as the generalized synchronous and asynchronous correlation spectra of the dynamic spectral intensity variations. The synchronous spectrum represents the simultaneous or coincidental changes spectral intensities at vi and V2, whereas the asynchronous spectrum represents sequential or unsychronized variations. 

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. 

The synchronous spectrum 0(vi, V2) represents the pattern similarity or in-phase nature of spectral intensity changes observed at vj and V2 along the perturbation variable t. On the other hand, the asynchronous spectrum V2) represents the dissimilarity or out-phase nature of intensity variation patterns. Although the procedure described is a highly simplified one, it should be applicable to many practical cases of spectral analysis. More general treatment of 2D correlation analysis, including the case for unevenly sampled spectral data, is discussed in Appendix F. 

Figure 21.8 shows the 2D IR correlation spectra of the N-isomeric form of HSA in a buffer solution, derived from the pH-dependent (pH 5.0,4.8,4.6, and 4.4) spectral variations. The synchronous spectrum shows a major autopeak around 1654 cm assigned to the a-helix. The analysis of several peaks in the asynchronous spectrum allows one to identify several bands at 1715, 1667, 1654, and 1641 cm The band at 1715 cm is assignable to a C=0 stretching mode of the hydrogen-bonded COOH groups of Glu and Asp residuals of HSA. On the other hand, the bands at 1667 and 1641 cm i are assigned to the -turn and -strand of HSA, respectively. 

The real part (vj, V2) of the complex cross correlation function is referred to as the synchronous spectrum, while the imaginary part 9 (vi, V2) is called the asynchronous spectrum. The practical significance of the correlation spectra related to applications in physical science and the step-by-step derivation of the above equation central to the two-dimensional correlation analysis are provided in this appendix. 

It should be pointed out that the correlation time r = 1 /(4s) used for the determination of the asynchronous spectrum in Equation (F14) is not a single fixed quantity on a time axis but a function of the Eourier frequency s. The asynchronous correlation function is a special... 

As the term in the brackets is the inverse Fourier transform of T(vi, 5), it becomes y(vi, 5). Furthermore, because the value of the dynamic spectrum is set to zero outside of the observation interval in Equation (FI), the integration with respect to t in Equation (F23) can be limited only to the observation interval. Thus, we obtain the final simplified expression for the asynchronous spectrum as... 

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. 

The collective summation by integration of all cospectra for the entire positive range of i yields the synchronous spectrum in Equation (FI3), and the same for quad-spectra to obtain the asynchronous spectrum in Equation (F14). 

The explicit analytical expressions given by Equations (FI 6) and (F24), obtained for the synchronous and asynchronous spectrum, are well suited for the efficient machine computation of correlation intensities from discretely sampled and digitized spectral data. If a discretely sampled dynamic spectrum y(v -, t,) with the total of n points of wavenumber Vj is obtained for m times at each point of time tj, with a constant time increment, that is, t,+i - t, i = At, the integrations in Equations (FI6) and (F24) ean be replaced with summations. 

---