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Jonathan W. Amy Jack W. Frazer G. Phillip Hicks
Donald R. Johnson Charles E. Klopfenstein Marvin Margoshes
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Harry L. Pardue Ralph E. Thiers William F. Ulrich
Fourier Transform Approaches to Spectroscopy Gary Horlick Department of Chemistry, University of Alberta, Edmonton, Alta., Canada
Fourier transformation techniques have already led to significant advances in methods of spectral data handling. Increase in the use of the Fast Fourier Transform program should facilitate further developments in this area and enhance its value to analytical chemists CPECTROSCOPISTS
HAVE, in the
past,
^ dealt primarily with spectra. How ever, a consideration of the Fourier transformation of a spectrum can often result in a more complete understand ing of several aspects of spectroscopy and spectroscopic measurements. The Fourier transform is basic to the very nature of a spectroscopic measurement since the dispersion step is, in effect, a Fourier transformation of the electro magnetic signal. The Fourier transform is intimately related to instrumental measurements through the convolution integral, and the important topic of spectral resolution falls in this area. Certain types of mathematical op erations, such as convolution, may be carried out on spectra using Fourier transformations. The calculations for this type of data handling are difficult to carry out. Recent advances in the machine calculation of Fourier trans formations have removed this prob lem, and wider use of the Fourier trans formation is sure to be seen in the data handling of spectra.
In some experiments the Fourier transformation of the spectrum may be the final desired result, rather than the spectrum. The work of Gordon on molecular correlation is an example of such a measurement (1). Finally, two fairly new instrumental areas are presently being developed that necessitate an understanding of the Fourier transform operation in order to understand the measurement and subsequent data analysis. These tech niques are Fourier transform spec troscopy in the optical region (3) and Fourier transform NMR spectroscopy (3). The measurement step in both these techniques results in the record ing of a signal that is the Fourier transform of the conventional spec trum. The above points indicate that ana lytical chemists need a basic under standing of Fourier transformations. A brief introduction to them is pre sented in this paper. Then several aspects of spectroscopy are interpreted on the basis of Fourier transformations. The coverage is not meant to be com prehensive but simply representative of this approach. Pictorial Fourier trans forms are used to illustrate several points. It is often easier to get an intuitive feeling for the mathematical operation that is taking place by look ing at a picture rather than an equa tion. The Fourier transform pairs illus trated pictorially in this paper are not schematic representations but are CALCOMP plots of the actual trans formations as carried out on a digital computer. Introduction to Fourier Transformations
This section provides a brief intro duction to Fourier transformations. Terminology that will be used later in the paper is defined, and two of the most important properties of Fourier transformations, with respect to spec-
troscopic application, are illustrated. There are several comprehensive treat ments of the theory of Fourier trans forms. A particularly useful source is the book by Bracewell (4). The Fourier integral is a mathemati cal means of relating two functions F(x) and F(v). It may be stated as: CO
F(v)e2 rixvdv
/
(1)
— m
An analogous integral exists such that F(v) = Γ J
F(x)e-2"*°dx
(2)
— CO
These two equations indicate the reciprocal property of the Fourier in tegral. In the case of Equation 1, it may be stated that F(x) is the Fourier transformation of F(v) and for Equa tion 2, F(v) is the Fourier transforma tion of F(x). Thus, functions F(x) and F(v) constitute a Fourier trans form pair. The exponential of Equation 2 may be written as: cos (2 -πχν) — i sin (2 ττχν). When F(x) is an even func tion, Equation 2 reduces to : F(v) = 2 I
F(x) cos 2 Trxvdx (3)
This equation is often referred to as the cosine Fourier transformation or just the cosine transform. When F(x) is an odd function, an analogous equation exists with the cosine term being re placed by the sine term. Both the sine and the cosine transforms have the reciprocal property indicated by Equations 1 and 2. There are no rigorous constraints on the units of the functions F(x) and F(v). However, one of the functions is usually a function of frequency. For the purposes of this paper, F(v) will designate the frequency dependent function and it will have units such as
ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
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61 A
Instrumentation
A
Β
c
0.062 sec
ο
TIME
0.125 sec 0
DOMAIN
1000 Hz
2000 Hz
FREQUENCY DOMAIN
Figure 1. Pictorial representations of the Fourier transformation of (A) 32 cycles, (B) 21 cycles, and (C) 10 cycles of a 1000-Hz cosine wave. Note the inverse dependence of the width of the frequency domain function on the length of the time domain function
A
ÉÉttHHUim
ο
Q0o2sec TIME DOMAIN
0J25sec 0
IfJOQHz
20QOHz
FREQUENCY DOMAIN
Figure 2. Pictorial representations of the Fourier transformation of a 1000-Hz cosine wave that has been damped in a linear (A), exponential (B), and Gaussian (C) manner. The respective functional dependencies of the frequency domain functions are sin 2 x/x2, Lorentzian, and Gaussian 62 A • ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
