(5, 6) indicates that niobium(V) exists principally as the dimer in aqueous HC1 solutions. The vapor phase osmometric data would admit the possibility of the Nb(V) being either monomeric or dimeric in DMF. The four particles obtained for each NbCls dissolved could be NbOC12+, C1-, and two HCl. [It has been reported that HC1 is a weak electrolyte in D M F (16).] The four particles could also be two HC1, ll/zC1- and [Nb2O2Cl3I3f. The fact that three particles were removed upon electrolysis at a platinum electrode and that 11/2 particles were removed at the mercury electrode is subject to ambiguous interpretation also. Either a monomeric or a dimeric Nb(IV) species might be expected to give the observed ESR spectrum. Both Cozzi and Vivarelli ( 4 ) and McCullough and Meites (5,6) indicate that Nb(1V) disproportionates to Nb(V) and Nb(II1) in aqueous HCl solution. Therefore, it would seem reasonable to account for the disappearance of the paramagnetic Nb(1V) species by a similar reaction. The fact that the original niobium reduction wave did not reappear, after electrolysis at the mercury cathode using potentials on the first polarographic wave, would indicate that the Nb(V), generated in the disproportionation reaction, is tied up in a form which is much harder to reduce than the original Nb(V) species. This gives added support to the Nb(V)-Nb(II1) adduct as a final product of the first reduction process. The known chemistry of niobium compounds would tend to favor dimeric structures throughout. If the third polarographic wave is examined carefully (Figure l), it appears to be two closely overlapping waves, each corresponding to a one-electron change. Exhaustive electrolysis at potentials on the third polarographic wave produced a white insoluble precipitate. Electrogravimetric analyses indicated this final reduction product to be NbO.
(16) D. S. Reid and C. A. Vincent, J . Electroanal. Chem., 18, 427 (1968).
With these considerations in mind, the following mechanism is reasonable for the second niobium reduction:
NbO(S)
+ 11/2 C1-
(15)
This mechanism is, by no means, the only possibility. As the disproportionation reaction, following the first reduction step, has a half-life of approximately four minutes, it is entirely conceivable that the unstable paramagnetic species could be reduced simultaneously with the Nb(V)-Nb(II1) adduct to give NbO(S). The mechanisms proposed are consistent with the experimental data. More involved studies of the species present in solution at various stages in the electrolysis might give more definitive data. RECEIVED for review May 15, 1970. Accepted July 22, 1970. This paper is based on a thesis submitted by Larry R. Sherman to the Graduate School of the University of Wyoming in partial fulfillment of the requirements for the Ph.D. degree. L. Sherman wishes to acknowledge and thank the Analytical Division of the ACS for a 1967 Summer Fellowship, the National Aeronautics and Space Administration for a Research Traineeship, and the Colorado-Wyoming Academy of Science for a 1968 Research Grant-in-Aid.
Basic and Practical Considerations for Sampling and Digitizing Interferograms Generated by a Fourier Transform Spectrometer Gary Horlick' and Howard V. Malmstadt Department of Chemistry and Chemical Engineering, University of Illinois, Urbana, 111. 61801
A number of problems associated with the sampling and digitizing of interferograms and their effect on the resulting spectra are investigated and described. These include studies of the sampling rate, the accuracy of the frequency axis, the effects of missed, extra, and bad points, and the resolution required for the analog-to-digital converter. These problems are all studied using computer simulation on real and synthetic interferograms, and spectra are calculated to demonstrate the erroneous effects that can result. Criteria are established for selecting the sampling interval and a number of examples are given to show how the interval and the spectral region under investigation determine the final labeling of the frequency axis. It is shown that the occurrence of a missed, 1 Present address, Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada. To whom requests for reprints should be sent.
extra, or bad point during the sampling and digitizing steps can distort the final spectrum. Thus these types of errors must be avoided if accurate spectra are to be calculated. Finally, it is shown that a lack of resolution in the analog-to-digital conversion step results in a loss of spectral resolution.
