Series Werner Herres and Joern Gronholz
Understanding FT·IR Data Processing Part 1: Data Acquisition and Fourier Transformation 1 Introduction Although Infrared spectroscopy Is one of the most powerful tools available to the analytical chemist and Is routinely used In research and application labs and for process control, the most advanced form of JRspectroscopy, Fourier Transform infrared Spectroscopy (FTIR), still holds some secrets for the chemist who is trained lo work with conventional grating instruments. One reason is surely that the generation of the spectral trace Is not straightforwardly controlled by setting appropriate knobs controlllng slit widths,
This Is the first of a series of three articles, describing the data acquisition and mathematics performed by the minicomputer Inside an FTIR spectrometer. Special emphasis Is placed on operations and artifacts relating to the Fourier transformation and on methods dealing directly with the lnterferogram. Part 1 covers the measurement process and the conversion of the raw data (the lnterferogram) Into a spectrum.
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scanning speed, etc. but involves a certain amount of mathematical manipulations such as Fourier transformation, phase
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correction, and apodization, which may introduce a barrier to understanding the FTIR technique. Despite this difficulty, moderately and low priced FTIR Instruments are now entering even routine labs, be
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cause of their clear advantages compared to grating spectrometers. Even in lowerpriced FTIR spectrometers, a laboratoryor dedicated computer Is the most Important component apart from the optics. As the quality of its software directly determines the accuracy of the spectra, It is recommended that the user be familiar with the principles of FT·IR data collection and manipulation. Unfortunately, there still seems lo be a lack of literature on FTIR at an Introductory level. Therefore, ·this series of articles allempts to compile the essential facts in a, hopefully, lucid way without too many mathematical and technical details and thus provide an insight into the Interrelation between FTIR hardware, the data manipulations involved, and the final spectrum.
Dr. Werner Herres and Dr. Joern Gronholz Bruker Analytlsche Messtechnlk GmbH, Wikfngerstr.
13, 7500 Karlsruhe21, WestGermany.
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Figure 1: A) Schematics of a Michelson Interferometer. S: source. D: detector. M1: fixed mirror. M2: movable mirror. X: mirror displacement. B) Signal measured by detector D. This ls the lnterferogram. C) Interference pattern of a laser source. Its zero crossings define the positions where the lnterferogram Is sampled (dashed lines).
_ ,Series of >J2. The complete dependence of l(x)on xis g)ven by a cosine function:
2 Raw Data Generation The essential piece of optical hardware in a FTIR spectrometer is the interferometer. The basic scheme of an idealized Michelson interferometer is shown in Figure 1. Infrared light emitted by a source (Globar, metal wire, Nernst bar ... ) is directed to a device called the beam splitter, because it ideally allows half of the light to pass through while it reflects the other half. The reflected part of the beam travels to the fixed mirror M1 through a distance L, is reflected there and hits the beam splitter again after a total path length of 2 L. The same happens to the transmitted part of the beam. However, as the reflecting mirror M2 for this Interferometer arm is not fixed at the same position L but can be moved very precisely back and forth around L by a distance x, the total path length of this beam is accordingly 2 • (L + x). Thus when the two halves of the beam recombine again on the beam splitter they exhibit a path length difference or optical retardation of 2 * x, I.e. the partial beams are spatially coherent and will interfere when they recombine.
The beam leaving the interferometer is passed through the sample compartment and Is finally focused on the detector D. The quantity actually measured by the detector Is thus the Intensity I (x) of the combined IR beams.as a function of the moving mirror displacement x, the socalled interferogram (Figure 1B). The interference pattern as seen by the detector is shown in Figure 2A for the case of a single, sharp spectral line. The interferometer produces and recombines two wave trains with a relative phase difference, depending on the mirror displacement. These partial waves interfere constructively, yielding maximum detector signal, If their optical retardation is an exact multiple of the wavelength,\, I.e. if 2•x= n• ,\(n = 0, 1,2, ... ).
(1)
Minimum detector signal and destructive Interference occur if 2 *xis an odd multiple
l(x) = S(v) • cos(2" · v ·x)
(2)
where we have introduced the wavenumber v = 1/A, which is more common In FTIR spectroscopy, and S (v) is the Intensity of the monochromatic line located at wavenumber JI. Equation (2) is extremely useful for practical measurements, because it allows very precise tracking of the movable mirror. In fact, all modern FTIR spectrometers use the Interference pattern of the monochromatic light of a HeNe laser to control the change in optical path difference. This is the reason why we included the Interference pattern of the HeNe laser in Figure 1C. This demonstrates how the IR interferogram is digitized precisely at the zero crossings of the laser interferogram. The accuracy of the sample spacing A. x between two zero crossings is solely determined by the precision of the laser wavelength Itself. As the sample spacing A. J1 in the spectrum is Inversely proportional to A. x, the error in A. v is of the same order as in A. x. Thus, FTJR spectrometers have a builtin. wavenumber calibration of high precision (practically about O.o1 cm·'}. This advantage is known as the Cannes advantage.
the mirror can be moved very fast, complete spectra can be measured In fractions of a second. This is essential, e.g. in the coupling of FTIR to capillary GC, where a time resolution of 1020 spectra per second at a resolution of 8 cm·1 is often necessary [1]. Finally, the Feliget and Jacquinot advantages permit construction of interferometers having much higher resolving power than dispersive instru'ments. Further advantages can be found In the IR literature, e.g. In the book by Bell [2].
4 Fourier Transformation Data acquisition yields the digitized interferogram l(x), which must be converted into a spectrum by means of a mathematical operation called Fourier transformation (FT). Generally, the FT determines the frequency components making up a continuous waveform. However, If the waveform (the interferogram} is sampled and consists of N discrete, equidistant points, one has to use the discrete version of the FT, I.e. discrete FT (OFT}: N1
S(k·Av) = "f:,l(nAx)exp(i27rnk/N) (3)
n=O 3 Advantages of FTIR Besides its high wavenumber accuracy, FTIA has other features which make it superior to conventional IR. The socalled Jacquinot or throughput advantage arises from the fact that the circu· lar apertures used in FTIR spectrometers have a larger area than the linear slits used In grating spectrometers, thus enabling higher throughput of radiation.
In conventional spectrometers the spectrum S (v) is measured directly by recording the Intensity at different monochromator settings JI, one v after the other. In FTIR, all frequencies emanating from the IR source Impinge stmuitaneously on the detector. This accounts for the socalled multiplex or Fellget advantage. The measuring lime in FTIR is the time needed to move mirror M2 over a distance proportional to the desired resolution. As
where the continuous variables x, J1 have been replaced byn · Axandk· Av, respectively. The spacing A. JI In the spectrum is related to Ax by
Av= 1/(N·AJ!)
The DFT expresses a given function as a sum of sine and cosine functions. The resulting new function S (k · A v) then consist of the coefficients (called the Fourier coefficients) necessary for such a developmen,Ailernatively, If the set S (k ·A v} of FourJ~t coefficients is known, one can easlly"%reconstruct the interferogram I {n · A.xi>hy combining all cosines and sines multiplle(j\ by their Fourier coefficients s (k. A v) 'and dividing the whole sum by points N. This is stated by the number the formula forthe inverse OFT (IDFT}:
o.t
/(n·AX) = N1.
\\
\
(1/N) "f:,S(k· A v)exp(.,,.i2,,· nk!N)
n=O 2
(4)
(5)
."Series amplitude. This Illustrates the need tor ADC's of high dynamic range In FTIR measurements. Typically, FTIR spectrometers are equipped with 15 or 16bit ADC's. For n = 0, the exponential In (5) Is equal to unity. For this case, expression (5) states, that the Intensity I (0) measured at the lnterferogram centerburst is equal to the sum over alt N spectral intensities divided by N. This means the height of the center
burst Is a measure of the average spectral Intensity. In practice, eq. (3) Is seldom used directly because It Is highly redundant. Instead a
number of socalled fast Fourier transforms (FTT's) are in use, the most common
4000
2000
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lt
of which is the CooleyTukey algorithm. The aim of these FTT's Is to reduce the number of complex multiplications and sine and cosine calculations appreciably, leading to a substantial saving of computer time. The (small) price paid for the speed Is that the number of lnterferogram points N cannot be chosen at will, but depends on the algorithm. Iii the case of the CooleyTukey algorithm, which Is used by most FTIR manufacturers with slight modifications, N must be a power of two. For this reason and from relation (4) It follows that spectra taken with lasercontrolled FTIR spectrometers will show a sample spacl'll! of Av.= m laserwavenumber/2* *N···
*
x~
Figure 2: Examples of spectra (on the left) and their corresponding lnlerferograms (on the right). A) One monochromatic line. B) Two monochromatic lines. C) Lorentzian line. D) Broadband spectrum of polychromatic source.
