chemiccrl in/trumentation
edited by FRANK A. SETTLE. JR. Virginia Military institute Lexington, VA 24450
'Fourier Transforms for Chemists Part I.
Introduction to the Fourier Transform
L. Glassw University of the Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa Mathematics has a life of its own-it does not require a tangible application in order to justify its existence. Such justification may, indeed, come in due course, as witness GaLois's early invention of Group Theory, which, taday, provides an underpinning for much of our molecular understanding. The physical scientist, however, can best take pleasure in mathematics when it is able to mimic natural processes successfully, or when it can replace natural procedures that are for some reason ohvsicdv inaccessible. The pleasure 1s all the greater when it rs powble to Inspert the meehamsm by whrch the mimicry occurs, thus promoting appreciation of its inner workings and limitations. TheFourier Transform (FT) is just such a procedure, for it mimics the p h & d processes by which signals and their spectra are interconverted. Although FT is most useful
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Leak Glawer received his first degree (in Chemical Engineering) from the Universityof CapeTown in 1956,and the PhDand DIC from imperial College. London, in Chemical Enginewing and in Chemical Physics, respectively (1960).He has been lecturer and Professor in Physical Chemis try at both Rhodes University and the University of the Witwatersrand, in South Africa, and Visaing Professw at Cornell University. Ithaca. NY (in 1986). During his academic career, hs b S been a vislting researcher at Princeton Universily (1968): Max-Pianck Institute for Biophysical Chemisby. Gbningen. West GBrmany (1972): University College of Wales. Aberyswh 11975k .. and C m l l Universrtv ll986171 . rl s principal pdbiicauons have been In me field 01 electrical propenies and hydrogen bondlng and, laterally in camp&tional chemisby. He has a strong interest in chemical education and CAI, particularly in regard to disadvantaged studems. He is presently Frolessor of Physical Chemislry at the University of the Witwatersrand. Johennesburg, and was Head of Department from 1982 to 1985.
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The substance of this DaDer was wesemed at a "mini-course" of the SA Chemical Institute. So* ern Transvaal section, in October 1985.
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Journal of Chemical Education
in promoting understanding of these processes and thus can, far example, be used in desienine better lenses-whether o ~ t i c aor l electromagnetic it really cwms into its own when it can rubstitute for missing phys. ical procedures. A prime example uf this is in X-ray crystallography where, by contrast to the optical case, there is no lens available to collect the scattered diffrartion from a crystal and recombine it to form an enlarged image of the seatterer. The Fourier transform can he used to construct (with some limitations) the image mathematically. Unfortunately, the process of Fourier transformation is tedious and it tended to be used in the past only when no substitute could be found. But the general availability of com~utersand the invention of the Fast ~ o u r i e ; Transform algorithm (which reduces comoutational lahor sienifieantlvl havemadeihe FTeasily acceasl6eand h&e resulted m the development of spectrometers that use the FT toconriderable advantage in sensitivity, speed, and resolution. At the same time, the ability of the F T ta mimic processes in the physical world can now be turned to advantage in another way, by resolving the component parts of the physical process from one another. So, one can enhance a spectral resolution that has been degraded by too slow a detector response or too wide an entrance slit, by actually extracting the true spectrum from the unwanted complexity of the p h y s i d process. In contrast to earlier times, it now often pays one to work in the FT domain even at the (formerly onerous) expense of having first to transform into the F T domain, and then retransform out of it. This paper will explore how the FT mimics spectral transformation, how this property can be exploited to advantage in spectroscopy, and how the F T can he used in data treatment. The table displays a numher of important FT seriallspeetral pairs, related by Fourier transformation, to illustrate the ubiquitous functionality of the Fourier operation.
