Application of Discrete Fourier Transform to Electron Spin Resonance Spectral Data Toshio Nishikawa and Karuo Someno National Chemical Laboratory for Industry, Honmachi, Shlbuya-ku, Tokyo, 15 1, Japan
The ESR spectrum can be treated as a wave of signals. This will be adequate to the flle search systems. The dlscrete Fourier transform has been applled to the data processing in ESR spectrometry. Normalization of the spectra for the signal height, the shift, and the signal-to-noise ratio can be achieved from the nature of Fourier algorithm. By using Fourier parameters, the computer storage may be reduced and the computing speed can be improved. This method is useful for storing and searching the data file of analytical spectra.
In file search systems for the spectral data in analytical chemistry, the computer storage is often a significant problem, with the growth of the amount of the spectral data. Among the various fields of spectrometry, there are often cases where the spectrum may be recognized as only the overall pattern rather than the individual values. I t is considered that an analytical spectrum can be processed as a wave, like speech-waves, seismic signals, electrocardiograms, or electroencephalograms. This technique may be expected to be useful to the collection and the search of spectral data in computers. ESR spectrometry seems to be one of the most suitable fields. In the case of complex organic free radicals, the ESR spectrum consists of the superposing sets of the hyperfine multiplets, and the knowledge of the spectral position is considered to be insignificant, because of the small difference in the g-values. In this sense, the ESR spectrum may be identified as only the pattern. Additionally, since the spectrum is characteristic of being center-symmetrical, such a technique will be advantageous. When an analytical spectrum is treated as a wave, the discrete Fourier transform (DFT) is often used as an orthonormal expansion. Data processing with the use of the DFT has appeared in several studies (1-4), with the development of the fast Fourier transform (FFT) algorithm (5-7). This paper describes an application of the discrete Fourier transform to ESR spectral data. I t is intended as the first step to the construction of an ESR data retrieval system. The processing to obtain the standard spectrum from the observed data, which vary due to the experimental conditions, is necessary to the file search. This can be achieved automatically by using the nature of DFT. I t has also an advantage that the use of the normalized DFT parameters can reduce the computer storage and improve the computing speed in the retrieval system.
were carried out with a FACOM 270-30 computer. The discrete Fourier transform was computed by the usual FFT method ( 5 , 6 ) .
DISCRETE FOURIER TRANSFORM AND NORMALIZATION PROCEDURE When the ESR spectral data may be considered as a time series, f ( n T ) ’ s 0, In IN - 1,consisting of N samples, the discrete Fourier transform is given as
where T is a sampling interval in the time domain and Q is a frequency increment, expressed as Q = 2*/NT, in the frequency domain ( 7 ) .On the other hand, the inverse discrete Fourier transform is given as
In these expressions, f ( n T ) and F ( K Q ) are generally complex numbers. For the storage of ESR spectral data, the normalization is necessary, because of the variety of the spectra, under the various conditions. Normalization for the Signal Intensity. The linearity of DFT gives the correction for the variation in the signal intensities of ESR data, using the relationship,
DFT {&T)}
= c D F T ~ ~ T ) }
The DFT term may be normalized by being expressed as the ratio t o the maximum value. Normalization for the Shift and the Deviation from Symmetry. A theoretical spectrum for an isotropic free radical solution ought to be center-symmetrical, when measured on a usual derivative-type spectrometer. However, the real trace on the recorder sheet is not always symmetrical, because of the difference of the setting of the magnetic field. The deviation from the symmetry will come from the noise. The normalization was performed by using the properties of DFT, as follows. As being experimental values, f(nT)’sare given as real numbers. If, in the ideal case, a set of f(nT)’s were given as an odd sequence with real numbers, the set of the DFT’s would result in an odd sequence with imaginary numbers. In fact, the sampled data are not purely odd, thus the set of the DFT’s is a sequence with complex numbers. The shifted data may be symmetrized by using the relationship in DFT:
EXPERIMENTAL A number of nitroxide free radicals in solution, as the trial samples, were prepared by the oxidation of the corresponding amines by perbenzoic acid. The ESR spectra were measured a t room temperature under the usual conditions, on a Varian E-12 spectrometer. The spectral data were digitized and accumulated on a JEOL EC-5 minicomputer, which is on-line equipped to the spectrometer. The output data set from EC-5 consists of the values of the ESR intensities, the number of which was tentatively adjusted. The other sophisticated calculations, using the above data set, 1290
ANALYTICAL CHEMISTRY, VOL. 47, NO. 8 , JULY 1975
(3)
and
where ( ) is used as a shorthand notation
(n - I ) =
H -
I (mod S).