5 Final Transmittance Spectrum To obtain a transmittance spectrum, the three steps shown in Figures 3 A, 8, Care
necessary (this example was taken from a GC run):
• an interferogram measured without The summation (5) is best illustrated In the
From Figure 2C we can, e.g., extract the
sample in the optical path Is Fourier
simple case of a spectrum with one or two monochromatic lines, as shown in Figures
general qualitative rule that a finite spectral line width (as Is always present for real samples) is due to damping in the interferogram: The broader the line the stronger the damping.
transformed and yields the socalled single channel reference spectrum R
2A and 28. For a limited number of func
tions like the Lorentzian in Figure 2C 1 the corresponding FT Is known analytically and can be looked up from an Integral
table. However, in the general case of measured data, the DFT and IDFT must be calculated numerically by a computer. Although the precise shape of a spectrum cannot be determined from the lnterfer
ogram without a computer, it may nevertheless be helpful to know two simple trading rules for an approximate description of the correspondence between I (n · Ax) and S(k·Av).
Comparing the widths at half height (WHH) of I (n · Ax) and S (k · A v), reveals another related rule: The WHH's of a 'humplike' function and Its FT are Inversely proportional. This rule explains why In Figure 20 the lnterferogram due to
a broad band source shows a very sharp peak around the zero path difference position x = 0, while the wings of the interferogram, which contain most of the useful
spectral information, have a very low
(v) of Fig. 3A. • an lnterferogram with a sample In the
optical path Is measured and Fourier transformed. This yields the socalled single channel sample spectrum S(v) of Fig. 38. S(v) looks similar to R(v) bul has
less intensity at those wavenumbers where the sample absorbs. • The final transmittance spectrum T(v) is definedastheratioT(v) = S(v)/R(v). This is shown In Fig. 3C.
Once the transmittance spectrum has been obtained 1 further data processing resembles that of digitized spectra from dis
persive Instruments.
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Figure 3: A) Single channel reference spectrum measured through an empty sample compartment. B) Single channel spectrum of absorbing sample. C) Transmittance spectrum equal to Fig. 38 divided by Fig. 3A.
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C)
B)
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Figure 4: Two closely spaced spectral lines at distance d (left) produce repeallve patterns at distance 1/d In the lnterferogram (right).
4
1000
5000
4000
3000
2000
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llf.IVENUHBERS CHl
Figure 5: A) First 2048 points of an lnterferogram consisting of a total of 8196 points. Signal In the wings Is amplified 100 times. B) FT of first 512 points of lnterferogram In Fig. 5a, corresponding to a resolution of 32 cm. C) FT of ali 8196 points of lnterferogram In Fig. 5a, corresponding to a resolution of 2 cm.
.Series 6 Resolution In FTIR
by zero filling it, I.e. one should choose a zero filling factor (ZFF) of two. In those cases, however, where the expected line width Is similar to the spectral sample spacing (as e.g. in case of gasphase spectra), a ZFFvalue of up to 8 maybe appropriate.
'·'
Figure 4 shows the lnterterogram corresponding to two sharp lines separated by a wavenumber distance d. Due to the separation d In the spectrum, the interferogram shows periodic modulation patterns repeated after a path length difference 1/d. The closer the spectral lines are, the greater the distance between the repeated patterns. This Illustrates the socaiied Rayleigh criterion, which states that In order to resolve two spectral Jines separated by a distance done has to measure the lnterferogram up to a path length of at least 1/d.
The Influence of zero filling on the appearance of water vapor bands ls demonstrated in Figure 6. Al the top, a · spectrum with no zero filling Is shown. The spectrum at the bottom Is zero filled using a ZFF of 8. While the lines of the upper spectrum look badly clipped, the lines are smooth In zero filled spectrum.
ll. 2
1600
For a practical measurement, which was done on a Bruker IFS88 using a broad band MCT detector, the influence of increasing the interferogram path length on the resolution is shown In Figures SA, B, C. The lnterferogram in Figure SA represents the first 2048 points from a total of 8196. Figure SB was obtained by transforming only the first S12 lnterferogram points, which corresponds to a resolution of 32 cm 1 • Figure SC exhibits the full 8196 point transform. It Is clearly seen that many
more spectral features are resolved In the case of a longer optical path.
7 Zero Filling
1550 1500 1450 'IAVENUHSERS CHI
!400
It should be noted, that zero filling does
not introduce Figure 6: Picketfence effect In bands due to water vapor. Top: no zero filling, bands look badly clipped. Bottom: spectrum zero fllled using a ZFF of 8.
any errors because the instrumental line shape is not changed. It Is therefore superior to polynomial interpolation procedures working In the spectral domain. Aliasing, leakage, apodlzation, and phase correction will be dealt with In the following installments.
The picketfence effect can be overcome by addin!hzelQ~
References (1] W. Herres, uCapfffary GC.fTIR Analysis of Vol· stiles: HRGC..FTIR" In Proceedings uAnalysls of Volatiles", P. Schreier (editor), Walter de Gruyter & Co., Berlin {19B4).
{2} A.J. Bell, u/nltoductory Fourier Transform Spectroscopy", Academic Press, New York (1972). (3] G.D. Bergland, IEEE Spectrum, Vol. 6, pp. 4152 (1969).
It should be noted that DFT only approximates the continuous FT, although it is a very good approximation if used with care. Blind use of eq. (3), however, can lead to three wellknown spectral artifacts: the picketfence effect, aliasing, and leakage. The picketfence effect becomes evident
when the interferogram contains frequencies which do not coincide with the frequency sample points k • A v. If, In the worst case, a frequency component lies exactly halfway between two sample points, an erroneous signal reduction by 36 O/o can occur: one seems to be viewing the true spectrum through a picketfence, thereby clipping those spectral contributions lying 'behind the pickets', i.e. between the sampling positions k .6. v. In practice, the problem is less extreme than stated above if the spectral components are broad enough to be spread over several sampling positions.
*
5
By Joern Gronholz and Werner Herres
UNDERSTANDING FTIR DATA PROCESSING PART 2: DETAILS OF THE SPECTRUM CALCULATION In the first part of this series, we covered the FTIR data acquisition and the Fourier transformation. This second part continues with the description of the mathematical operations performed by an FTIR minicomputer to compute the spectrum from the interferogram.
1 Aliasing
In
part 1 [l] it was shown that sampling the continuous interferogram and the use of the discrete version of the Fourier transformation (the DPT) may produce artifacts, such as the picketfence effect, unless special precautions are taken. Another possible source of error due to the use of the DPT is aliasing. To understand aliasing, we recall the basic DPTequation
L; exp(i27fk n!N)I(n • il.x)
exp i27rk
= (exp i27r) = 1 *k = 1
(I)
n"'O
*
* *k
=
S(k)
(2)
(3)
about the. !c:>c~lled 'folding'  or G!l)Jyquist' :wavenimiber ,;•, .JP,...• (N/2)· fiv Dv~ f'1 b'f ;;;·ll(2. Ax) . (4)
which describes how a spectrum sampled at wavenumbers k · !J.v can be computed from an interferogram sampled at optical path differences n Furthermore, one sees that Equa· Ax. In practical calculations both (1) is not only valid for indices tion n and k will run from 0 to N1, k from 0 to N  1 but for all integers i.e. the DPT produces N (generally including negative numbers. In parcomplex) output points from an inif we replace k in Equation ticular, put interferogram of N (generally (I) by k + m · N, we get the equareal) input points. If we expect the tion spectrum to be of the form shown in Figure IA, we will find that the DPT S([k + m · N]) = S(k) (5) yields not just a single spectrum but rather the spectrum plus its mirror which states that the mirrorsymimage, as given in Figure 1B: the metrical Npoint sequence of Figure first N/2 points represent the ex 1B is endlessly and periodically reppected spectrum, the second part, licated as indicated in Figure 1C. This replicat.ion ~f th~ original starting with the index k = N/2 E~pectrum and its mtrror image on equals its mirror image. For if (he wavenumber axis is termed practical computations this means "''aliasing'.