s e c - 1 or c m - 1 . Often the function F(v) is said to be in the frequency domain. The function F{x) then becomes a function of time or distance and will have units of sec or cm. The function F(x) is said to be in the time or space domain. Thus, very simply, F(x) is a waveform and F(v) is a spectrum, and the transform relationships provide a means of converting from one domain to the other. The integrals in Equations 1, 2, and 3 exist for any physically realizable functions F(x) and F(v)—i.e., any waveform, in general, is composed of several frequencies. Fourier transformation is simply a technique for sorting out the intensities and frequencies present in any given waveform. It is possible to solve this integral (Equation 3) for some simple functions. A common waveform is a cosine wave of finite length. This type of waveform occurs often in the physical world. Its transform is easy to calculate and illustrates several important characteristics of Fourier transform pairs. When we substitute F(x) = cos 2 •κχυ' into Equation 3, the following simplified equation is obtained:
=
fa 2 «(.'-»)
(4)
2 T(V' — v) where v' is the frequency of the cosine wave, and χ is the length of the cosine wave. This equation represents the spectrum of the cosine wave. This transform is shown pictorially in Figure 1A. A finite length (32 cycles) of a 1000-Hz cosine wave was transformed, and the resulting spectrum is shown im mediately to the right. Note that the function described by Equation 4 has a finite maximum at ν = ν', and nega tive maxima on each side with inten sities of about 20% of the central maxi mum. The width of this function de pends inversely on x, the length of the original cosine wave. This is shown in Figures IB and 1C where the Fourier transformations of 21 cycles and 10 cycles, respectively, of the same 1000-Hz cosine wave are illustrated pic torially. This inverse dependence of the width of the frequency domain function on the length of the time do main function is a very important characteristic of Fourier transform pairs. A second important characteristic to note is that the functional dependence obtained for the spectrum of the cosine waveform is determined by the form of the truncation applied to the cosine wave. For the cases illustrated in Fig ure 1, the truncation was abrupt. The pictorial Fourier transform pairs for three other common truncations of a cosine wave are shown in Figure 2. A
Instrumentation linear truncation of a cosine wave re sults in a sin2x/a:2 functional depen dency for the spectrum (Figure 2A), an exponential truncation in a Lorentzian functional dependency (Figure 2B), and a Gaussian truncation in a Gaus sian functional dependency (Figure 2C). These dependencies can be shown mathematically by solution of Equa tion 3. However, the illustration of the Fourier transforms pictorially ef fectively indicates the main properties of the transforms. The book by Bracewell contains a dictionary of pictorial Fourier transforms for many additional functions (4) • Equations 1, 2, and 3 cannot easily be solved except for relatively few simple waveforms or spectra, such as those illustrated in Figures 1 and 2. For general waveforms and spectra, the Fourier transformation is usually performed on a digital computer. The waveform or spectrum is sampled, and the evaluation of the Fourier trans form takes the form of a summation. In 1965 Cooley and Tukey rediscovered and developed a technique for per forming this summation efficiently. This is often referred to in the litera ture as the Fast Fourier Transform or the Cooley-Tukey Algorithm (δ, β). The development of this computer pro gram has greatly facilitated the use of Fourier transformations in many data-handling situations. The Fast Fourier Transform was recently the topic of a special issue of the IEEE Transactions on Audio and Electroacoustics (7). Utilizing the basic ideas covered in this section, we can discuss several as pects of spectroscopy by considering the time domain function in addition to the frequency domain function (i.e., the spectrum.) The Fourier transform provides the link between the two domains. Fourier Transform Approaches to Spectroscopy
Frequency Decoding. The fre quency-dependent nature of the inter actions of electromagnetic radiation with matter provides the chemist with a vast amount of information. This information is most usefully interpreted in the form of a spectrum, a plot of the intensity of electromagnetic radiation as a function of frequency. However, this information is encoded in an elec tromagnetic waveform. To obtain the spectrum, this electromagnetic wave form must be analyzed for its fre quency content—i.e., the frequency in formation must be decoded. This step amounts to taking the Fcurier trans
