Fourier Transforms for Chemists

topic/ in edited by

FRANK A. SETTLE. JR. chemica 1 in/t rum en tat ion virNia Lexington. Military vA Institute 24450

Fourier Transforms for Chemists Part Ill. Fourier Transforms in Data Treatment

L. Glasser University of Witwatersrand, 1 Jan Smuts Avenue, Johannesburg 2001, South Africa

Properties of the FT The hehsvror of a spectral funcrion is de-

termined by the hehavror of the funcrion in the serial domain through a many-to-one mapping, and vice versa. One may thus picture' a function as "accompanied by a coun- terpart in another domain, a kind of shadow which is associated uniquely with the fune- tion through the FT, and which changes as the function changes." This is very helpful when considering relations between and changes to functions.

Table 1 lists some important properties of FT pairs that will be helpful in attempting to use the FT. I t will be noted, as we pro- ceed, that these properties of the mathemat- ical formulation have identical counterparts in the physical behavior of F T systems.

Addillvlty An elementary example is that the F T is

additive-just as a mixture of absorbing components gives an additive spectrum, as described by Beer's Law. This follows di- rectly from the definition of the FT and the distributive nature of integration:

If

and

that is,

f(x) + g(x) - F(s) + G(s)

The substance of this paper was presented at a "mini-c0urre"of the SA Chemical Institute, South- ern Transvaal section. in October 1985.

* Bracewell. R. N. The Fourier T r a n s f m and its AppIic~tims, 2nd ed.; McGraw-Hill: New York, 1986.

Table 1. Properties of Fourler Transforms

f(x) FW

Scaling

Shift Modulation Convolution Autocowelation

Shlfllng A shift of origin, for example, a shift of the

zero of time by an amount, xo, does not alter the fundion f(r) but does shift the time reference plane and, hence, the phase for each component wave of the function. Since the different comoonents have different fre- quencies, the phase shift for each will he differen!, in pmportmn to i t s frequrucy, a: vir., the phase shift will be sxo cycles Thus, the phasor becomes

and has shifted in phase by 211szo. Since the magnitude of the phasor is unity, no change in the scale of the transform occurs.

Shifting in frequency causes a corre- sponding phase shift in the time domain. The process of frequency shift is used in radio; when an audio-frequency signal is used to modulate the amplitude (AM) of a high-frequency carrier wave, the modulat- ing signal is shifted in frequency and can be transmitted.

Audio signal: g(r) * G(s) Carrier wave: eos 2 n s ~

Modulated: f(x) = g(x). eos 2 n s g - - g(x)(e'2"" + e-'znw 112

Fourier transform: F(s) = f(x)e-'2n"dr -

Here, two sidebands have been introduced (in addition to the carrier frequency), and the signal of frequency s has been shifted into the region of the carrier frequency, so. Such a procedure is used in FT-NMR, where the pulse applied is actually a gated (or switched) cosine that yields broad-band radiation centered around the frequency of the exciting cosine.

(Continued on page A308)

Journal of Chemical Education

Symmetry The FT and its inverse are imperfectly

symmetrical (transforms are +i and -i), so that it is important that the sequence of transformation is maintained; however, i t is unimpo~tant which is regarded as the forward and which the inverse transform, as long as they are used consistently.

If the relation: f(x) - F(s) holds, then we can find the inverse transform by a trivial Process, for: F(x) - f(-s), that is, if the spectral function is regarded as being in the serial domain, then its transform is just the reflection in the origin of the original serial functibn. For even functions, this means that the forward and inverse pairs are invariant.

Proot:

Substitute -I for x :

and interchanges and r:

Following a line of reasoning of this kind, it is possible to generate the relations con- tained in Table 2 for even and odd functions, complex, real or imaginary. It should be noted that a general function is neither even nor odd, hut can always he split unam- biguously into even and odd parts.