I
Joern Gronholz and Werner Herres, Bruker Analytische Me.fttechnik, GmbH, Wikingerstr. 13, Karlsruhe 21, FRO. 6
calculated if the spectrum does not overlap with its mirrorsymmetrical replicate (alias). Nl>"•overlap will occur.if·4he•spectrum·is zet
one obtains the mathematical description of mirror symmetry S([Nk])
N1
S(k· il.v) =
that a DPT of an Npoint interferogram yields only N/2 meaningful output points. The second set of N/2 points is redundant and therefore automatically discarded. This behavior is also easily derived from Equation (1), if one substitutes index k by Nk. Using the identity
I.I Alias Overlap
From Figures 1B and IC it is clear that a unique spectrum can only be
(6)
1/(2. Ax)
Here, we recall from [1] that /J.v is related to Ax by !J.v
=
1/(N ·Ax)
(7)
If; ·however, the'· ~pectrum contains ..a.non·zero ·Contribution e.g. 200...cnF'. above the.folding wave. number·vr,this•will•be'folded back' below• Pf. and appear at the wrohg position; i,e; P['200 cm' 1• This is the possible artifact due to aliasing. The finer the interferogram .sample spacing Ax is, the further apart are the aliases and the lower the danger of alias overlap. However, small Ax means also an increased number of points N and therefore higher storage needs and larger computation times. For a given wavenumber range, the FTIR software has therefore to choose the maximum sample spacing for which still no overlap occurs. In part I we explained that in PTIR, the sampling positions are derived from the zero crossings of an HeNe laser wave having a wavelength >.. of 1/15800\cm. As a zero crossing occurs every A./2, the minimum possible sample spacing Axmin is 1/31600 cm. With Equation (4), this corresponds to a folding wave"
n111uber 'of 15800~ cm 1, i.e. the maximum bandwidth which can be measured without overlap has a width of 15800 cm 1• A larger range can be covered, if the laser frequency is electronically doubled (frequency multiplication). Very often, the investigated bandwidth is much smaller than 15800 cm1, e.g. in midIR, \Vhere Vmax is generally less than 4500 cm  1 and especially in the farIR with wavenumbers below 200 cm  1• In these cases, one can choose & to be an mfold multiple of dxmin· This leads to an mfold reduction of the interferogram size.
\Vhere Vmin =
0, is termed 'under
sampling'. It should be noted that undersampling enables measurements with a Vmox higher than the original laser wavenumber, because only the difference vru  VfL, and not the absolute values of the folding wavenumbers is to be considered in the sampling condition
1 llx ,;; ~2~[v_ru___v_fL~I
(10)
An advanced FTIR software package will automatically account for proper sampling and undersampling, such that only the upper and lower limits of the desired spectral range need to be specified. The user only needs to make sure
1.2 Undersampling
An even greater reduction of the data slze is possible, if the spectrum is zero below a lower band limit Vmin and if
Vmin
is not zero as assumed
above. If the spectrum bandlimits Vmin and Vmax lie bet,veen lo\ver and upper folding wavenumbers VfL and vru, which are related by n1
VfL = ~n
Vru
(n = 1,2,3,. .. ) (8)
it will look as indicated in Figure ID for the case n = 4. The upper folding wavenumber Pru must be a natural fraction (or integer multiple) of the HeNe laser wavenumber: Vru = fr' 15800
(fr= .. ., 1/J, 1/2,1,2,3,. .. ).(9) If we now further increase the sample spacing by a factor n, the aliases of Figure 1D will overlap appreciably, thus filling the previously empty range from 0 to VfL with n  1 further copies of the original spectrum. This is shown in Figure IE. As all copies are identical (except that their absolute wavenumber scaling and their direction on the vaxis can differ from the original), we need not calculate the spectrum at its true position by an Npoint FT, but can rather calculate the alias of lowest wavenumber by an NI npoint FT and correct its wavenumber scaling afterwards. This further nfold reduction of interferogram size compared to the conventional case,
D
0
0
v...
ll..
Figure I: Effects of Sampling. A) Expected Shape of the Spectrum. B) The DPT Yields the Spectrum and Its Mirror Image. Only the First N/2 Points Contain Useful Information. The Second N/2 Points are Redundant and Discarded. C) Aliasing:
Figure JB is Endlessly Replicated on the Wavenu1nber Axis. Aliasing Causes Errors, if the Spectrum is Nonzero up toa Wavenumber Vmaxand ifvmrixlsabove the Nyquist Wavenumber vf' This Happens if the Sampling Condition fil: < 11(2 · vm,,) is Violated. D) Spectrum is Zero Belolv vmin and above Vmru;" This Allows/or Undersampling. E) Undersampling Produces Spectrum Aliases. The DPT Calculates the Alias of Lolvest Wavenumber Instead of the Original. The Spectrum must be Zero outside The Bandpass Defined by the Upper and Lower Folding Limits.
7
·that the investigated spectrum is, a digitized version of a continuous really zero outside the range vfL to ,interferogram. Leakage is caused by Pru J:>y il!serting either, optical ,or the truncation of the interferogram at finite optical path difference. The electronic filters. iifl For technical reasons, the sample proper mathematical term to despacing must often be increased in scribe the effect on the spectra of ~ steps of powers of two. The possible ·truncating theinterferogramis 'con1\ folding wavenumbers for this comvoli\tilln'. 'i\mon case are given in Table I. 2.1 Convolution
'ii
2 Effect of the Finite Record Length: Leakage
Unlike the picketfence effect and aliasing, leakage is not due to using
Mathematically, an interferogram IL(x), truncated at optical path difference x = L can be obtained by multiplying an interferogram Ii (x)
of infinite extension by a 'boxcar' function BX (x), which is zero for x > L and unity for x ~ L, i.e.
IL (x) =Ii (x) · BX(x) .
According to the convolution'' theorem of Fourier analysis, the Fourier transform of a product of two functions is given by the convolution (here indicated by the ' o' symbol of their individual Fourier transforms, i.e. if Si (v) and bx (v) are the Fourier transforms of Ii (x) and BX (x), respectively: +oo
Si (v)
=J
bx (P)
=J
exp(i2'1!'vX) · Ii(x)dx
Table 1: Possible Folding Wavenumbers Depending on the Sample Spacing.
Actual sample spacing is 2* • SSP • (l/31600) [cm]. Laser wavenumberis assumed to be exactly 15800 cmI, BWis the maximum possible bandwidth, i.e. the difference between upper and lower folding limit Pn; and vfL. Undersampling occurs, if n from Equation (8) is > I. Only cases up ton= 4 are shown. 2• •SSP
n
BW[cmIJ
vfL[cmI]
Pru [cmI]
I
15800.000 15800.000 15800.000 15800.000
0.000 15800.000 31600.000 47400.000
15800.000 31600.000 47400.000 63200.000
I I I
2 3 4
2 2 2 2
2 3 4
7900.000 7900.000 7900.000 7900.000
0.000 7900.000 15800.000 23700.000
7900.000 15800.000 23700.000 31600.000
4 4 4 4
l 2 3 4
3950.000 3950.000 3950.000 3950.000
0.000 3950.000 7900.000 11850.000
3950.000 7900.000 11850.000 15800.000
8 8 8 8
2 3 4
1975.000 1975.000 1975.000 1975.000
0.000 1975.000 3950.000 5925.000
1975.000 3950.000 5925.000 7900.000
16 16 16 16
2 3 4
987.500 987.500 987.500 987.500
0.000 987.500 1975.000 2962.500
987.500 1975.000 2962.500 3950.000
32 32 32 32
2 3 4
493.750 493.750 493.750 493.750
0.000 493.750 987.500 1481.250
493.750 987.500 1481.250 1975.000
I
I
64
I
64 64
2 3 4
246.875 246.875 246.875 246.875
0.000 246.875 493.750 740.625
246.875 493.750 740.625 987.500
128 128 128 128
l 2 3 4
123.437 123.437 123.437 123.437
0.000 123.437 246.875 370.312
123.437 246.875 370.312 493.750
64
8
\
(11)
(12)
+oo
exp(i2'1!'PX) . BX(x)dx
00
then we get the following relation for the Fourier transform SL (v) of the truncated interferogram IL (x) +oo
SL (P) =
J
exp(i2'1!'PX) . IL(x)dx
00
+oo
J
exp(i2'1!'vx)Ii(x)BX(x)dx
00
(13)
+oo
J
Si(k)bx(vk) dk Si(v)
o
bx(v) .
The computation of a convolution integral B(v) o C(v) as in Equation (13) can be visualized by the following procedure:  Put the function B onto the kaxis with its origin at k= 0. This is B(k). Do the same for function C(k).
 Move the origin of function C(k) to another position k = v and reflect it about this position. This yields C(vk).  Multiply the displaced and reversed function C(vk) by B(k) and measure the area under the product function. This is the value of the convolution integral for one position.  Repetition of the above three steps for all positions yields the wavenumber decomplete pendence of the convolution integral.
