Two dimensional correlation spectroscopy applications in vibratioal and optical spectroscopy  Isao N.
ISBN 0471623911
Download (direct link):
It can be easily pointed out by observing Equation (2.40) that the Hilbert transform h(t) may be regarded as the convolution integral between the two functions, g(t) and 1/t. From the convolution theorem,10 the Fourier transform of the Hilbert transform h(t) becomes proportional to the product of the Fourier transforms of g(t) and 1/t.

Æ
—²Ø
H(co) =/ h(t) eimt dt
— Æ
Æ
1 1 f
  Qimt dt ?
Ï J — Æ t J —
Q~l0Jtdt g(t)Q~lC0t dt (2.41)
00
34
Principle of Twodimensional Correlation Spectroscopy
Thus, the Fourier transforms of functions g(t) and h(t) are related by
H(m) = i sgn (a) ? G(a)
—Glm(a) + iGRe (a) a > 0
0 if a = 0 (2.42)
Glm(a) — iGRe(a) ax 0
where GRe(a) and Glm(m) are the real and imaginary component of the Fourier transform of the function g(t). The value of signum function sgn(a) becomes — 1, 0, or +1, respectively, if the variable a is smaller than, equal to, or greater than 0.
The above result provides a simplified view of the Hilbert transform relationship. The Hilbert transformation of a function shifts the phase of each Fourier component of a function forward by n/2 if a > 0, and backward by n/2 if a < 0. Thus, the functions g(t) and h(t) are orthogonal to each other and have the relationship
/»TO
g(t) ? h(t) dt = 0 (2.43)

J —c
2.5.4 ORTHOGONAL CORRELATION FUNCTION AND ASYNCHRONOUS SPECTRUM
Let the orthogonal spectrum IE (v2, t) be the timedomain Hilbert transform of the dynamic spectrum y(v2, t).
1 fc y(v2, t') z(v2,t) =  pv/ (2.44)
* J—c t' — t
Thus, the orthogonal spectrum is a function where the phase of each Fourier component of the temporal variation of the dynamic spectrum is shifted by n/2. The twodimensional orthogonal correlation function between different spectral intensity variations observed at v1 and v2 for a period between Tmin and Tmax is given by
1 r Tmax
D(r) = _ y(v\, t) ? z(v2, t + r) dt (2.45)
T max T min J Tmin
where t is the correlation time. The orthogonal correlation function defined above is nothing but a crosscorrelation function between the dynamic spectrum and orthogonal spectrum. By applying the WienerKhintchine theorem (see Appendix), the orthogonal correlation function can be directly related to the Fourier transforms of the dynamic spectrum and orthogonal spectrum as
1 fc ~
°(r)= Y*(co) • Z2(co) e^ dm (2.46)
2*(Tmax Tmin) J —to
CrossCorrelation Analysis and 2D Spectroscopy
35
where Z2 (rn) is the Fourier transform of z(v2, t). By setting t = 0, and by using the relationship Z2(rn) = i sgn (m) ? Y2(m) based on Equation (2.42) and the definition of the cross spectrum (Equation 2.31), the above equation reduces to
i f ^ y y
°(0) = sgn(co)Y*(co) ? Y2(co) dco
2n(Tmax Tmin) J—<x>
1 r
= — I i sgn (co) S(co) dco
2n J<x,
1 f
= — dco = ^(vi, v2) (2.47)
n Jo
Thus, the asynchronous 2D correlation spectrum can also be computed directly from the dynamic spectrum and the orthogonal spectrum.
1 p Tmax
y(v\,v2) = _ y(v1,t)z(v2,t)dt (2.48)
T max T min J Tmin
This result shows that the asynchronous correlation intensity is equivalent to the time average of the product of dynamic and orthogonal spectrum measured at two different spectral variables, v1 and v2. Note that the evaluation of Equation (2.48), like that of Equation (2.39), does not require the Fourier transformation of dynamic spectral data.
2.5.5 DISRELATION SPECTRUM
Another convenient way to estimate the asynchronous spectrum by circumventing the need to transform spectral data into the Fourier domain is to compute a special type of a heuristic 2D spectrum known as the disrelation spectrum.1 The disrelation spectrum A(v1, v2) is given by
02(V1, V2) + A2(V1, V2) = 0(V1, V1) ? 0(V2, V2) (2.49)
The total joint variance of the spectral intensity changes during the observation period, which is represented as the product of the two autopower spectrum intensities at v1 and v2 on the righthand side of the above equation, is separated into two parts: the correlated portion and the disrelated portion. It can be easily observed that the absolute value of the disrelation spectrum intensity may be directly calculated from a set of only synchronous correlation intensities. By rearranging Equation (2.49) one obtains
A(vi, v2) = sgn(/c)y0(vi, vi) • <D(v2, v2)  <D2(vi, v2) (2.50)
36
Principle of Twodimensional Correlation Spectroscopy
where k is some constant to determine the sign of this spectrum, which is tentatively given by the slope of the cross correlation function (Equation 2.30) evaluated at t = 0.
k = dC(T)/dT t =0 (2.51)
The 2D disrelation spectrum can often be substituted effectively for the asynchronous spectrum, as long as the time dependence of the dynamic spectrum is not very complex. Even though the mathematical form of the disrelation spectrum in Equation (2.50) does not look at all like the form given by Equation (2.48), A(vi, v2) can often serve as an excellent substitution for ty(vi, v2) to highlight the basic asynchronous features of different spectral intensity variations. The similarity between the two types of 2D correlation spectra becomes especially noticeable when the dynamic spectrum y (v, t) is changing in a relatively monotonic fashion with respect to t .It was found in many instances that the only difference between ^(v1, v2) and A(v1, v2) is a simple proportionality constant independent of the spectral coordinate. In such cases, the contour map representation of A(v1, v2) becomes virtually indistinguishable from that of ^(v1, v2).