If f(x) = E(x) + O(r) then E(x) = [f(x) + f(-x)]/2 and O(x) = [ f (4 - f(-x)1/2

As an example, consider a cosinusoid in the serialdomain; it is even, and its transform is the even impulse r air, I k (s). I t follows that . . the transform of a pair ojpulses, symmetri- cally placed about x = 0 in the serial domain, is just a cosine in the spectral domain. On the other hand, a serial domain real sinusoid has the odd, imaginary transform i I,&). Hence, the transform of the real, odd impulse pair I(x) is a sinusoid with imaginary values, for:

Scaling (Similarity Theorem) If - F(s)

f ( d , . 8

Proof:

Table 2. Symmetry Properties of Fourier Transform Pairsa

Functhn FT

fld F(4

Real and even Real and even Real and odd Imaginary and odd Imaginary and even Imaginary and even Comoiex and even Camolex and even Complex and odd Complex and odd Real and Complex and

asymmenical hermitian Imaginary and Complex and

a~ymmetrlcal antihermitian Real even plus Real

imaginary odd Real add plus lrnaglnary

imaginary even Even Even W d Odd

A M Bracewell. R. N. me FWIBT Transfnm and I& APPIIUI~IME; 2nd ed.; Mdjraw-Hill: New York. 1986.

This theorem provides an important and useful relation. If the period of a grating decreases, then the spread of its spectral orders increases; however, the total energy in the spectrum remains constant and so the intensity of each order decreases to compen- sate. The theorem also provides a basis for the use of the reciprocal lattice in X-ray crystallography; the reciprocal lattice is a fundamental construct used in the analysis of X-ray diffraction patterns.

Some FT Palrs Figure 1 provides some pictures of FT

pairs that are useful in interpreting F T op- erations. These pictures are often the pre- ferred way of tackling an F T problem in a qualitative fashion.

Derivative Theorem

If f(x) - F(s) then f'(z) - i2nS(s)

= lim e'2nhT(s) - F(s)

Ax

A308 Journal of Chemical Education

Imag inary M a g n i t u d e Power

F.T. k x +

Rea l

F.T. I *

Figure 1. Fourier transform pairs:

Band-pass "White" spectrum

(even impulse pair)

(add impulse pair)

(square wave) (triangular wave) (Gaussian) (Larentria")

(* denotes averaged)

(a i l From Bracewell, R. N. TheFowi~TransformsndltsAppl;~al&~, 2nd ed.; MdXaw-Hill: New York, 1986. Adspled wlththe permission ofthe publisher and theauthm. (k-h) From Mar- shall. A. G.. ~ d . ~wrier . mdamard, andti;fb& rransforms m Chemis1w Plenum: ~ e w ~ o r k . 1982. ~dapted with the permission of the publisher and me editor.

Volume 64 Number 12 December 1987 A309

where we use the series expansion of the exponential. This result means that the transform (spectrum) of the derivative is simply the original spectrum multiplied by iZns, which attenuates the lower frequen- cies (suppressing any zero-frequency eam- pouent) and enhances the higher frequency components. As is common experience, dif- ferentiation of a signal decreases the signal- to-noise ratio, where the noise appears in the higher frequency regions. I t is, there- fore, common practice to suppress (filter out) high-frequency signals before differen- tiation.

Convolution Convolution of two functions is a signifi-

cant concept in many physical situations; i t describes the process which causes "blur- ring" or spreading of a signal, or is repre- sented by a running mean, and leads to re- duction in the resolution of a spectrum. Convolutiou itself does not involve FT, but it is its behavior under FT that renders it important here, together with its close representation of real physical processes.

In order to describe and justify the process of convolution, consider the followine. A system can oftenbe described. in electry- calkrms, as a filter that modifiks an input signal, Vl(t), so generating an output signal,

( Filter I We will consider only linear filters, that is, ones in which superposed input signal yield superposed output signals, which are also time-invariant, that is, the filter eharacter- istics are stable in time. If the input signal, Vl(t), is an impulse (a delta function), then the filter output can be considered to be a

characteristic of the filter itself, viz., its impulse response, I@). I(t) often has the form of an exponential (or sum of exponentials) decay (Fig. 2a).

If Vl(t) is a sequence of impulses, we can picture V2(t) as a sequence of impulse responses (Fig. 2h). If the responses overlap, then the overall response will be the sum of the individual responses (by the linearity assumption) (Fig. 24. Finally, a contiuuaus function Vl(t) can he considered a series of closely spaced impulses, with V&) consist- ing of the sum of the various, appropriately delayed, impulse response functions. The process of summing each term with appro- priate delay is very easily simulated by re- versing I(t) andoverlapping it an V,(t) (Fig. 2d), then adding the product of each inter-

section of the functions. The sum represents the superposition of signals, or the convolution of the two functions (Fig. 3).

The convolution integral is

= f(x) * g ( 4 = g(x) * f(x)

(commutation)

that is, to yield the value at a particular point, x, of the integral, reverse g(x) and center it a t x on f(x), then sum the products of the superposed functions. The net result, h(x) or V2(t), is a running weighted mean or smoothing of f(x) hy the function g ( r ) .