According to Equation (13), the spectrum SL(v) of a finite interferogram can thus be obtained by convolving the spectrum Si(v) corresponding to infinite optical path difference (and hence to infinite resolution, see part 1) with the 'instrumental lineshape' (ILS) function, bx(v). This enables a clear description of a measured Iineshape in terms of a natural lineshape (NLS) due to physical linebroadening, the ILS representing solely the contribuof finite resolution. The analytical form of the ILS corresponding to boxcar truncation can be easily derived from the Fourier integral of a unity operand using a finite integration range. The result is the well known sine function: bx(v) =L ·sine (2vL) =L · sin(27rvL)/(27rvL), (14)
which is plotted in Figure 2. One sees, besides a main maximum centered about v = 0, numerous additional peaks, called side lobes or 'feet'. These side lobes cause a 'leakage' of the spectral intensity, i.e. the intensity is not strictly localized but contributes also to these side lobes. The largest side lobe amplitude is 22 OJo of the main lobe amplitude. As the side lobes do not correspond to actually measured in
forination but rather represent an artifact due to the abrupt truncation at x = L, it is desirable to reduce their amplitude~Jlll!PtOJ.'~~··~pJ!;b<
lobes than the sine function. Numerous such functions exist. An extensive overview of their individual properties can be found in the review by Harris [2]. In Figures 3B 3D, four examples of such functions and their Fourier transforms are plotted together with the boxcar cutoff in Figure 3A. The analytical forms of these apodization functions are
,;~~t••~i!~~~'~:a~ d!Z~!IO!J:'
, (originating from the Greek,word C<1i'OO, which means 'removal of the feet').
2.2 Solution to Leakage: Apodization
Triangular (TR):
The solution to the problem of leakage i to truncate the interferogram less abruptly than with the rectangular or 'boxcar' cutoff. This is equivalent to finding a cutoff or apodization function with a Fourier transform which shows fewer side
TR(x) = 1n/L (n=0,1, .. .,L)
(15)
Trapezoidal or fourpoint (FP): = 1 (n=0, 1, .. .,BPC)
FP(x)
1 [n BPC]l[BPDBPC] (n =BPC, .. ., BPD) (16)
A BOXCAR
B
TRIANGULAR
c TRAPEZOIDAL
SIHC2bU
"" 

a.s11t
D
H.RPPGENZEL
E 3TERH. BLRCKH.RNHRRRJS
,,1!
Figure 2: Fourier Transfor1n of the Boxcar Cutoff. known as the Sine Function. Largest Side Lobe is 22 % of the Main LobeAmp/itude. L = Opitca/ Pathlength Difference.
Figure 3: Several Apodization Fu"nctions (left) and the 'Instru1nental Lineshape' Produced by Them (right). The Cases A D Are Commonly Used in FTIR. 9
This is a boxcar function between 0 and breakpoint BPC, and a triangular function between breakpoints BPC and BPD. In Figure 3C we chooseBPD=L. Hamming or HappGenzel (HG): HG(x) = 0.54 + 0.46 COS('ll"nlL) (n=0, 1, ... ,L) (17) This is a cosine halfwave on a boxcar pedestal. The amplitude at the boundary x = L is not zero but still 8 % of the amplitude at the origin. The parameters 0.54 and 0.46 have been chosen for optimum suppression of the first, largest side 'Johe. Three and fourterm BlackmannHarris (BH) BH(x) =AO+ Al COS('ll"nlL) + A2 COS('11"2n/L) + AJ CoS('11"3n/L) (n=0, 1, ... ,L) (18) This set of windows is a generalization of the HappGenzel function. The coefficients have been optimized numerically to trade main lobe width for sidelobe suppression (see [2]):
AO Al A2 A3
3termBH
4termBH
0.42323 0.49755 0.07922 0.0
0.35875 0.48829 0.14128 0.01168
The threeterm EHwindow is plotted in Figure 3E. 2.3 Apodization and Resolution As expected, Figure 3 reveals that all apodization functions produce an ILS with a lower sidelobe level than the sine function. Ho\vever, one also sees that the main lobes of all ILS's in Figures 3B  3E are broader than that of the sine function in Figure 3A. The width at half height (WHH) of the ILS defines the best resolution achievable with a given apodization function. This is because if two spectral lines are to appear resolved from one another, they must 10
be separated by at least the distance of their WHH, otherwise no 'dip' will occur between them. As side lobe suppression always causes main lobe broadening, leakage reduction is only possible at the cost of resolution. The choice of a particular apodization function depends therefore on what one is aiming at. If the optimum resolution of 0.61/L is mandatory, boxcar truncation ( = no apodization) should be chosen. If a resolution loss of 50 % compared to the boxcar can be tolerated, the HappGenzel, or even better, the 3term BHapodization is recommended. If the interferogram contains strong lowfrequency components, it may show an offset at the end, which would produce 'wiggles' in the spectrum. To suppress these wiggles, one should use a function which is close to ·zero at· the boundary, such as the triangular, trapezoidal, or BlackmanHarriswindows. As the ILS produced by the BlackmanHarris function shows nearly the same WHH as the triangular and HappGenzel function (roughly 0.9/L), but at the same time, the highest side lobe suppression and is furthermore nearly zero at the interval ends, it can be considered the top performer of these three functions. In practice, the shape of a spectral line measured at finite resolution is always a mixture of natural and instrumental lineshape. This is demonstrated in Figures 4A4C, where the same NLS (in our case a Lorentzian), recorded at different resolutions, is plotted. The ILS corresponds to boxcar truncation. One sees that a lineshape close to the NLS can only be observed if the width of the ILS is small compared to theNLS. At the end of the discussion of apodization, it should be noted that an instrumental lineshape with side lobes is of course also imposed on spectra from dispersive instruments. The ILS produced by the slit of a grating spectrometer corresponds to the ILS caused by triangular apo
dization. The difference between FTIR and dispersive spectroscopy concerning apodization is that an FTIR spectroscopist can choose the optimum ILS for his specific needs, while the 'dispersive spectroscopist' cannot.
3 Phase Correction The last mathematical operation to be performed during the conversion of an interferogram into a spectrum is phase correction. Phase correction is necessary, because the FT of measured interferogram generally yields a complex spectrum C(v) rather than a real spectrum S(v) as known from conventional spectrometers. A complex spectrum C(v) can be represented by the sum
a
C(v) = R(v)
+
il(v)
(19)
of a purely real part R (v) and a purely imaginary part I(v) or, equivalently, by the product
~);S(vJ'exfltltfi(JJ)r
<20>
of the true 'amplitude' spectrum S(v) and the complex exponential
exp(ic/>(v)) containing the wavenumberdependent 'phase' cf>(v). The aim of the phase correction procedure is to extract the amplitude£• spectrum S(v) from the comple'j!•··~ output C(v) of the FT. This can done either by calculating the squa~ root of the 'power spectrum' P(v) c!
bN
C(v) · C*(v):
'[1, '/
S(v) = [C(v) · C*(v)J112 = [R2(v) + l2(v)] 112
!~.
(21)1
or by multiplication of C(v) by thel1, inverse of the phase exponential and~ taking the real part of the result:
J;
S(v) = Re[C(v) exp(ic/>(v))]
(22f
The phase c/>(v) in the exponential exp(ic/>(v)) can be computed from the relation cf>(v) = arctan[I(v) I R(v)] .
(23)
Equations (21) and (22) are equivalent, if one deals with perfect data free of noise. However, if noise is present, as is always the case with measured data, noise contributions
computed from Equation (21) are always positive and in the worst case, a factor or ...fi larger than the correctly signed noise amplitudes ~g111puted from Equation (22). ~!Sf&ft\~ dure'(22).·isknown.1!$dmultipll~1ttive
phase ··correction' · ON·theI<«Mer'tzmethod~
height 0.5 and a ramp through the origin, as indicated in Figure 6B. Because of the symmetry properties of the FT, the even boxcar function contributes only to the real part of the FT and has the effect of multiplying all contributions from
PJP/2 to +PIP/2 by 1/2, while the odd part of the ramp contributes only to the imaginary part of the FT. The odd part of the ramp is added to avoid a step at + PIP/2, which would produce 'wiggles' in the spectrum.
t3J.
3.1 Reasons for the NonZeroPhase
The reason for getting a complex spectrum out cif the FT is that the in: put to the FT is not mirrorsymmetrical about the point x = 0. The asymmetry of the FT input originates from three different sources: a) None of the sampling positions coincides exactly with the proper position of zero path difference. This is generally the case and causes a phase linear in v.