formation of the electromagnetic wave form. In the electronic region this is relatively easy, as tunable components and systems are available that respond specifically to the actual frequencies of the electromagnetic waveform. A radio is an excellent example of such a sys tem. Such components and systems are not yet available that respond in this fashion to the very high frequency electromagnetic waves that constitute the optical region. In the optical region, somewhat in direct approaches must be used to carry out the Fourier transformation of an electromagnetic waveform. Prisms and gratings are, in a sense, powerful Fourier transformers. They decode the frequency information present in the electromagnetic waveform. The frequencies are spread out in space along the focal plane of an optical in strument (such as a spectrograph or monochromator) to form a spectrum. Another approach is to use a Michelson interferometer to generate a signal, called an interferogram, from the elec tromagnetic radiation. The Fourier transformation of this signal must be taken by the experimenter in order to obtain the spectrum of the original electromagnetic radiation. This step is usually performed on a digital com puter. The more implicit presence of the Fourier transform step in this ap proach has resulted in the technique's being called Fourier transform spec troscopy. However, as can be seen from the above discussion, Fourier transformation is fundamental to all spectral determinations. Fundamental Line Shapes. As was noted earlier, a damped or truncated cosine wave and the line shape of its corresponding spectrum are Fourier transform pairs. A real electromagnetic wave has a finite length and is damped or truncated in a specific fashion. Thus the fundamental width of a spectral line depends on how long the wave is and the shape of the line depends on the manner in which the wave is damped or truncated. Classical radia tion theory leads to the conclusion that an emitted light wave, in the case of an unperturbed radiation lifetime, is exponentially damped (8). The Fourier transformation of an ex ponentially damped cosine wave is a line with Lorentzian functional depend ency. This is the well-known line shape in the case of radiation damping. This unperturbed line width and shape are seldom observed because of various line-broadening interactions, such as Doppler and collisional broad ening. Doppler broadening leads to a Gaussian line shape, normally a couple of orders of magnitude wider than the natural line width. Collisional broad-
Instrumentation
ening leads to a Lorentzian line shape again—in general, significantly wider than the unperturbed line width. However, care must be exercised in concluding the type of electromagnetic signal that results in a specific line shape. In the case of collisional broadening, the Lorentzian line shape is not the result of an exponentially damped cosine wave but the result of the summation of several sinusoidal waves abruptly truncated by collisions. Spectral Resolution. What shape does a spectroscopic instrument impose on an infinitely narrow spectral line? This is determined by the resolution or instrumental function of a spectroscopic measurement system. In prism and grating instruments, the resolution function may take two limiting forms. In the diffraction limited situation, the resolution function takes the form of a sin2 x/x2 function (9), and in the slit-width limited situation, it may take the form of a triangular function. The width of the resolution function imposes a limit on the resolution of the spectroscopic instrument, and the
shape of the resolution function limits the ability of the instrument in accurately measuring spectral line shapes. If the resolution function is significantly wider than the width of the line being measured, the observed line width and shape will be that of the resolution function rather than that of the line itself; if they are approximately equivalent in width, the measured line shape will be a composite of the two, and only if the resolution function is significantly narrower than the observed line will the actual line shape and width of the source be measured. The effects of the resolution function on the resulting spectrum are described by the convolution integral. The concept of convolution is generally useful in describing the effects of any particular instrument on an observed parameter {4,10). However, it is often difficult to intuitively visualize the convolution operation. For the example stated above, the resolution function can be thought of as a scanning function that takes a "weighted average," or "running mean" of the spectrum to generate the observed spectrum. This
SPECTRUM
RESOLUTION FUNCTION
type of terminology is frequently used to describe convolution. The convolution operation may take on additional meaning and often ease of interpretation when it is realized that convolving two functions is equivalent to multiplying the Fourier transformations of the two functions. This is illustrated pictorially in Figure 3A. The upper section of Figure 3A depicts the effects of convolving a Gaussian spectral line with a sin2 x/x2 resolution function to generate an observed spectral line. The lower section of Figure 3A depicts the convolution operation as a multiplication of the Fourier transforms of the respective functions. In this case, the resolution function is narrower than the spectral line, and little broadening or distortion is observed. Figure 3B depicts the same convolution but with a wider sin2 x/x2 resolution function. In this case the observed line is severely widened and distorted to the point of taking on the shape of the resolution function. That this should happen can be readily appreciated by noting the multiplication of the respective Fourier transforms
OBSERVED SPECTRUM
CONVOLUTION
FOURIER TRANSFORMATION
FOURIER TRANSFORMATION FOURIER TRANSFORMATION
MULTIPLICATION
CONVOLUTION
FOURIER TRANSFORMATION
MULTIPLICATION
FOURIER TRANSFORMATION
FOURIER TRANSFORMATION
•
Figure 3. Convolution of a single Gaussian line spectrum by a sin2 x/x a resolution function (two different widths) to generate an observed spectral line. In each case the convolution is also depicted as a multiplication of the Fourier transforms of the respective functions ANALYTICAL CHEMISTRY, VOL 43, NO. 8, JULY 1971