Figure 2. A waveform as constituted fromaseries of impulse functions: (@the impulse function. I(0: (b) ase- ries of widely spaced impulse functions; (c) a sequence of impulse functions summed to yleld the output function, V(q: (d) convolution of i(0 on a set of impulses ( i to iv), yielding the output value ( i + ii + iii + iv).

i i i

Figure 3. Convolution: (a) of two exponentials; (b) graphical construction for (a); (c) a spectral pattern (ii). i v broadened (iv) by the slit function (iii) convolved with it. (a. b) From Bracewell, R. N. TheFourier Trmsfonn

andlts Applications, 2nd ed.; McGraw-Hill: New York. 1986. Adapted with the permission of the publisher and the author. (c) From Marshall, A. G., Ed. Fourier, Hadamard, and Hilbert Transforms in Chemistry: Plenum: New York. 1982. Adapted with the permission of the publisher and the editor.


impulse transfer response

I(t) - T(dfunction Basically, correlation is equivalent to con-

volution hut without the reversal.

that is, the FTof a convdution msimplv the product of the FT's of the convohing func-

z g(x)*f(x)

If f(x) = gb) , then this cross-correlation becomes an autocorrelation.

If f(x) - F(s) then P(x)*f(x) -(F(s)lZ

iions.

ProoE

Consider a double-beam spectrophotome- ter. The slit and monochromator represent the filter; the reference beam gives the transfer function, and the sample beam gives the output signal. So, the ratio of the two is the true output signal, deconvolved from the nonuniform spectral intensity of the input signal and various instrumental effects. This is an example of a physical process of deconvolution.

Proof:

(separating variables)

(invoking the shift theorem)

= F(s)G(s)

This expression is simply the most direct version of the convolution theorem, which can be expressed in many different ways.

The usefulness of this result is immedi- ately apparent from the fact that we can use it to decode the input signal (or its spectrum) from a known output signal with the aid of the impulse response of the filter.

Vdt) - s1(4 * X

Correlation (Auto-Correlation and Cross Correlation)

Correlation is another imnortant Droce-

This is the auto-correlation. or Wiener- ~ ~~~ ~- ~~~~ ~ ~ ~ ~ ~

Khinchint theorem (Fig. 4,. itsphysicalval. Ire Ires in the intrrpretatim of hsl >,which

dure that, in thc present context, is important lrecause of its hehavior under YI' and because of its impmtant physical interprr-

is the spectral power or intensity of a signal.

(Continued on page A312) tation.

J I \ : I I I

-1" I . (0 I-L-17 d

Figure 6. Ccnvolutlon and aversampling: the specbal orders are well separat- ed. From Brigham, E. Orsn. The Fast Fourier Transform: Prentice-Hall: Engle- w ~ d Cliffs. NJ. 1974: p 84. Adapted by permission of the publisher.

Figure 5. Convolution in the serial domain, and its conespandence to multipil- cation in the specnai domain: (a) sampling function; (b) slit function: (c) sampling spectrum: (d) slit spectrum: (e) grating function: (I) spctrai envelope. From Brigham, E. Oran. The Fast Fourier Transfwm; Prentice-Hail: Engiewwd Cliffs. NJ. 1974: p 62. Adapted by permission of the publisher.


Figure 4. Autocorrelation: (a) II(a*IIM - sine s (b) A(x)*A(x)- stncz s. From Bracewell, R. N. The Fourier Transform and its Applications, 2nd ed.; McCaaw-Hill: New Yoh. 1986. Adapted with the permission of the publisher and author.

Most spectrophotametric detectors (e.g., the eye) measure intensity and, integrating over time. ienore nhase information. This . " process corresponds to an auto-correlation of the input signal. A sine and a cosine have the same auto-correlation, with a maximum of the function a t theorigin, where thefunc- tion coincides with itself, and reducing posi- t iw values in positive and negative direc- t i o n ~ The phnzs infornmation iz <men nut reauired but. when i t ia. elahwatr efforts m& he needed to regenerate it.

Correlation is an important procedure in the understanding of diffusional processes,

t "('I

where motion of one particle is imparted to another by impact, and the correlation of motion describes the rate of decay of asignal as it traverses the system

Applications The convolution theorem provides us

with an opportunity to give an almost com- plete explanation of the process of genera- tion of a spectrum (Fig. 5) by a grating. The slit function, x ( t ) , of the grating is repeated by convolution with the "sampling" function, h(t), to generate the grating function, h(t) * x(t). In the transform domain, the slit spectrum, X(0, is multiplied with the sampling spectrum, H(0 , to yield the spectral envelope H(f)X(f). The envelope squared gives the intensity of the various orders (and is clearly generated by the shape of the slit function) while the positions of the various spectral orders are generated by the period- icity of the sampling function.

