INST R 0 ME NTAT I0 N
Advisory Panel Jonathan W. Amy Jack W. Frazer G. Phillip Hicks
Donald R. Johnson Charles E. Klopfenstein Marvin Margoshes
Harry L. Pardue Ralph E. Thiers William F. Ulrich
I
Fourier Transform Approaches to Spectroscopy Gary Horlick
Department of Chemistry, University of Alberta, Edmonton, Alta., Canada
Four ier transformat ion 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 PECTROSCOPISTS HAVE, in the past, Sdealt primarily with spectra. However, a consideration of the Fourier transformation of a spectrum can often result in a more complete understanding 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 electromagnetic 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 operations, 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 transformations have removed this problem, and wider use of the Fourier transformation is sure to be seen in the data handling of spectra.
I n 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 techniques are Fourier transform spectroscopy in the optical region (2) and Fourier transform N M R spectroscopy ( 3 ) . The measurement step in both these techniques results in the recording of a signal that is the Fourier transform of the conventional spectrum. T h e above points indicate that analytical chemists need a basic understanding of Fourier transformations. A brief introduction to them is presented in this paper. Then several aspects of spectroscopy are interpreted on the basis of Fourier transformations. The coverage is not meant to be comprehensive but simply representative of this approach. Pictorial Fourier transforms are used to illustrate several points. It is often easier to get an intuitive feeling for the mathematical operation that is taking place b y looking a t a picture rather than an equation. The Fourier transform pairs illustrated pictorially in this paper are not schematic representations but are CALCOMP plots of the actual transformations as carried out on a digital computer. Introduction to Fourier Transformations
This section provides a brief introduction 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 treatments of the theory of Fourier transforms. A particularly useful source is the book by Bracewell ( 4 ) . The Fourier integral is a mathematical means of relating two functions F ( z ) and F ( v ) . It may be stated as:
F(z) =
s-mm
F ( v ) e2 *izvdv
(1)
An analogous integral exists such that
F(v)
=
J m -m
F(z)e-2rimciz
(2)
These two equations indicate the reciprocal property of the Fourier integral. I n the case of Equation 1, it may be stated that F ( z ) is the Fourier transformation of F ( v ) and for Equation 2, F ( v ) is the Fourier transformation of F ( z ) . Thus, functions F ( z ) and F ( v ) constitute a Fourier transform pair. The exponential of Equation 2 may - i sin be written as: cos (2 ( 2 xsv). When F ( z ) is an even function, Equation 2 reduces to:
F(v)
= 2
I=
F ( z ) cos 2 *zudz
(3)
This equation is often referred to as the cosine Fourier transformation or just the cosine transform. When F ( z ) is an odd function, an analogous equation exists with the cosine term being replaced 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 ( s ) 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
61A
I
I
C
1
I
0
A
0.062sec TIME DOMAIN
0.125sec0
V
Y
I000 Hz 2000 H t FREQUENCY DOMAIN
Figure I . Pictorial representations of the Fourier transformation of (A) 32 cycles, (8) 2 1 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
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 sinax/x*, Lorentzian, and Gaussian
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ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
sec-1 or cm-l. Often the function F ( v ) is said to be in the frequency domain. T h e function F ( z ) then becomes a function of time or distance and will have units of sec or cm. The function F ( z ) is said to be in the time or space domain. Thus, very simply, F ( z ) 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 P ( z ) and F(v)-Le., 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. I t s transform is easy to calculate and illustrates several important characteristics of Fourier transform pairs. When we substitute F ( r ) = cos 2 am’ into Equation 3, the following simplified equation is obtained:
F(v) =
sin 2 a z ( d - v ) 2 a(v’ - v )
(4)