A
D
Figure 4: Measured Lineshape Function Corresponding to a Lorentzian Natural Lineshape (D) and Boxcar Truncation, Depending on the Resolution. Resolution IncreasesfromA toD.
b) Only a 'onesided' interferogram is measured, i.e. only one side is recorded to its full extent, the other consists only of a few hundred points. c) The interferogram may be 'intrinsically' asymmetric. This may be due to wavenumberdependent phase delays of either the optics, the detector/amplifier unit, or the electronic filters. Figure 5 shows how the phase contributions from a) and b) can be calculated from a short doublesided portion of length PIP (PIP/2 points on either side of the centerburst). This yields, after iipodization, zero filling, and FT gives a low resolution phase spectrum. As the phase is a slowly varying ful\ction of v (except in the regions of beamsplitter or filter cutoff), a low resolution phase spectrum is sufficient. During the phase correction this is expanded to the size of the full array by interpolation. The steps in the computation of the full array are shown in Figure 6. After apodization, the short doublesided part is multiplied by a ramp to avoid counting it twice. The effect of this ramp can be understood by decomposing it into a boxcar of
c
B
A
8
"'
'
'
'" '
y
c
D
+ lr'AVENUHBERS
Fig~re5:
Phase Computation. A) The Complete 'OneSided' Interferogram. B) PIP Poiiftsofthe Short DoubleSided Part Around the Centerburst are Used to Compute the Phase Spectrum. C) The DoubleSided Part is Apodized, Zero Filled and Fourier transformed. D) The Phase Spectn11n is Computed from the Complex Output of the FT via Equation (23). This is Later Interpolated to Full Resolution.
11
I ,. ! 1,
1,:
R '(v)
Having multiplied the data array by the ramp and the apodization function, it is zero filled and Fourier transformed. Using Equation (22) the resulting complex array C(v) is then multiplied by the complex exponential exp( i(v)), which is computed by interpolation from the low resolution phase spectrum. After the phase correction, the corrected real part R ' (v)
~
R(v)cos((v))
 J(v)sin((v)) represents the final singlechannel spectrum S(v) which is stored for further processing. The corrected imaginary part I' (v) I' (v)
~
R(v)sin((v)) + I(v)cos((v))
originates from the antisymmetric contribution of the cutoff at
d=Jeven
d= r:+,jodd
PIP/2 and the ramp as is shown in Figure 7. I' (v) would be zero, if a doublesided interferogram had been used. The corrected imaginary part is normally skipped or not even calculated to save computer time. The fact that I' (v) is non zero after the phase correction demonstrates that the direct calculation of S(v) from the power spectrum via Equation (21) will yield erroneous results in the case of onesided interferograins, because this procedure corrects for all contributions to I(v), including those from the cutoff at PJP/2. This method is only properly applicable to doublesided interferograms. With this last topic we have now completed the discussion of the standard operations necessary to convert an interferogram into its spectrum. In the next installment, we will deal with several FT or interferogrambased techniques which are very useful in IR and GCIR spectroscopy. o
k
·~·~ iua
an
HU VAVfXUMBEltS CitI
l~U
OU
HU !l!U t.O.Vf!
=
•••n
r +
UH
\
·~ uu
21u
Jen
1aaa
YllYHIJll!(llS Cll•J
Figure 6: Computalion of 1he Final Spec/rum. A) The OneSided lmerferogram is Apodized to Reduce Leakage. BJ Afler Apodizalion, !he DoubleSided Par! of Lenglh PIP ls Mul1iplied by a Ramp. The Ramp can be Decomposed imo an Even and an Odd Par!. The Even Par! Prevems !he PIPRange from Being Coun1ed 1kice. The Array is !hen Zero Filled and Iransformed. CJ Real Pan of !he D(TOutpul before Phase Correc1ion. D) Imaginary Par! of the DFTOutp~I before Phase Correclion. E) Final Spec/rum after Phase Correclion Usil:!g !he In1erpola1ed Phase Spec/rum of Figure 5D According to Equaiio\(22). 12
"'
.Figure 7: Represemation of a OneSided Cutoff Fune/ion by 1he Sum of an Even and an Odd Par!. The Even Part Corresponds to a DoubleSided lnterferogram. The Odd Part Contribu1es Solely to !he Imaginary Part of 1he Complex Spectrun1.
References [1] W. Herres and J. Gronholz. Con1p. App/, Lab. 2 (1984) 216. [2] F.J, Harris, Proceedings of the IEEE, 66 (1978) 51. 13] L. Mertz, Infrared Phys. 7 (1967) 17.
Joern Gronholz and Werner Herres
l
UNDERSTANDING FTIR DATA PROCESSING PART 3: FURTHER USEFUL COMPUTATIONAL METHODS In Parts 1 and 2 of this series lVe covered the standard operations of FTJR, fro1n data acquisition to the final spectru111. This third and last part continues with a discussion of additional useful techniques and closes lvith a remark about the data system.
n Parts 1 and 2 2] of this Itionseries we dealt with data acquisiin a Fourier transform in[l,
frared {FTIR) spectrometer and described the processing of the raw data to generate the final spectrum. To perform the necessary calculations in a reasonable time, the minicomputer of an FTIR spectrometer needs sufficient online computing power and is thus also very well suited to do more than just calculation of FFT's, phase correction, and ratioing of spectra. In fact, the considerable inherent numbercrunching capability inside an FTIR spectrometer is sometimes called the fourth advantage of FTIR. In this last part of our series we discuss some examples of such additional data processing in both domains: the frequency domain and the interferogram domain.
I The Problem of "Ghost" Interferograms or Fringes The appearance of sinusoidal modulations, called 'fringes' on the baseline of IRspectra is well known to spectroscopists. These
Joern Gronholz and Werner Herres, Bn1ker Analytische Messtechnik GnzbH, Wikingerstr. 13, Karlsruhe 21, West Gerrnany.
fringes or 'channel spectra' result from multireflections of the IR beam between the surfaces of a planeparallel device in the spectrometer's optical path, such as the plane parallel sample itself or a liquidcell window. They can disturb the 'useful' spectral information quite seriously, as is shown for the absorbance spectrum of a silicon wafer in Figure IA. From Figure 2 in [l] it is readily concluded that fringes of constant frequency in the \vave number domain must result from a single 'spike' or a narro\v 'signature' in the corresponding interferogram. Indeed, the sample interferogram from which the absorbance spectrum in Figure lA was calculated, shows a 'ghost' interferogram or 'echo peak' ai an offset of 4206 interferogram points from the centerburst (see Figure JB). The lnterferogram was acquired at a resolution of 2 cm 1 and thus consists of 8192 points. Fourier transformation of the full interferogram including the echo peak yields the spectrum in Figure lA, whereas reduction of the resolution to 4 cm 1 (equivalent to an interferogram of 4096 data points) truncates the interferogram just before the echo peak yielding the 'fringefree' spectrum of Figure JC.
This example shows that one can easily get rid of the fringes by discarding all interferogram points
from the echo peak to the end. This can be achieved either by choosing a lower optical resolution during data acquisition or by apodization with a trapezoidal function (see Figure 3 of [2]) chosen such that all points from the echo peak to the end are set to zero.
1.1 Mathematica/Description of the Fringes The single channel spectrum S(P) of the IRradiation trans
mitted through a multireflecting plate can be calculated from the emptychannel background spectrum B(v), the halfspace power reflectance R(v), the refractive index n(v), the absorption coefficient a(v), the phase change ¥'(v) occurring at every internal reflection and the thickness d of the plate. The result is the wellknown Airy formula [3] which we have extended to include absorptive intensity loss and cast into a form better suited for Fourier transformation: S(P)
~So+ 2So!;
{R(v)exp(ad))k
k=I
x cos(4?Tkn(v)d + 2k<0(v))
(I)
with So(v) ~ B(v) (IR(v))'exp(ad) (2) I R2 (v) exp(2ad)
Equation (1) shows nicely that the resulting spectrum can be represented by the sum of a fringefree spectrum S0 (v) and an infinite number of interference terms. Due to the factor (R(v) exp(ad))k, the intensity of the kth order inter13
ference term decreases 'vith ascending order k because Rk(P) is always less than l. The interferogram corresponding to S(•) can be calculated from eq. (l) by inverse Fourier transformation. For the sake of simplicity we assume that the refractive index n (•) and the phase \O(P) within the argument of the cosine interference terms do not depend on P. For this simplified case one gets
0.7
A 0.35
0.0
I(x)
1200
1000
800
~
Io(X)
600
WRVENUMBERS CM1
+ 2 ~J,(x) o lo[x(2nkd + 2k\O)] k=l
(3)
+ 2 ~J,(x) o lo[x+(2nkd + 2k,,)] k=l
B
'
I' 11
11
ti
I
i II
x
Ii 0.5
c 0.375
0.25 1200
1000.
800
600
WRVENUMBERS CM1 Figure 1: A) Absorbance spec/nun of a silicon lvajer calculated using the full length (BK points ~ RES 2 cm1) of the interferogramfrom Fig. IB. B) Satnple interferogram of a silicon lVafer. Spectral resolution corresponds to 2 cm1 (= BK points). The distance between thes1nall 'ghost' interferogra1n on the right and the centerburst is 4206 points (yscale expanded for clarity). C) Absorbance spectrun1 of a silicon wafer calculated fro1n the interferogranz of Fig. IB using only 4K data points (resolution = 4 cin 1).