sure amplitude that varies with time at a given point in space), or in space (a densitometer measures the s ~ a t i a variation l of optical density of a plate, in one or in two spatial dimensions). The alternate way of describing a physical process is to repreaent it as a spectrum, where the spectral amplitude is a function of a (time-inverse) frequency or a (space-inverse) spatial frequency (there exists no independent term for the latter, hence we introduce the term "undulancy"). In some situations, the serial descriptions are the more familiar, in others the spectral. We sense a lighthouse lamp as flashing on and off periodically-aserial behavior-hut the beams of light that it emits are not ohsewed, in terms of their wave nature, as fading and rising periodically in intensity; rather, we sense the color of the light, whieh is-as we know from the rainhow-a spectral phenomenon. In the spatial domain, we recognize a picture by the spatial arrangement of its picture elements ("pixels") but describe a piece of cloth as having a specified number of threads per unit lengthclearly, a spatial frequency or undulancy description. Baron Jean Baptiste Joseph Fourier (1768-1830),a French engineer, teacher, diplomat, Egyptologist and administrator, first suggested that the mathematical functions that are used to describe physically feasible phenomena may be constructed by adding together simple periodic (sine, or equivalently, cosine) waves, whether the function whieh is to be constructed is itself periodic or not. That, at first blush, is quite an extraordinary claim. But, consider a wind instrument, whieh, when blown, produces a sound intensity that rapidly builds up and more slowly dies away when the blowing stops; the ear has little difficulty in distinguishing the different pitch (spectral) comnonents of the sound. each a oeriodic sign&, together producing the sound transient of the blown instrument.
Serlal-Spectral Domains
Fourier theory is ~iimplestin the c a w of a regular o r periodic signal, where the period.
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In essence, the mathematics of F T recognizes that there are two equivalent ways of describing many types of physical processes. One way treats the process as serial, one event following another, either in time (far example, a sound wave has a certain pres-
Perlodlc Signals
(Continued on page -4230)
Some Natural Fourier Transform Pals and Their Relatlonshipa
ic function is constructed by summing together a series of sine waves: a fundamental having the same period as does the signal, plusan infinite series of higher harmonics of the fundamental, of appropriate amplitudes and phases. There may also be a constant component to correct the average level of the Fourier series to that of the signal: f(x) = a,
+
z
la,, eos 2nns+
"-1
+ b,, sin 2 n n s g J
Field (1) Time Pulse Specborn% by (0.g.. - NMR) Irreversible proca~sas(e.g., diflusion, conduction)
where so = fundamental frequency, n = order of harmonic ( n = *I, *2,. . .),and x represents the independent variable in the serial domain. It follows that
Transform Process and direction
Pulse techniques
Spectral Domain
Mathematical
Spectrum (8.g.. NMR)
~
Pulse techniques (i.e.. injection of Pulse) Time evolution of system
Information processing (e.g.. wmmunications)
Signal with noise Filter4 signal
Geophysi~~ (e.g.. metemlogy)
Time series
(2) spam Spectral Analysis
+ b, sin 2Ilns+)
Serial Domain Feahlre
Displacement Interterometry (e.g., FT IR)
Spectrum
W i n g or Prism
lmmterencaof radiation,
Spatial wquence (8.g.. mirrw displacement in Mi&elson interferometer)
interference1Math
fa cuss in^
SpecbYm
Spechum t
Diffraction pattern
small obien
Image
The coefficients a,, b., e,, d, represent the unknown amplitudes of the nth harmonic, and 8, its unknown phase. The determination of these quantities represents the central problem of harmonic analysis. Fourier discovered that these coefficients could be evaluated from a knowledge of the function f(x) and of the period, xa, of the function, where xo = l/so: f(x)dx
x-ray Crystal Structure Analysis
Crystalline material C ~ s t a~truchlre I
X-ray interference Math
Information processing (8.g.. picture processing)
Signal with noise (e.g.. out-~f-focus oichnel Enhanced image
Math
Optical Diflractian
Photographic image
-.
7
Enhanced image
C
* Math 4
Object Rewnslr~~ted image
(mean value of f(x))
*
Optical diffractometar Light scattering. interierence Light scattering. interference
(3)Spam/Tim * Molecular D y m ics
x rns 2IIns+dx = d.
+ d-,
[n > 11
Correlation function (radiation scarierino from fluid. or mam. model)
-
= c, sin 8.
Diffraction pattern
Math
Ma*
\
',
_
*
-
7
Spenrum
Spatial specbum
Hologram
Time-dependent radial distribution function.. trans~on . ~Mttticients (e.g.. diffusion coenC cient, dielectric relaxation)
.Note: the WEBand Urns pa* may also be handled separately, m yield only spaoe paramstera 1e.g.. radial diaalbvtlml hmnion) or time parameters (e.9.. dieienic relaxation parameters).
x sin 2Ilns+dx = i(d,
- d_J
[n 2 11
= c. cos 8,
I
and
% i
j
x exp (-i2Ilns+)dx
[Id = 0.1,2. ..I Thus, we have a mathematieal relation between the two domains, serial and spectral. This relation is shown rather vividlv, in Figure 1. as a three-dimensional re~re&ati& of amplitude against either time or frequency, for a periodic signal.