WITH THE AVAILABILITY of commercial instrumentation Fourier transform spectrometers are becoming more common. The main distinguishing feature of this spectrometric technique is that the spectrum is obtained by taking the Fourier transformation of an interferogram which is generated by a twobeam interferometer ( I , 2 ) . Considerable data handling is necessary with Fourier transform spectrometry and many (1) Gary Horlick, Appl. Spectry., 22,617 (1968). ( 2 ) H. A. Gebbie, Appl. Opt., 8 , 501 (1969).
ANALYTICAL CHEMISTRY, VOL. 4 2 , NO. 12, OCTOBER 1970
1361
apparently minor errors can result in completely erroneous spectra. The interferogram must, in general, be digitized because the data handling steps necessary for the reduction of a n interferogram t o a spectrum normally require a digitized interferogram. An interferogram can only be digitized at a certain finite number of sampling points. It is necessary that the resulting set of discrete digital values be a n accurate representation of the original continuous interferogram. A number of questions then arise. How often must the interferogram be sampled? What is the effect of the sampling rate on the frequency axis of the final spectrum? What is the effect of a missed or additional sampling point? What is the effect of a n erroneous value being assigned t o a sampling point? What is the effect of the resolution of the analogto-digital converter making the conversion? In order to obtain a practical understanding of the erroneous effects that can result from the improper sampling and digitization of a n interferogram, the problems posed by the above questions were investigated using experimental interferograms in conjunction with computer simulation. It should be noted that the considerations investigated and discussed here are directly applicable, with little modification, to the sampling and digitizing of the free induction decay signal obtained in the Fourier transform N M R experiment ( 3 , 4 ) .
A
-.
. .
.
- .........
-~
. . . . . . .
B
C
.........
~
..
,
I
......
S
...
8
.
I
... .I.-- .....
;
......
.
,
I
-.., ...
' ... _ _, ................ .
...... ',.- .... I1...... .Yl~~~l~~ll!. , ,
./
. .......,
Figure 1. Partial interferograms for a He-Ne Laser ( A ) , an iron hollow cathode lamp (B), and an 8400-A interference filter (C)
X
Y
SPECTRUM
A
Figure 2. Simulated aliasing
+
I
I
4
I
i
+
I
+
I
+
I
+ I
5mrec.
Figure 3. Aliasing of a 175-Hz sine wave to a 25-Hz sine wave by undersampling 1362
EXPERIMENTAL Interferogram Measurements. The experimental interferograms were measured with a Fourier transform spectrometer system constructed at the University of Illinois (5). Interferograms of a He-Ne laser, a n argon-filled iron hollow cathode lamp, and the transmission of a n interference filter were measured in the visible and near infrared spectral regions. Portions of the continuous interferograms for each of these three sourcp [He-Ne laser ( A ) , iron hallow cathode lamp (B), 8400-A interference filter transmission (C)] are shown in Figure 1. These illustrate typical interferograms t o be expected from these sources. Only very small portions of the interferograms for the He-Ne laser and the iron hollow cathode lamp are shown in Figure 1, representing a change in retardation of about 44 p in both cases. The complete interferograms were measured with a change in retardation of about 1120 p. In the case of the 8400-A interference filter, the complete interferogram is shown (120-p retardation change). The line shape of a n interference filter transmission approximates a Lorentzian. Thus, the interferogram would be expected t o be a n exponentially damped cosine wave as observed. Computer Calculations. All spectra were calculated on a n IBM 360 computer utilizing a software system developed for the reduction of interferograms t o spectra (5). This included apodization, Fourier transformation using the CooleyTukey algorithm, and the calculation and utilization of phase information. All spectra were plotted using the standard CALCOMP plotting system. SAMPLING RATE AND THE EFFECTS OF UNDERSAMPLING How often must the interferogram by sampled? This is one of the first questions that must be asked when making a spectral measurement with a Fourier transform spectrometer. The answer t o this question is found in information theory (3) R. R. Ernst and W. A. Anderson, Reu. Sei. Zitsfrum., 37, 93 (1966). (4) R. R. Ernst, "Advances in Magnetic Resonance, Vol. 2," J. S . Waugh, Ed., Academic Press, New York, N. Y., 1966, p 1. ( 5 ) Gary Horlick, Ph.D. Thesis, University of Illinois, Urbana, Ill. 61801.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
14220
14574
14928
15282
15636
15990 WAVENUMBERS (ern-')
16344
I6698
17052
17406
17760
Figure 4. Spectrum of a He-Ne laser
