# Two dimensional correlation spectroscopy applications in vibratioal and optical spectroscopy - Isao N.

ISBN 0-471-62391-1

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2.5.1 CROSS-CORRELATION FUNCTION AND CROSS SPECTRUM

The classical cross-correlation function between distinct dynamic spectral intensity variations observed at v1 and v2 along t for a fixed period between Tmin and Tmax is given by

-Vf

Tmin J Tmi„

C(t) = -------------- I y(vi,t) ? y(v2,t + r)dt (2.30)

Tmax Tm

where t is the correlation time. The function compares the time dependence of two separate functions (i.e., intensities of dynamic spectra measured at two different wavenumbers) shifted by a fixed constant t . The corresponding cross spectrum S(m'), which is the Fourier transform of the cross-correlation function, is given by

w = w-Sw

Tmax Tmin

= fw(V1,V2) - lfw(V1,V2) (2.31)

The real and negative imaginary component of the cross spectrum, fm(v\, v2) and fm(v\, V2), are referred to, respectively, as the cospectrum and quad-spectrum, which can be expressed in terms of the real and imaginary components of the Fourier transform of the dynamic spectrum intensities as

<PJvi, v2) = \ [?*» • Y^(co) + Ylm(co) ? ?2In»] (2.32)

T max T min

fJVl, v2) = 1 [Ylm(co) • Y^(co) - TR» • T2In»] (2.33)

Tmax - Tmin

32

Principle of Two-dimensional Correlation Spectroscopy

It is necessary to point out that the term spectrum (e.g., cross spectrum, cospectrum, and quad-spectrum) used here specifically refers to the Fourier domain spectral representation of time-dependent signals. This mathematical spectrum should not be confused with the optical intensity spectrum measured with an electromagnetic probe. From Equation (2.5) the cospectrum and quad-spectrum are related to the synchronous and asynchronous 2D correlation spectra by

1 fto

4>(u, v2) = - <pm(v1,v2)(ko (2.34)

n Jo

1 fto

^(vi, v2) = - f0)(v1,v2)(ko (2.35)

n Jo

In other words, the cospectrum and quad-spectrum represent the individual Fourier component of the synchronous and asynchronous 2D correlation spectrum. In turn, the synchronous spectrum and asynchronous spectrum may be regarded as the integral sums of the contributions of cospectrum and quad-spectrum over the entire positive Fourier frequency of the external variable.

2.5.2 CROSS-CORRELATION FUNCTION AND SYNCHRONOUS SPECTRUM

We now derive an important relationship between the cross-correlation function and synchronous 2D correlation intensity. By applying the well-known Wiener-Khintchine theorem,10 the cross spectrum can be directly related to the time-domain Fourier transform of dynamic spectral intensity variations (see Appendix).

1 fto ~ ~

C(r) = Y*(co) • Y2(co) e,a,r dr» (2.36)

2n(Tmax Tmin) J —to

By setting the correlation time to t = 0, Equation (2.36) reduces to

1 f to ~

C(°) = _ Y^co) • Y2(a>) dm (2.37)

2n(Tmax Tmin/ J—to

Since the imaginary component of a cross spectrum must consist exclusively of an odd function, the integration over the symmetric range of rn from —to to +to leaves only the real component

C(0) = 1 Re ( [ Yi{(o) • ?2» da,} (2.38)

n(Tmax Tmin) lv0 J

The notation Re{} stands for the real part of a complex number. According to Equation (2.5), the above expression is identical to the real part of the generalized 2D correlation function. Thus, the synchronous 2D correlation intensity

Cross-Correlation Analysis and 2D Spectroscopy

33

0(vi, v2) can be directly calculated from the cross-correlation function C(t) with t = 0. Most importantly, since the determination of the cross-correlation function according to Equation (2.30) does not require the use of the Fourier transformation of dynamic spectra, the synchronous 2D correlation spectrum can be directly computed as

1 C Tmax

®(vi,v2) = ----------_ y(vt> t) ? y(v2, t)dt (2.39)

T max T min J Tmn

This result shows that the synchronous 2D correlation intensity is nothing but the time average of the product of dynamic spectral intensity variations measured at two different spectral variables, v1 and v2.

The manipulation of a classical cross-correlation function of dynamic spectral intensity variations yields only a synchronous 2D correlation spectrum. An asynchronous 2D correlation spectrum must be calculated by other means if one wishes to circumvent the use of the Fourier transformation of dynamic spectra. Although a heuristic approximation using a disrelation spectrum discussed later, for example, provides a reasonable estimate of an asynchronous 2D correlation spectrum, a more reliable computational method based on a rigorous mathematical derivation is desired. Such a method may be developed by utilizing the time-domain Hilbert transformation of dynamic spectra.2

2.5.3 HILBERT TRANSFORM

For a given analytic function g(t), the Hilbert transform h(t) of the function is given by

1 n(f)

h(t) = -pv d f (2.40)

Ï J-æ t' - t

The integration symbol pv / denotes that the Cauchy principal value is taken, such that the singularity at the point where t' = t is excluded from the integration. It is well known that the Hilbert transform operation is closely associated with the Kramers-Kronig analysis of various spectra which are coupled by the orthogonal dispersion relationship.

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