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Journal of Chemical Education
Figure 1. DecomposRion of a wavefarm in born time and hequency. From a leaflet f w the Micro FFT Analyser. 50340. Adapted by permission of scientific Atlanta. Spectral Dynamics Division.
As we see from these mathematieal expressions, the spectral coefficients consist of integrals of cosine (or sine) functions of x , weighted hy f(z); in effect, f(x) determines how much of that particular frequency component, nso, is to be added into the Fourier sum, just as the original signal can be regarded as constructed by the summation of its spectral components.
Aperiodic Signals If the function to be dealt with is aperiodic, then the equations above must be generalized. We may visualize the generation of an aperiodic function by considering a periodic function, and allowing its period to lengthen indefinitely, when the fundamen-
tal frequency w i l l decreane correspondingly toward z e n w n e may i m w e a wiRlitar that grows ever longer, with an accompanying decrease in the fundamental frequency of the vibrations of its strings. The inverse relationship between wavelength and frequency follows from the defmition,
where so is a frequency or undulancy, and has dimensions of cycles per serial unit (e.g., cycles per second) and xo is a period that has dimensions of serial units per cycle (e.g., seconds per cycle). The decrease in the fundamental frequency also results in a decrease in the frequency difference between adjacent harmonies:
In the limit, as the period becomes infinitely long, the discontinuous frequency variable (nso)will become a continuous quantity (s), with infinitesimal difference between adiacent harmonics. The infinite sum of the series synthesis of f(r) will then heeome an
The pair of equationscontsiningF(s) isusually rpgarded as constituting the Fourier
transform pair, which transform between the serial function, f(r), and the spectral function, F(s). As in the periodic case, the determination of the various coefficients is the central problem of Fourier analysis. Then A(s) = F(s)
+ F(-s)
= D(s) sin [$(s)) -Ahsorption Spectrum B(s) = iF(s) - F(-s)] = D(8) co8 [8(8)] -Dispersion Spectrum D2(s) A2(s) + B2(s) = F(s) P(-S) -Power Spectrum
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where * denotes a complex conjugate (i.e., i is replaced by -9. DZ(s)corresponds to the intensity of the signal, as recorded by most physical detectors, and it is noticeable that B(s) is missing from the expression for Dz(s). Thus, such detectors lose the phase information (8), which cannot be regained directly. This constitutes the "phase problem" of X-ray crystallography, which prevents the complete automation of "yatal structu~e determination.
Explalnlng the Domaln Transtormatlon The mathematical form of the FT does not demonstrate directly its capability of domain transformation, and it seems worthwhile to attempt ademonstration of how the complex exponential "pi& out" the components of f(x) at each frequency, s, to yield the soectrum. Fh). The trandurm consists of three part.-a complex exponent~al,which is the Fourier kernel, exp (-i2Esx); the weighting factor (f(x) for the forward transform, F(s) for the inverse transform); and the integration, which corresponds to summation of the separate real and imaginary components, but with a continuous variable. The complex factor describes a unit vec-
Figure 2. Real and imaginary components of a phasor.
tor that rotates (a phasor, Fig. 2) with the given cyclic frequency,~,m afwction of the serial parameter, x (e.g., time). Thus, the product, f(x) eap (-i2nxs), may be pictured as a helicoid with instantaneous radius f(z) and pitch (or period) xo = 11s (Fig. 3). The sign of the phasor exponent determines the senae of rotation, clockwise or cowtercloekwise (Fie. 4). For the sake of simplicity, we depict the helicoid for f(x) =constant (Fig. 5).Without loss of generality, we may take the constant to be unity since, if f(x) = A, then
(Continued on page A232)
Figure 3. (left) " R i n d by M. C. Escher--a phasor (of varying radius) extended into an helicoid in me serial domain. Copyright M. C. Escher Heirs, c/o Cordon Art. Baarn. Holland. Figure 4. (above) "8ond of Union" by M. C. Escher-posnive and negative frequency phasns(of varying radii)extended intohelicoids in the serlal domain.CopyrigM M. C. Escher Heirs, C/OM a n An. W n , Holland.
Volume 64
Number 10 October 1987
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Figure 5. (above) me helicoid ol constant radius: exp (-8). The helicoid is a function of a complex varia h l ~hut may b c resolved into iui real and imnginarycomponentslFig. 6 ) by pr