(6, 7). An interferogram, or any waveform that is a function of time or distance, must be digitized at a sampling rate that is twice the bandwidth of the system in order that the spectrum can be accurately recovered. A number of experimental parameters and components in the Fourier transform spectrometer system determine this bandwidth. The spectral region under investigation is the primary factor determining the range of frequencies that constitute the bandwidth of the system. In the case of a rapid scan instrument (1 sec) in the middle infrared (40-4 p ) the frequency content is in the low audio range (25-250 Hz). For slow scan instruments, such as the one constructed for this investigation, the frequency content is sub-audio. However, in each case the physical distance that the mirror must be moved to modulate any particular wavelength is the same. Thus, to determine the sampling interval, the amount the retardation must change in order to modulate the highest optical frequency (shortest wavelength) passed by the system (6) R. Bracewell, “The Fourier Transform and Its Application,” McGraw-Hill Book Co., New York, N. Y., 1965. (7) T. Kobylarz, Electronics, 41 (8), 124 (1968).
14220
13866
13512
13158
12304
12450 WAVENUMBERS
through one-half period is calculated and then the interferogram is sampled at this interval. This becomes a sampling rate if the velocity of the mirror is taken into consideration. Note that the sampling interval calculated on this basis (proper sampling of highest frequency) involved the simplifying assumption that the bandwidth of the interferogram extended from some maximum frequency to zero frequency. This is, of course, not true in real practice. The spectral region actually measured, Le., the bandwidth, will be limited by the components in the Fourier transform spectrometer system. These will include the optical transmission of the interferometer as determined by the beamsplitter material, the beamsplitter substrate, the compensator, the reflectivity of the Michelson mirrors, and any input and output mirrors or lenses; the spectral and frequency response of the detector; the frequency response of the detector amplifier; and the time constant of the final readout device. The optical and electronic filters used in the instrument never have perfectly sharp bandwidth cutoffs. As a result there is the possibility of aliasing (6, 7) of high frequency noise and unwanted signal. Aliasing refers to the improper
12096
11742
11388
11034
10680
(sm-11
Figure 5. Spectrum of an iron hollow cathode lamp ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
1363
14220crn-’
SPECTRUM A
0 . 7 ~SAMPLING
INTERVAL
7110crn-1
Figure 6. Absorption spectra of an 8400-A interference filter sampling of frequencies greater than those sampled correctly by the sampling interval used. These improperly sampled high frequencies show up as spurious low frequencies in the final spectrum. This is shown in Figure 2. Spectrum A resulted from the Fourier transformation of a synthetic interferogram which contained three frequencies each with a different amplitude. The interferogram was sampled at a rate such that all frequencies from zero to point X were sampled correctly. Then every other sampling point was dropped, the spectrum was calculated again and Spectrum B resulted. Now the interferogram is sampled so that frequencies out to point Y are sampled correctly. Peaks 2 and 3 now appear as completely spurious low frequencies and their position can be predicted as a folding over about the central point of the original frequency axis, when, as was done in this case, the number of sampling points is exactly halved and their spacing doubled. Peak 1 remains at its proper position on the frequency axis as it is still sampled correctly in the interferogram at this lower rate. Thus if an interferogram signal has a maximum frequency of 200 Hz (point X)it must be sampled at a rate of at least 400 Hz. In the case of undersampling as described above a signal of 175 Hz (peak 3) appears as a signal of 25 Hz; Le., it has an alias of 25 Hz. A 125-Hz signal (peak 2) has an alias of 75 Hz. This can readily be appreciated by a simple exercise. A 175-Hz sine wave sampled at a 400-Hz sampling rate is shown in Figure 3A. If every other point is dropped out simulating the undersampling of Figure 2 and the remaining points are connected (Figure 3B), a 25-Hz sine wave is the result. If this same exercise is repeated for a 125-Hz sine wave, a 75-Hz sine wave will be the result. It should be noted that if Peak 1 (Figure 2) did not exist, the fold over of Peaks 2 and 3 would not be serious as it occurs in an accurately predictable manner. This simply illustrates that it is the bandwidth and not necessarily the highest frequency of the waveform that is important when determining the sampling interval. If no low signal or noise frequencies exist, then the high frequencies of interest can be allowed to fold over without introducing any error. However, care must be exercised in ensuring the absence of spurious fold over and also in the proper labeling of the frequency axis in order that the interferogram can be transformed to an accurate spectrum. 1364