where Y’ is the frequency of the cosine wave, and z is the length of the cosine wave. This equation represents the spectrum of the cosine wave, This transform is shown pictorially in Figure 1-4. A finite length (32 cycles) of a 1000-Hz cosine wave was transformed, and the resulting spectrum is shown immediately to the right, Note that the function described by Equation 4 has a finite maximum a t Y = v’, and negative maxima on each side with intensit’ies of about 20% of the central maximum. The width of this function depends inversely on r, the length of the original cosine wave. This is shown in Figures 1B and 1C where the Fourier transformations of 21 cycles and 10 cycles, respectively, of the same 1000-HZcosine wave are illustrated pictorially. This inverse dependence of the width of the frequency domain function on the length of the time domain 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 Figure 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 results in a sin2dz2 functional dependency for the spectrum (Figure 2A), an exponential truncation in a Lorentzian functional dependency (Figure 2B) , and a Gaussian truncation in a Gaussian functional dependency (Figure 2C). These dependencies can be shown mathematically by solution of Equation 3. However, the illustration of the Fourier transforms pictorially effectively indicates the main properties of the transform. 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 transform takes the form of a summation. In 1065 Cooley and Tukey rediscovered and developed a technique for performing this summation efficiently. This is often referred to in the literature as the Fast Fourier Transform or the Cooley-Tukey Algorithm (5, 6 ) . The development of this computer program 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 aspects 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 frequency-dependent nature of the interactions of electromagnetic radiation with matter provides the chemist with n 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 electromagnetic waveform. To obtain the spectrum, this electromagnetic waveform must be analyzed for its frequency content-Le., the frequency information must be decoded. This step amounts to taking the Fcwier trans-
formation of the electromagnetic waveform. I n the electronic region this is relatively easy, as tunable components and systems are available that respond spccifically to the actual frequencies of the electromagnetic waveform. A radio is an excellent example of such a system. Such components and systems are not yet available that respond in this fashion to the very high frequency clectrornagnetic waves that constitute the optical region. I n the optical region, somewhat indirect, approaches must be used to c:trry 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 instrument (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 electromagnetic 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 computer. The more implicit presence of the Fourier transform step in this approach has resulted in the technique's being called Fourier transform spectroscopy. However, as cdn 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 t'ransform 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 radiation 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 exponentially damped cosine wave is a line with Lorentzian functional dependency. This is the well-known line shape in the erne of radiation damping. This unperturbed line width and shape are seldom observed because of vwrions line-broadening interactions, such as Doppler and collisional broadening. Doppler broadening leads to a Gaussian line shape, normally a couple of orders of magnitude wider than the natural line width. Collisiona,l broad-
Instrumentation
I
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. I n 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. I n prism and grating instruments, the resolution function may take two limiting forms. I n the diffraction limited situation] the resolution function takes the form of a sin2 2 / 2 2 function ( 9 ) , and in the slit-width limited situation, it may take the form of a triangular function. The wicih of the resolution function imposes a limit on the resolution of the spectroscopic instrument] and the
shape of the resolution function limits t,he 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, t,he observed line width and shape will be that of t,he resolution function rather than that of the line itself; if they are approximately equivalent in width, t,he 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 t,he 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, I O ) . However, it is often difficult t o intuitively visualize the convolution operation. For the example stated above, the resolution funct’ion can be t,hought of as a scanning function t,hat takes a “weighted average,” or “running mean” of the spectrum to generate the observed spectrum. This
type of terminology is frequently used to describe convolution. The convolution operation may take on nddit,ional 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 effect,s of convolving a Gaussian spectral line with a sin2 xlx2 resolution function to generate an observed spectral line. The lower section of Figure 3.4 depicts the convolution operation as n multiplication of the Fourier transforms of the respect,ive functions. I n 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 2 / 5 2 resolution function. I n 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