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Instrumentation SPECTRUM
RESOLUTION FUNCTION
OBSERVED SPECTRUM
CONVOLUTION
FOURIER TRANSFORMATION
FOURIER TRANSFORMATION
i
FOURIER TRANSFORMATION
'MM
MULTIPLICATION Figure 4. Convolution of a Gaussian line spectrum by a triangular resolution function to generate an observed spectrum. equivalent Fourier transform route of the convolution operation is also illustrated
(lower section of Figure 3B). The multiplication of the Gaussian damped cosine wave by the short linear trunca tion function results in a damped co sine wave with considerable linear char acter. This indicates that the observed spectral line will have a significant amount of sin2 xl x2 functional depen dence. Thus, the broadening and dis tortion of a spectral line when it is convolved by the resolution function of a spectroscopic instrument are readily understood on the basis of. the two simple properties of Fourier trans form pairs discussed in the first sec tion—namely, the inverse dependence of the width of the frequency domain function on the length of the time do main function and the dependence of the shape of the frequency domain func tion on the form of the truncation ap plied to the time domain function. The effects of convolving a spectrum by a resolution function are further illustrated in Figure 4. The format of this figure is analogous to that of Figure 3. In this case a spectrum con sisting of wide and narrow Gaussian lines is convolved with a triangular resolution function to give the ob served spectrum. This figure depicts the well-known situation where a nar row line in a spectrum may be severely distorted by a particular slit width, and a wide line in the same spectrum will not be significantly distorted (11). Again this can be readily appreciated by thinking in terms of the equivalent Fourier transform route of convolution. Note that in the lower section of Fig ure 4, the multiplication of the Fourier transform of the spectrum by the Fourier transform of the resolution function results in little, if any, trun cation of the Gaussian damped cosine wave due to the wide line, while that due to the narrow line is truncated. As mentioned by Savitzky and Go66 A
•
lay (10), the observed spectrum is further convolved by time constants in the measurement electronics. The effects of this convolution could also be treated using Fourier transforma tions. However, the examples dis cussed in conjunction with the resolu tion function serve to indicate the ap proach. Thus, a number of instru mental effects on spectra can be ap preciated, understood, and interpreted on the basis of a Fourier transform ap proach rather than by a direct appli cation of the convolution integral. Also, an understanding of this ap proach leads to the development of data-handling techniques that can be performed on a spectrum to minimize or remove instrumental effects and also to the performance of operations on spectra not readily possible with hard ware but easily implementable with software. Data Handling Based on Fourier Transformations. Certain types of data-handling operations can be car ried out on spectra, by utilizing Fourier transformations. In this section a pos terior convolution of observed spectra will be mentioned briefly to indicate one approach. In the last section it was seen that the line shape in the observed spectrum was determined by convolution of the real spectrum by the resolution func tion. This convolution can be per formed on a digitized spectrum using a digital computer. Thus, the line shape in an observed spectrum can be modified. For example, side lobes on a line shape function are undesirable if a small peak occurs close to a large peak, the side lobes easily being mistaken for real peaks. In this case the ob served spectrum could be convolved on a computer with a mathematical reso lution function that results in a line shape function of minimal side lobes.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
The
This could be easily carried out using the Fourier transform route of the convolution operation. The Fourier transformation of the observed spec trum is simply multiplied by the ap propriate truncation function. This type of data handling is used exten sively in Fourier transform spectros copy and is called apodization. This simple example illustrates a type of data-handling operation that is possible with spectra when utilizing Fourier transforms and convolutions. For the most part, extensive use of Fourier transformations in spectral data handling has not yet been made by analytical chemists. The Fast Fourier Transform program should facilitate further developments in this area. References
(1) R. G. Gordon, J. Ckem. Phys., 43, 1307 (1965). (2) G. Horlick, Appl. Spectres., 22, 617 (1968). (3) R. R. Ernst, "Advances in Mag netic Resonance, Vol. 2," J. S. Waugh, Ed., Academic Press, New York, N.Y., 1966,ρ 1. (4) Ron Bracewell, "The Fourier Trans form and Its Applications," McGrawHill, New York, N.Y., 1965. (5) G-AE Subcommittee on Measure ment Concepts, "What is the Fast Fourier Transform?", IEEE Trans. Audio Electroacoustics, ATJ-15 (2), 45 (1967). (6) L. Merfz, Appl. Opt., 10, 386 (1971). (7) IEEE Trans. Audio Electroacous tics, ATJ-17 (2), 65-186 (1969). (8) W. Kauzmann, "Quantum Chemis try," Academic Press. New York, N.Y., 1957,ρ 556. (9) R. A. Sawyer, "Experimental Spec troscopy," Dover Publications, New York/N.Y., 1963, ρ 33. (10) A. Savitsky and M. J. E. Golay, ANAL. CHRM., 36, 1627 (1964).
(11) W. E. Wentworth, J. Chew,. Educ, 43, 262 (1966).