EFFECT OF SAMPLING INTERVAL ON FREQUENCY AXIS PRESENTATION It is important to see how the choice of sampling interval determines the labeling of the frequency axis because the validity and accuracy of the final spectrum is only as good as this knowledge of the frequency axis. With most Fourier transform spectrometers, it is not possible to have complete freedom of choice of the sampling interval. It is usually set by the angular size of step for a stepping motor drive, or the wavelength of a monochromatic line chosen as a reference wavelength. The reference line used in the interferometer constructed for this work was a neon line at 7032.4 A (14220 cm-l). If one sampling point is taken for each cycle of the reference line, that is for every 0.7-p change in retardation, wavelengths up to 1.4 microns (71 10 cm-l) can be sampled correctly. Thus a bandwidth of 7110 cm-l can be covered. If modulation frequencies resulting from wavelengths shorter than 1.4 microns are optically or electronically removed, then the frequency axis of the final spectrum will extend from zero to 7110 cm-1. With appropriate filtering, this bandwidth could equally well cover the regions of 14220 to 7110 cm-l, 28440 to 21330 cm-I, or 14220 to 21330 cm-’. These constraints are rigorous with respect to this specific sampling interval (0.7 p ) and the 7110 cm-I bandwidth cannot be centered at any other frequency positions. If the frequency of the reference line is divided by two, the sampling interval becomes 1.4 p and the bandwidth that can now be covered is 3555 cm-l. The regions that can be covered in the infrared and visible are listed in Table I. Again it should be emphasized that these are the only regions that can be covered starting with the 0.7-p reference line, dividing its frequency by two and thus sampling at 1.4-p intervals. To utilize any specific bandwidth, all wavelengths outside of it must be filtered out in some fashion in order to avoid aliasing. Thus starting with a certain fixed sampling interval, a specific bandwidth can be covered and that bandwidth is centered at certain specific spectral regions. Spectra of a He-Ne laser, an iron hollow cathode lamp, and the transmission of an interference filter have been measured and calculated in order to illustrate the above points. The spectrum that was calculated from an interferogram measured using a small He-Ne laser as a source is shown
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
2
Table I. Bandwidths Covered with a 1.4-p Sampling Interval Region No. Region, crn-l Region, p 1 0- 3,555 a-2.8 1 . 4 -2.8 2 7,110- 3,555 3 7,110-10,665 1.4 -0.94 4 14,220-10,665 0 . 7 -0.94 5 14,220-17,775 0.7 -0.56 6 21,330-17,775 0.47-0.56 7 21,330-24,885 0.47-0.4 8 28,440-24,885 0.35-0.4 Table 11. Argon Emission Lines Peak No. 1 2 3 4
I
3
4
5 6 7 8 9 10
11 12 14 15 16
**-rrr
&
Figure 7. Effect of missed ( A ) and extra ( B ) points on the spectrum of a He-Ne laser in Figure 4. The interferogram (See Figure 1) was a slowly damped sinusoidal waveform and was sampled at 1.4-p intervals, thus the plot covers a bandwidth of 3555 cm-l. The resolution was approximately 8.4 cm-I (3 A) as the interferogram was double sided and 1710 points long. Triangular apodization was used. The wavelength of the laser source is 6328 A (15,803 cm-1) which falls in region 5 of Table I. The simplicity of the laser source provides the bandwidth limiting in this case. A check on the validity of this choice of axis labeling is provided by calculating the wavelength of the laser line from the plot parameters. The plot is 1024 points long and the maximum of the laser line is at point 456. Assuming that the plot starts at 14220 cm-I and is 3555 cm-’ long, the wavelength of the laser line is calculated to be 6328 (15804 cm-I). The spectrum of an iron hollow cathode lamp (Argon filler gas) in the near infrared is shown inoFigure 5 . The resolution was approximately 10 cm-l (4 A) as the interferogram (See Figure 1) was double sided and 1540 points long. The interferogram was sampled at 1.4-p intervals, and thus the bandwidth is again 3555 cm-l. However, the bandwidth is now limited by a Corning glass filter (No. 7-69) and the spectral response of the Si cell detector (8)
A
(8) W. L. Wolfe, Ed., “Handbook of Military Infrared Technology,” Office of Naval Research, Department of the Navy, Washington, D. C., 1965.