14
Ii.'
This result shows that the cosine interference terms of the spectral domain correspond to additional ghost interferograms in the interferogram domain, appearing symmetrically on both sides of the centerburst at distances 2nkd + 2k\O. The shape of these echo peaks is given 1ly convolution of the undisturbed main interferogram J0 (x) with the 'interferogram' h(x) which is the inverse FT of the kth power of the product of the reflectance spectrum R (P) and the transmission spectrum T(P) exp(ad). For the same reason as given above, the intensity of the kth ghost interferogram is at least smaller by a factor Rk than the main centerburst. Due to the wavenumber dependence of n(P) (i.e.: to dispersion) and of \O(v) which we explicitly neglected in the derivation of eq. (3), the echo peaks are generally additionally broadened and distorted and may be thus further reduced.
1.2 Use ofFringes for Thickness Determination Eq. (3) shows, that the distance between the main centerburst and the echo peaks is directly proportional to the product of refractive index n and thickness d. If the average refractive index n is known, one can therefore calculate
the thickness of the fringeproducing element. This possibility is, e.g., often used in semiconductor quality control to determine the thickness of epitaxial layers deposited on doped substrates. Conversely, if both n and dare known, the offset X between main peak and first echo peak can be calculated as:
Table 1: Location of the echo peak as a function ofsubstrate thickness. Offset of the echo peak relative to the centerburst as a function of the thickness of four co1n1nonly used substrates, Last co/unzn contains the maximum possible resolution/or lYhich still no interferences occur assuming 250 points before and 2* *N~250pointsafterthecenterburstinallcases. Bandwidth= 7900cmI throughout. Value ofrefractive indexn corresponds to JOOOcm1.
Substrate
=
(n = 1.53)
2·n·d
or in points
N= 2·n·d(&)I
(4)
with (&)I = 15800 cmI for midIR bandwidth, 07900cmI, The problem of ghost interferograms is of course not restricted to silicon wafers but can also be encountered in measuring standard alkali halide pellets or cast films on, e.g., KBr crystals. Sometimes spectra look 'noisy' at longer wavelength whereas expansion reveals fringes resulting from 'too perfect' a preparation, i.e. from planeparallelism of the pellet. In Table 1 the distance N between the interferogram centerburst and the first echo peak has been compiled for several thicknesses of four substrates commonly used in FTIR. This table also sho,vs the maximum resolution which does not include the echo peak. It may be helpful in finding the proper combination of sample thickness and resolution for a given substrate.
1.3 Possibilities for Eliminating Fringes The question might arise whether there is some method of eliminating all kinds of fringes which plague the spectroscopist when measuring polymer films. Figure 2A shows the transmission spectrum of some technical polyethylene (PE)type film. Although the modulations are very pronounced in the spectrum; the corresponding echo peak in the
Offset from centerburst
KBr X=N·&
Thickness
AgCl (n=
1.98)
ZnSe (n=2.40)
Si (n= 3.42)
[cm] 0.05 0.1 0.15 0.2 0.3 0.5 0.05 0.1 0.15 0.2 0.3 0.5 0.05 0.1 0.15 0.2 0.3 0.5 0.05 0.1 0.15 0.2 0.3 0.5
sample interferogram of Figure 2B can hardly be detected, being due to a much smaller reflectance R (v) of PE as compared to silicon and to stronger dispersion, which additionally broadens the echo peak and reduces its size. Furthermore, the echo peak is located only 267 points away from the centerburst because the film is only about fifty microns thick. For this latter reason, truncation of the interferogram ( = reduction of resolution) is here obviously no solution because the resulting resolution would be too poor. In those cases where reduction of resolution cannot be applied, there are at least two other solutions to the fringe problem: an experimental and a numerical solution.
[points] 2417 4834 7252 9669 14504 24174 3128 6257 9385 12514 18770 31284 3792 7584 11386 15168 22752 37920 5403 10807 16211 21614 32421 54036
Maximum resolution \Vithout interferences
[cmI] 8 4 4 2 2 8 4 2 2 1 8 4 2 2 1 0.5 4 2 1 0.5
1.3.J Experimental Solution of/he Fringe Problem From Eq. (!) it is clear that all interference terms vanish if the reflectance R (v) of the sample can be made zero. Experimentally, this is indeed possible using polarized IR radiation, as is sho\Vn in Figure 3A, where the same polymer film was measured. A KRS5 polarizer was used to provide an IR beam polarized parallel to the plane of incidence, while the film itself was oriented to the incident light at the Brewster angle. As the reflectance of this type of polarized radiation approaches a minimum at the Brewster angle (or is ideally zero), multireflections and thus 'channel spectra' are also minimized. This experiment makes it possible to obtain a difference interferogram 15
...............................................
1. 05
A 0.525
0.0 4000
3000
2000
1000
(Figure 2B minus Figure 3B) which reveals the structure of the echo peak more clearly (see Figure 4). Another experimental solution to the fringe problem would be a measurement using ATR (attenuated total reflection). However, unlike the Brewster angle measurement, ATR would change the relative intensities due to the wavelengthdependent depth of penetration.
VAVENUHBERS CH1
1.3.2 Numerical Solution of the Fringe Problem
Figure 2: A) Transmission spectru1n of a technical packaging fihn (1neasured on a standard Bruker IFS85 FTJR spectrometer). B) Sample interferogram of the packaging film corresponding to Fig. 2A, ysca/e expanded, only first 1024 points plotted. Insert shows ax and yscale expanded plot of the segment containing the signature.
1. 05
A 0.525
0.0 4000
B
'1
3000
2000
VAYENUHBERS CH1
~w~111 ·~~i11Mr1~~·~x
Figure3: A) Transmission spectrum of sanze poly1ner film as in Fig. 2A but using polarized light at the Brewster angle (see text). B) Sample interferograrn of po!y1ner film measured using polarized light at the Brelvster angle. Plotted part and expansion as in Fig. 2B.
16
The undesired oscillations due to multireflections are spread over a large part of the spectrum but are confined to a small region in the interferogram domain. Numerical correction of the fringes is therefore easier in the interferogram domain. As we have learned from Eqs. (1) and (2), removal of the fringes from the spectrum is equivalent to removal of the echo peaks from the interferogram. It has been shown by Hirschfeld and Mantz [4] that this can be done by substituting the regions around the echo peaks by zeroes, by a straight line, or by another reasonable guess of the undisturbed 0th order interferogram. Using the interferogram of Figure 2B as an example, the substitution of the echo peak by a straight line is demonstrated in Figures 5AC. One sees that, in contrast to the 'clean' experimental solution of Figure 3A, not all oscillations are removed but they are drastically reduced in size such that small peaks which were hidden under the oscillations are now clearly detectable. Although substitution by a straight line is a bruteforce method, the result is quite useful but must always be regarded with caution as additional artifacts, like creation or cancellation of line splittings, can easily be introduced. This numerical solution of the fringe problem can only be applied if the echo peak is intense enough to be detected and if it is far enough away from the centerburst.
While the smallness of the echo peak can be overcome by calculating its position from Eq. (5) if both n and dare known, violation of the second condition leads to severe baseline distortions and cannot be recommended. Before leaving the interesting field of ghost interferograms, it is worthwhile noting that a 'simple' absorbancemeasurement of, say, a KBr pellet is not adequately described by the familiar formula T(v)
= exp(a(v)d)
with transmittance T(v) and absorbance a(v)d, but must even in the simplest case of no multireflections be substituted by the numerator of Eq. (2). Figure 4: Difference interferograrn, Fig. 2B 111inus Fig. 3B, sho1ving the echo peak 1nore clearly. The insert sho1vs the signature xscale expanded.
T(v)
= (1R(v))'exp(a(v)d)
(5)
A 1. 05
B 0.525
D 0.0 4000
3000
2000
1000
WAYENUHBERS CH1
x+ 1. 05
Figure 5: A) Part of the san1ple interferogra111 front Fig. 2B containing the signature. B) Sa111e part of interferogra1n as in Fig. 5A, but points 466 to 479 (14 points) are substituted by a straight line to suppress the si'gnature. C) Saine as Fig. 6B but points 466 to 514 are substituted (49 points). D) Transn1ission spec/nun calculated fro1n the interferograrn of Fig. SB. E) Trans111ission spectru1n calculated ·fro1n the interjerogra1n of Fig. SC.
0.525
E
0.0 4000
3000
2000
1000
WRYENUHBERS CH1
17
This means that one must not forget at least two reflections at both boundaries of the sample. The factor (1  R) * * 2 causes an intensity reduction of the transmitted radiation, but may also lead to changes in position and intensity of single lines if R(v) is significantly structured. If the pellet has planeparallel surfaces, multireflection comes into play and the full Eq. (2) (divided by the empty channel background spectrum B(v}) must be used. Thus, from a transmission
measurement one always gets a mixture of reflection and absorption properties unless the reflectance contributions are explicitly corrected for. This single or multireflection correction, i.e. extraction of the absorbance a(v)d from equations (5) or (2) is another task which could be routinely performed by the FTIR spectrometer's minicomputer.