Figure 3. Convolution of a single Gaussian line spectrum by a sinz x/xZ resolution function (two different widths) to generate an observed spectral line. In each case the convolution is also depicted as a mulmtiplication of the Fourier transforms of the respective function9 ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
e
65A
Instrumentation
Figure 4. Convolution of a Gaussian line spectrum by a triangular resolution function t o generate an observed spectrum. equivalent Fourier transform route of the convolution operation is also illustrated
(lower section of Figure 3 B ) . The multiplication of the Gaussian damped cosine wave by the short linear truncation function results in a damped cosine wave with considerable linear character. This indicates that the observed spectral line will have a significant amount of sin2 x/x2 functional dependence. Thus, the broadening and distortion of a spectral line when it is convolved by the resolution function of a spectroscopic instrument are readily understood on the basis o f . the two simple properties of Fourier transform pairs discussed in t’he first section-namely, the inverse dependence of the width of the frequency domain function on the length of the time domain function and the dependence of the shape of the frequency domain function on the form of the truncation applied to the time domain funct’ion. 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 spect,rum consisting of wide and narrow Gaussian lines is convolved with a triangular resolution function to give the observed spectrum. This figure depicts the well-known situation where a narrow 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 ( 12 ) . .4gain this can be readily appreciated by thinking in terms of the equivalent Fourier transform route of convolution. S o t e that in the lower section of Figure 4, the multiplication of the Fourier transform of the spectrum by the Fourier transform of the resolution function results in little, if any, truncation of the Gaussian damped cosine wave due to the wide line, while that due to the narrow line is truncated. -4s mentioned by Savitzky and Go66A
lay (IO), the observed spectrum is further convolved by time constants in the measurement electronics. The effects of this convolution could also be treated using Fourier transformations. However, the examples discussed in conjunction with the resolution function serve to indicate the approach. Thus, a number of instrumental effects on spectra can be appreciated, understood, and int,erpreted on the basis of a Fourier transform approach rather than by a direct’ applicat,ion of the convolution integral. Also, an understanding of this approach 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 hardware but easily implementable with software. Data Handling Based on Fouiier Tiansformations. Certain types of data-handling operations can be carried out on spectra. by utilizing Fourier transformations. In this section a posterior convolution of observed spectra will be mentioned briefly to indicate one approach. I n the last, section it was seen that the line shape in the observed spectrum v;as determined by convolution of the real spectrum by the resolution function. This convolution can be performed 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 smnll peak occurs close to a large peak, the side lobes easily being mistaken for real peaks. I n this case the observed spectrum could be convolved on a computer with a mathematical resolution 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 spectrum is simply mult,iplied by the appropriate truncat,ion function. This type of data handling is used extensively in Fourier transform spectroscopy and is called apodizatiori. This ,simple example illustrates a type of dats-handling operation that is possible with spect,ra when utilizing Fourier transforms a.nd 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 . Chem. Phys., 43, 1307 (1965). (2) G. Horlick, A p p l . Spectros., 22, 617
(1968).
(3) R. R. Ernst, “Advances in Magnetic Resonance, Vol. 2,” J. Waugh,
s.
Ed., Academic Press, New York, N.Y.,
1966, p 1. Ron Bracewell, “The Fourier Transform and Its Applications,” McGrawHill, Sew York, X.Y., 1965. ( 5 ) G d E Subcommittee 0.11 Measurement Concepts, “What is the Fast Fourier Transform?”, I E E E Trans. (4)
Audio Electroacoustics, AU-I5 (2), 45
(1967).
(6) I,. Rlertz, A p p l . Opt., 10, 386 (1971). ( 7 ) .JEEE Trans. Audio Electroacoust ~ s AU-17 , (23, 65-186 (1969). (8) W. Kaiizrnnnn, “Quantum Chemistry,” Academic Press. New York, N.Y., 1957, p 556. (9) R. A. Sxwvycr, “Experimental Spec-
troscopy,” Dover Publications, New York, N.Y., 1963, p 33. (IO) A . Savitsky and M. ,J. E. Golay. ASAL. CHF:M.,36, 1627 (1964).
(11) W. E. Wentworth, J . C h e m Educ., 43, 268 (1966).