Figure 6 demonstrates that the sampling process in the signal domain generates repeated copies of the spectrum. If the sampling frequency is reduced through the Ny- quist frequency, the spectra approach (Fig. 7) and overlap (Fig. 8) each other, to yield an aliased spectrum.

Data Treatment

Filtering Perhaps the simplest application of the

F T to data is to generate an objective filter

Fiqure 7. Convolblon and samplmg at lhe hyqulsl trequency; the band-lmited spectra are just in contact. From Bugham. E Oran The Fasr Fowrer Trans form Prentiea-hall' Eng ewood C Its. NJ. 1974: p 81. Aaapled Dy Permlrslan of the publisher.


in thespertraldomnin.Thus,onr con gener- arcrhezpertrum ofasignnl, hy C'I',anil then cut out any f r ~ q u e n q region simply hy s ~ t - ting such components to zero. An inverse transform will generate the filtered signal. In this way, i t is possible to remove high- frequency noise, and so enhance the signal- to-noise ratio of the signal, or a base-line drift can he corrected by deleting low-fre- queney components. Such procedures are also performed in the signal domain direct- ly, convolving the signal with some sort of weighted running mean function. The most familiar example of this is due to Savitzky and Golay, who published a series of nu- merical coefficients far least-squares poly- nomial smoothing, for smoothed deriva- tives, and for integrals. While this is a very successful procedure, i t is useful to look at the spectra of the Savitzky-Golay coefficients (Fig. 9) to realize that the operation is not simple hut involves a complex grouping of spectral pass- and stop-bands.

It should be noted that i t is unwise simply to truncate a spectrum in afinite amplitude region in order to filter a signal; truncation corresponds (in the serial domain) toconvo- lution with a sinc function, which generates oscillations ("leakage" or side-lobes) in the serial function. It is more correct to cut the signal spectrum off more gently, in a procedure called "apodization" (''removal of the feet") (Fig. 10). Various apodizing (window) functions are available, of which one of the simpler and more useful is the "shifted cosine" or Hanning function,

Fig~re 8. Convo ution ana unaersampling: the spectra over sp andarea iased. From Brlgnam. E Oran The Fasf Fourrer Transform, Plent ce -rial. Englswwa Cliffs. KI. 1974; p 82. Adapted oy permss on of me pub rher

where05 t 5 T,

This (or its spectral equivalent) has a b c d smooth roll-off in itsspectrum (Fig. 11), and much lower side-lobes than has the sinc2 I function which results from a simple trunea- tion of the signal.

Acknowledgment The research upon which this publication

is based is supported by a grant from the Foundation for Research Development of the South African Council for Scientific and Industrial Research.

The ready agreement of many publishers and authors to allow republication of their material is gratefully acknowledged; charges have been levied by Prentiee-Hall and by Cordon Art (who have reduced their fee for publication in an educational journal).

a 4~ a * sponn'ng Figure a/ 9. frequency Seven- and responses twenty-ane-polnt (columns Savitzky-Golay b and dl: Cquadratidcobic filter ibL functions (columns least+quares a and smoofh: c) wim 11-quart*/ their

quintic least-squares smooth. From Betty. K. R.; Horlick. G. Anal. Chem. 1977, 49. 351. Reprinted with Permission from Analytical Chemistv. Copyright 1977 American Chemical society

Figure 10. Apodiratian or windowing: (a) noisy signal and is spectrum; (b) window function for the signal, to enhance signal-lo-noise ratio in the spec- hum at the expense of resolution: (c) window function, to enhance resolution at the expense of signal-to-noise ratio. From Marshall. A. G.. Ed. Fourier, Hadamard, and Hilbert Transforms in Chemishy; Plenum: New York, 1982. Adapted with the permission of the publbher end edilor.

- F gLre 11. Tne banning w ndaw tunct on (or 'ratsea cosine 'I, ana 01s Four er transform From Brigham. E. Oran. The FastForrrier Transform. Prent cs-dal : EnglewoodCifk. h.. 1974. p 144 Adaptedoy wm#Swmof t m p.01 She,


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Fourier Transforms for Chemists