It is man
dantory if one is interested in precise determination of optical constants.
2 Interrelation of Smoothing and Apodization In Part 2 of th is series we already described apodization at some length. Apodization means multiplication of the interferogram by a decaying function. Its effect on the spectrum is a suppression of side lobes at the expense of decreased resolution. We also showed that multiplication of the interferogram I(x) by a function a(x) is equivalent to convolution of the spectrum S(v} by the Fourier transform A (v) of the apodization function. Three of the special apodization functions discussed (HappGenzel, 3 and 4term BlackmanHarris) consisted of a sum of cosine functions as N
BH(x) = AO
+ 2;An cos(n · "· x/L). n=l
It is instructive to calculate the Fourier transform of such a func18
tion and look at the corresponding convolution in the spectral domain in more detail. One gets S' (n · Liv) = AO S(n · Liv)
+ +
Al(S((nl)·Llv)
+
S((n+l)·Llv))
without floating point hardware support. As spectral smoothing is so widespread and well known, we omit a detailed description and rather turn to the question of what happens if smoothing is done in the interferogram domain.
A2(S((n2) ·Liv)+ S((n+2) ·Liv))
2.1 Smoothing in the Interferogram Domain: + AN(S((nN) ·Liv)+ S((n+N) ·Liv)) Digital Filtering (6)
Hence, in order to calculate one
point S' (n • .:l.v) of the spectrum corresponding to the apodized interferogram one must:
 multiply the ordinate S(n · .:l.v} of the nonapodized spectrum by AO,  multiply the left and right next neighbors of S(n • .:l.v) by AI,  multiply the left and right second next neighbors by A2 and so forth (up to N = 3 for 4term
BH)  sum the intermediate results. This is S' (n · .:l.v). This shows that the points of the spectrum S' (n · .:1.v) are a weighted mean of adjacent points of the nonapodized case S(v), the coefficients An of the apodization function being the weighting factors. As taking a weighted mean of spectral data amounts to nothing but smoothing, we conclude that apodization in the interferogram domain is equivalent to smoothing in the spectral domain. (Note that this special family of apodization functions would easily allow one to perform the apodization after the FT by summing neighboring points of the nonapodized spectrum multiplied by appropriate factors.) Smoothing is an essential tool for reducing the noise in a spectrum. In addition to apodization, smoothing is mostly done by the famous SavitzkyGolay procedure [5], the merit of which is that (although a leastsquares method and therefore datadependent) the weighting coefficients used in the averaging process are fixed integers. This means that smoothing can be programmed in integer arithmetic leading to short computation times even on machines
The effect of smoothing an interferogram on the corresponding spectrum is demonstrated in Figure 6AC. Figure 6A shows an unmodified single channel spectrum, Figure 6B shows the spectrum corresponding to the same interferogram smoothed using a 9point SavitzkyGolay window. Figure 6C is the ratio of Figures 6B and 6C. One sees that the effect of smoothing the interferogram is to decrease the intensity of the spectrum towards higher wavelengths, i.e. it behaves like a lowpass filter with the frequency response of Figure 6C. Accordingly, convolution in the interferogram domain is termed 'digital filtering'. Digital filtering is closely related to apodization although the aims of both procedures are completely different. While during apodization one multiplies the interferogram by the apodization function to achieve a certain line shape (convolution), digital filtering means that one convolves the interferogram by a filter function to achieve a certain (multiplicative) frequency response. In practice, the filter function is seldom of the SavitzkyGolay type (this was simply used as an example) but is derived from the desired frequency response by inverse Fourier trans
formation of the desired spectral bandpass in case of nonrecursive
filters. The advantage of digital  compared to analog  filtering lies in its great flexibility, because almost any conceivable frequency response can be modeled. The only problem is that, even today, where highspeed multipliers/adders are
o.z
A
'· l
available, the time needed for computing the necessary convolutions is often still too long for realtime filtering in highspeed applications with data rates > 100 KHz. For slowscanning FTIR spectrometers, however, digital filtering can be appropriate and is then another task to be performed by the spec . trometer's minicomputer.
forms of S(v), S' (v), and L (v), respectively. As an easy example we consider the case of an absorbance spectrum S(v) consisting of a single Lorentzian line which may be represented by convolution of an infinitely sharp line (a delta function) at v= v0 S'(v)
=
3 Deconvolution
'
2000
4000
6000
\l'RVENUMBERS CMI
•• 2 +      >     +     +    
B
Deconvolution is another interesting FTbased method closely·related to apodization and convolution. The aim of deconvolution is to decrease the widths of all lines in a limited spectral region or, equivalently, to enhance the apparent resolution of the spectrum. The method is based on the assumption that the investigated spectrum S(v) may be represented by a convolution S(v)
4000
6000
liRVEtlUHBERS CMI
l.'
S' (v) O L(v)
(7)
+~·+<
c
from which the broadening effect of L(v) can be easily cancelled by division by l(x). The deconvolved spectrum S' (v) is then obtained by another forward FT to the spectral domain. I(x), I' (x), and l(x) represent the inverse Fourier trans
0.5
2000
4000
60(Hl
WRYENUMBERS CMI
(9)
with a line shape function L(v)
= 3!!!_ :
S(v)
=
a2
+
(10)
v2
S'(v)
o L(v)
= b(vvo) o ahr a2 + vi
(11)
ahr a2 + (vvo)2
As was shown in Figures 2A, C of[!], a sharp line corresponds to a cosine function in the interfer ogram domain
of a deconvolved spectrum S' (v) containing sharp lines and a func · I' (x) = cos(hvox) (12) tion L(v) which is responsible for the line broadening. From the dis\vhereas a Lorentzian at \Vavecussion of apodization in Part 2 of number v0 corresponds to a cosine this series we know that convoludamped by (i.e. multiplied by) an tion in the spectral domain correexponential decay function sponds to simple multiplication in the interferogram domain. Thus, l(x) = exp (  2'r a fxJ) (13) inverse FT of Eq. (7) yields the product and is thus represented by the I(x) = I' (x) I (x) (8) product
'· l
2000
=
b(vvo)
Figure 6: A) Singlechannel sanzple spectru1n of the polymer Ji/Jn used for Figs. 2, 3. B) Sa111e spectru1n as in Fig. 6A but after s1noothing the interferogra111 by a 9point SavitzkyGolay window. The intensity tolvards higher lYavenu1nbers is decreased. C) Ratio of Fig. 6B and Fig. 6A, showing the frequency response of the SavitzkyGo/ay 9point digital filter (see text).
I(x) = I' (x) · l(x) = cos(hvox)
exp(2" a fxf)(l4)
Deconvolution ( = removal of the damping function l(x)) is here obviously achieved by multiplication of I(x) by the inverse of l(x) !/l(x) =
exp(+ 2,,. a fxf)
(15)
yielding first the Fourier representation I' (x) of the deconvolved spectrum S' (v) and · after another forward FT  the deconvolved spectrum S' (v) itself. Applied to experimental data, deconvolution is less trivial than it appears from the synthetic case 19
above because the proper form of the line shape function L(v) _is not known a priori. L(v) can only be approximated by making a more or less reasonable assumption of its form (e.g. Lorentzian or Gaussian, or a mixture of both) and by estimating its width from the narrowest line in the investigated spectral area. Another complication arises from the noise which is always present in experimental data as it is strongly amplified by the removal of the damping function. It has been shown by Kauppinen el al. [6] that noise effects can be partly suppressed by apodization of J(x) by a triangular function which truncates I(x) at a point x = Ebefore the end at x = L. If it happens that the assumed L(v) exactly equals the true L(v), the shape of the lines in the deconvolved spectrum S' (v) will be exactly of the type L'(v)
=
Esinc2 (irvE)
due to the apodization. An example of what one typically may expect from deconvolution is given in Figures 7AE. By an inverse FT, the original spectrum of Figure 7A is transformed to the interferogram domain (Figure 7B) and enhanced (multiplied) by the function of Figure 7C yielding Figure 7D which, after another forward FT, results in the resolutionenhanced spectrum of Figure 7E. _ On comparing Figure 7A with Figure 7E, one sees that the original spectrum has been resolved into four components \Vith an indication of a fifth component near 1450 cm 1 • The question, whether all of the smaller three components are true or artificial is not easy to solve. Additional oscillations may easily be introduced by 'overdeconvolution', i.e. by overestimating the widths of the line shape L(v), as shown in Figure 9. In case of doubt, one should always underestimate the proper line width; this leads to lines broader than the achievable minimum but also with less artifacts. 20
B
><
c
D
11
FT
1460
1440
1420
1400
.VAVENUMBERS CM1
Figure 7: Individual steps of deconvolution. A) Spectral region to be deconvolved with broad, overlapping bands. BJ Result of an inverse FT of data from Fig. JA. CJ Plot of the function 1/l(vJ used to amplify the wings of the interferogramlike quantity fron1 Fig. 7B. This function is the product of a n1onotonically increasing exponential function and a decreasing triangular apodization function (see text). DJ Result of multiplying the functions from Figs. 7B and 7C. EJ Final result of the deconvolution procedure obtained by a forward FT of the datafron1 Fig. 7D. The individual lines are clearly sharpened.