Table I. P a t t e r n P a r a m e t e r s (PP)of the Largest Twenty Terms 1“
No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Q
K
2
PP
K
PP
K
K
PP
IF~I
~ ( k ~ =1 )
exp(iB,),
ek =
tan-‘(lm(Fk)/Re(Fk))(7)
PP
K
PP
1.0000 4.9162 0.7367 0.4597 4.4217 4.2521 4.2244 4.2132 0.2085 4.1678 0.1570 0.1264 0.0907 0.0789 4.0734 0.0724 0.0677 4.0638 0.0612 -0.0593
41 4 40 38 44 43 37 39
1.0000 0.7675 -0.7133 4.6523 4.5633 0.5336 0.4175 4.3981 4.3919 0.3277 4.2913 0.2542 0.1977 0.1889 4.1794 0.1520 0.1408 -0.1159 4.0418 0.0415
K
PP
40 1.0000 42 1.0000 26 1.0000 26 1.0000 41 59 0.8114 4.9327 15 4.9339 15 4.8967 12 0.8021 20 4.6304 38 12 0.8384 12 0.7986 54 -0.6102 4 37 4.5213 37 4.5469 37 4.5373 60 4.4965 40 23 4.4186 23 4.4481 16 0.4383 40 0.3992 16 0.4385 23 4.4085 114 21 0.4671 39 4.4277 39 16 0.3869 40 0.3912 40 0.4045 11 4.3337 2 0.3506 5 11 4.3717 11 4.3418 29 4.3025 29 4.3227 6 0.3448 3 29 4.3327 22 0.3277 22 0.2999 22 0.2941 15 0.3387 45 36 0.2557 41 4.3295 2 36 0.2950 24 4.2469 24 4.2708 36 0.2417 24 4.2484 115 4.3149 115 25 4.2429 51 4.2184 51 4.2264 55 0.2331 43 51 4.2089 27 4.2091 25 4.2251 16 4.1872 46 114 4 0.1609 25 4.1951 30 0.2066 0.1517 36 30 0.1602 30 0.1884 34 0.0424 35 4.1487 6 28 4.1589 28 4.1847 28 4.1730 57 4.1342 116 17 0.1552 41 4.1655 17 0,1713 172 4.1019 196 38 4.0954 195 27 4.1487 4 0.1625 4 0.1678 13 0.1267 17 0.1489 41 4.1504 3 0.0873 173 The numbers in the headings refer to the traces in the figures. K denotes the order of DFT term. 26 15
These expressions state that cycling 1 samples from the end of a sequence to the beginning of the sequence is equivalent to multiplying the DFT by a linear-phase function; the shift in the time domain corresponds to the rotation of an argument of complex number in the frequency domain. For the sake of simplified calculation, the complex F ( k Q ) is written as
6
5
4
3
la
Ib 8
2a
7
42 9 6 91 36 46 12 35 94 174 142
\
I
1
1
2b
When the absolute value of F ( k Q ) has a maximum at k = and the argument 0 is expressed as O, the step angle r is defined as
K,,
Y =
(*n/2) -,,,Q kmax
(8)
+
where the sign or - is selected, according to whether the spectral trace is advanced or delayed with respect to its center. Using the relationship, Equations 5 , 7 , and 8, the new P(KQ)’s are obtained from the F(hQ)’s for all k , as given by
P(kS2) = F(kS2) exp(irk)
(9)
This processing means that any shifted trace can be normalized by such rotation of DFT vectors in the complex plane, and that the maximum term may be converted to be purely imaginary. After the conversion, the P ( k Q)’sare the odd sequence, almost all of which are purely imaginary, and the real parts of which are negligibly small. It is considered that the larger term of the absolute value in the DFT sequence may make the larger contribution to the spectral trace. Therefore, the partial set of the imaginary part of the sequence, P ( k Q)’s,consisting of some numbers of the largest terms in the absolute value, may be used as the index to extract the feature of the spectral pattern. For the convenience of the following discussion, these terms of the sequence are named as pattern parameters. Once the pattern parameters are determined, the inverse discrete Fourier transform (IDFT) can reproduce the original ESR spectrum by using them, which will be demonstrated below.
,
I
Figure 1. Original spectrum patterns ( l a and 2a), and DFT plots ( l b and 26),for diisopropyl nitroxide radical, in the case of different signal heights and shift
RESULTS AND DISCUSSION In order to examine the efficiency of this method, the ESR spectra of several nitroxides were measured under various conditions. As the DFT has to be calculated by fast Fourier transform (FFT), the number of samples, N , should be a power of 2. The number of sampling points was taken as 512 over a whole spectrum, for the time being, although some discussions will be given below. The observed spectra of diisopropyl nitroxide are shown as l a and 2a in Figure 1. In Figure 1, 2a is contrasted with la, with respect to the difference of signal height and, simultaneously, the shift of ESR traces, due to the different field. The largest twenty values of the pattern parameters are shown in Table I. The plots of the pattern parameters are shown as l b and 2b in Figure 1. As the original data are given as the real numbers, the pattern parameters are plotted over 256 points, because the half sequence of the DFT is sufficient from the properties that the absolute values of the DFT are even and the arguments are odd. ANALYTICALCHEMISTRY, VOL. 47, NO. 8, JULY 1975
1291
IC
L-50
L-LO
lb
la
I
1
3a
i=20
L=30
3b I
