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

ISBN 0-471-62391-1

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3.1.1 SYNCHRONOUS SPECTRUM

A discrete set of dynamic spectra measured at m equally spaced points in time t between Tmin and Tmax is represented by

where the 7th point in t is given by tj = Tmin + (TmaK - Tmin)(j - 1 )/(m - 1). We assume that the reference spectrum y{v), typically the average spectrum given by y{v) = YCj=i tj)/m, has already been subtracted from raw data as y{v, tj) = y{v, tj) — y{v). By transforming the integral in Equation (2.39) to a discrete summation form, the synchronous 2D correlation intensity may be directly calculated from the dynamic spectra by

For a set of data collected at varying intervals not equally spaced in time, it is necessary to modify Equation (3.2). The computation of 2D spectra for unevenly sampled data set will be discussed later.

Two-Dimensional Correlation Spectroscopy-Applications in Vibrational and Optical Spectroscopy

I. Noda and Y. Ozaki © 2004 John Wiley & Sons, Ltd ISBN: 0-471-62391-1

yj(v) = y (v,tj) j = 1, 2, ??? ,m

(3.1)

j=1

40

Practical Computation of 2D Correlation Spectra

It is often convenient to represent a discrete set of data in terms of a matrix notation. The column vector depicting the dynamic spectra of Equation (3.1) is given by

(v,h) y(v, t2)

y (v) =

y(v, tm)

(3.3)

By using this notation, the synchronous 2D correlation spectrum is concisely represented by the inner product of two dynamic spectrum vectors:

^(V1, V2) =

1

m — 1

y(V1) y(V2)

(3.4)

It is worth pointing out that the discrete form of a synchronous 2D correlation spectrum (Equations 3.2 and 3.4), derived from data observed at v1 and v2 over a period between Tmin and Tmax with an equal increment, results in the expression similar to that of statistical covariance of the spectral intensity with m — 1 degree of freedom.

3.1.2 ASYNCHRONOUS SPECTRUM

By adopting Equation (2.48) for a discrete set of dynamic spectra, one obtains the computational formula for an asynchronous spectrum.

1m

^(Vi, V2) = ——y^5v(v0 -Zj(v2) (3.5)

m j=1

The discrete orthogonal spectra Zj(v2) can be directly obtained from the dynamic spectra yj(v2) by using a simple linear transformation operation

m

Zj(V2) =^2 Njk ? yk(V2) (3.6)

k=1

where

N. =\ 0 if j = k (37)

jk \ 1/n(k — j) otherwise ( . )

By using the matrix notation of Equation (3.3), the asynchronous correlation spectrum can be written as

1

'I'Oh, v2) = ry(vi)1 Ny(v2)

m—1

(3.8)

Unevenly Spaced Data

41

where the Hilbert-Noda transformation matrix N is given by

0

-1

_ l 2 _ 1

3

1

0

-1 _ 1

2

1

2

1

0

-1

(3.9)

The above transformation method for carrying out the discrete Hilbert transform is robust and reliable, as long as the functional form of dynamic spectra is in alignment with Equation (2.1).2

3.2 UNEVENLY SPACED DATA

Spectra analyzed in the previous sections are assumed to be sampled with a fixed increment along the external variable to create a discrete data set consisting of equally spaced spectral traces. The results in Equations (3.2) and (3.5) are applicable only for such evenly spaced data. There are, however, many occasions in the real world where spectral measurements are taken with unevenly spaced increments of the external variable. The 2D correlation analysis of such unevenly spaced data sets cannot be effectively carried out by the computational method used earlier. One way to circumvent this limitation is to convert the unevenly spaced data into evenly spaced data through interpolation or a curve-fitting procedure. It would be much more convenient, however, if one could directly analyze the original data without such data conversion. Fortunately, only a modest modification to the original computational procedure is required to obtain the 2D correlation spectra reflecting the effect of uneven sampling of spectral data.4 This section provides a general method to efficiently calculate the synchronous and asynchronous 2D correlation spectra from spectral data collected with arbitrary and irregular increments.

For unevenly spaced data, the reference spectrum y{v), taken as the discrete

time-average of the observed spectral data, can be calculated by a simple numer-

ical integration method.

m

• (tj+1- j-1)

y(v) = ^--------------------- (3.10)

y^Stj+1- tj-1) j=1

We must also define two additional points in time t0 and tm+1 located outside the observation period as

t0 = 2t1 — t2 (3.11)

lm+1 2tm tm-1

(3.12)

42

Practical Computation of 2D Correlation Spectra

The denominator of Equation (3.10) then becomes

m

Y(tj + 1 - tj_x) = 3tm — tm — 1 + t2 — 3t1 (3-13)

j=1

If the discrete data are collected at an equal increment At, such that

At = tj — tj-1 = tj+1 - tj

= txh (3-14)

m — 1

then Equation (3.10) simplifies to the familiar straight average value of all data points

1m

y(v) =-y>(v) (3.15)

m

j=1

The synchronous and asynchronous 2D correlation spectrum calculated for unevenly spaced spectral data set are given by

1m

4>(vi, v2) = —-------— y'y>(vi) • yj(v2) ? (tj+i - tj_x) (3.16)

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