4 Spectrum Simulation Deconvolution allows one to determine peak positions, relative in
tensities, and the number of individual components contributing to a certain spectral area. It should not be confused with spectrum simulation, which works in the spectral domain and which tries to represent a given spectrum by superposition of individual lines, the parameters of which (position, height, width, type) are optimized such that the deviation between experimental and simulated curve approaches a minimum. It should be noted, however, that deconvolution is well able to provide good starting values for the parameters of a spectrum simulation. This has been documented in Figure 8 which represents the result of an automatic leastsquares spectrum simulation using the example of Figure 7A and the results from the deconvolution as starting values
1460
1440
1420
1400
\iRVEtlUHBERS CM1
Figure 8: Result of a spectrum sitnulation using the exa1nplefrom Fig. 7A. The experimental curve is here approximated by superposition of three lines whose para1neters (height, position, width, type) have been optitnized auto111atically in the leastsquares sense.
steps of computation typical for FTIR spectroscopy like apodization, FT, phase correction, and ratioing to a stored background spectrum and is therefore a demanding task for the spectrometer's minicomputer. If it is to be done in realtime, with both high time resolution and sufficient spectral resolution, it generally needs the number crunching capability of an additional FT or ar5.2 GC Traces Directly from the Interferogram
Besides window GC traces also nonfrequency selective 'total' IR chromatograms similar to a GCFID trace can be calculated in various ways [7,8]. From the algorithms working directly on the interferometric data, a vector projection technique using the GramSchmidt orthonormalization procedure became fairly popular as 'GramSchmidt' technique and
basis vectors represents the starting condition of pure carrier gas. During the subsequent GC run the same segment from each acquired interferogram is used as a sample vectors. This sample vector is projected onto the orthonormal set of basis vectors and compared to its projection p by taking the
WRVEtlUMBERS CM1
for the variation parameters. This shows that both methods complement each other nicely. Deconvolution and spectrum simulation are mostly applied to absorbance spectra but they could equally well be used for analyzing other data like chromatograms.
5.2.1 GramSchmidt Technique
In the GramSchmidt technique a reference set of M interferograms is collected directly before the GC run when only carrier gas is leaving the GC column. A segment of each reference interferogram of N points is extracted, treated as an N
ray processor.
Figure 9: Exa1nple of overdeconvolution using the data ofFig. 7A (see text).
shall be treated below, Extracting chromatographic information directly from the interferogram without FT is fast and thus also possible on minicomputers without dedicated FTprocessor.
dimensional vector r0 , and used to construct a set of M
6 Generation of GCIR Chromatograms The combination of fastscanning spectrometers with gas chromatography (GC) and especially with capillary gas chromatography (HRGC) has evolved to the powerful hyphenated techniques GCFTIR and HRGCFTIR. In these techniques, the FT
M { ~~
IR spectrometer is used to measure
complete interferograms of the gas leaving a GCcolumn at constant time intervals. From the acquired interferograms different kinds of chromatographic traces may be generated.
0
+
N
5.1 Spectral Window Chromatograms
Spectral
window
chromat
ograms monitor the change in absorbance in discrete spectral re
gions. This calculation involves all
Figure 10: Visualization of the GramSclunidt para1neters. 0: offsetfro111 the centerburst in points. N: n111nber ofpoints per vector (vector dimension). M: nzunber of reference and basis vectors (see te~t).
21
s P
t
s
I I
I
I I I
I I
b2 Figures 11: Explanation of the GramSchmidt trace calculation lVifh two basis vectors. bi, b2 : basis vectors. s: sample vector. p: projection ofs onto twovector basis set. The 111agnitude of the vector differences p is the GratnSchnzidt trace.
r,
vector difference sp. This procedure is explained in Figure 11 for the case of just two basis vectors. While p represents the part of s due to pure carrier gas, the difference s  p represents deviations therefrom, i.e. the value of s  p constitutes the actual point of the chromatogram trace. A practical example of the GramSchmidt trace is shown in Figure 12 in comparison with a conventional GCFID trace. One sees the excellent agreement between the two chromatograms and also the excellent sensitivity of the GramSchmidt method. The achievable sensitivity depends on the setting of the GramSchmidt parameters, namely the number of points N per vector, their offset 0 relative to the centerburst and the number M
 The optimum number of basis vectors depends on the spectrometer's stability and should be found between 10 and 20.  The optimum number of points per vector N depends on the spectra of the investigated components and is expected to be between 80 and 200. As the computation time increases roughly proportionally to the product of Mand N, it may be advisable to use values smaller than the optimum ones if very high time resolution is necessary. Although our short overviews of the various kinds of computations to be performed by the minicomputer of an FTIR spectrometer is, of course, far from complete, we shall now end this series \vith some concluding remarks about the required performance of the data system.
6 Typical Data System Requirements The development of fast scanning interferometers in connection with HRGC and the availability of extensive libraries have increased the demand for speed and storage capacity enormously in recent years.
To get an idea of the speed and storage requirements, \Ve consider a HRGC run with 10 20 scans per second at a resolution of 8 cm  1. Each scan consists of 2048 data points of 16 bit each. The incoming raw data are written into the computers RAM by DMA either in automatic hardware coadd mode (get current content of a computer word, add new data, write result back, advance memory address) or in simple replace mode at a rate of about I 00 KHz. If every single scan is to be stored on disk, this corresponds to an average disk transfer rate of 2040 Kwords/second (= 60120 Kbytes/second with 24 bits/word). A disk of 20 Mbyte capacity would be full after 5  IO minutes of GC run. During all this DMAactivity a GramSchmidt and possibly several window traces must be computed and displayed. In case of \vindo\v traces a complete spectrum calculation must also be performed. In order to meet the speed requirements, not only is sufficiently fast hardware needed but also and even more important  an operating system with high disk throughput and real time capability. It should be noted that most commercially available operating systems for personal computers would not be suited for HRGC because their file organization is inadequate (no contiguous files), which drastically reduces the transfer rate. In connection with the necessary dedicated hardware this explains why the m1mcomputer of an FTIR spectrometer is more than just an 'overexpensive personal computer'.
' The required disk space also seems to be everincreasing, especially if large spectral libraries of several tens of thousand spectra are to be stored on a disk which is also used for GCIR. Therefore, if optical disks with capacities of several gigabytes become reasonably priced in the next few years, their use might well be interesting,
I
MINUTES
Figure 12: Example of traces fron1 a capillary GCFTJR separation of oil destillate. Bottom: Gra1nScl11nidt IR chro111atogran1. Top: GCFID detector trace acquired during the san1e GCIR run. FID (fla111e ionisation detector)
n1ounted behind the entire GCIR interface.
Modern FTIR scanners already allow for more than 50 scans per second and new 16bit ADC's permit sampling rates of 500 KHz, which calls for the development of faster hardware and software. The future will show where these developments will lead. References
[1] W. Herres and J. Gronholz, Comp. Appl. Lab. 4 (1984), 216. [2] J, Gronholz and \V. Herres, In
struments & Co1nputers 3 (1985), 10. [3] M. Born and E. \Volf, "Principles of Optics", Pergamon, Oxford (1970).
[4) T. Hirschfeld and A.W. Mantz, Appl. Spectr. 30 (1976), 552. [5] A. Savitzky and M.J .E. Golay, Anal. Chem. 36 (1964), 1627. [6] J.R. Kauppinen, D.G. Moffat, H.H. Mantsch, and D.G. Cameron Appl. Spectr. 35 (1981), 35. [7] D.A. Hanna, G. Hangac, B.A. Hohne, G.W. Small, R.C. Wieboldt, and T.I. Isenhour, J. Chromatogr. Sci. 17 (1979), 423. [8] P.M. Owens, R.B. Lam, and T.L. Isenhour,, Anal. Chen1. 54 (1982), 2344. [9) W. Herres, to be published. [10] D.T. Sparks, P.M. Owens, S.S. Williams, C.P. Wang, and T.L. Isenhour, Appl. Spectr. 39 (1985), 288.
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