Measured values cm-I A 13324 7505 13307 7515 13095 7636 12946 7724 12578 7950 12488 8008 12474 8017 12338 8105 12321 8116 12099 8265 11866 8410 11734 8522 10971 9115 10960 9124 10839 9226
Listed values, A 7505.1 7514.6 7635.1 7723.7, 7724,2 7948.2 8006.2 8014.8 8103.7 8115.3 8264.5 8408.2 8521.4 Fold over 9123.0 9224.5 ~
to region 4 of Table I, 14220 to 10665 cm-I (0.7 to 0.94 p). Again a check of the validity of this axis choice can be provided by calculating the wavelengths of the major lines present in the measured spectrum and comparing these with literature values. A listing of the wavelengths of the major lines as calculated by the computer when the interferogram was’ transformed to the spectrum is contained in Table 11. The maximum of each peak was found, and using this value, the wavelength of the line was calculated as for the laser line. A more accurate determination of the peak maximum could be made using a program that fits the region around the peak with a cubic equation and then differentiates the equation to find that maximum (9). Also contained in Table I1 is a listing of major Argon emission lines in the near infrared as tabulated by Zaidel (10). All the lines are argon lines and essentially all the wavelengths match within the limit of the plot accuracy, which is y e part in a thousand (1024 words) or approximately 2.2 A. The side peak at 9115 A (10970 cm-l) is the result of the fold over (aliasing) of a strong argon emission line at 9657.8 A. Three absorption spectra of an interference filter are sh9wn in Figure 6. The peak of the transmission is at 8400 A as measured with a Cary 14. The source was a tungsten bulb. The bandwidth of the system is limited by the spectral simplicity of the interference filter transmission and the spectral response of the Si cell detector. Each spectrum resulted from the transformation of an interferogram which was sampled at three different intervals. The interferogram (See Figure 1) was about 120 p long and the same length was used in each case. Therefore the resolution (about 170 cm-l) is the same for each spectrum. Pertinent information about the plots and a list of the wavelengths of the (9) R. N. Jones, Appl. Opt., 8, 597 (1969). (10) A. N. Zaidel’, V. K. Prokof‘ev, and S . M. Raisku, “Tables of Spectrum Lines,” Pergamon Press, Ltd., Oxford, England, 1961.
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
1365
Spectrum
Table 111. Summary of Parameters from Figure 6 Sampling interval, p Bandwidth, cm-1 Axis, cm-l
Transmission peak, A
7110 3555 1777
8412 8407 8405
B
0.7 1.4
C
2.8
A
transmission peak as calculated by the computer are summarized in Table 111. This last example clearly shows how the choice of sampling interval and the resulting frequency axis presentation are intimately related. In addition all the examples illustrate that some a priori knowledge of the spectral bandwidth under investigation is necessary in order to accurately calculate the resulting spectrum. EFFECT OF MISSED AND EXTRA SAMPLING POINTS It is possible in some experimental systems for a sampling point to be missed or for an extra sampling point to be taken. Thus it was necessary to determine what effect such occurrences would have on the resulting spectra. It
14220-7110 14220-10665 10665-12442
was felt that computer simulation of a missed or extra point in a normal interferogram would provide the best practica1 indication of the effects to be expected. Interferograms of a He-Ne laser, an iron hollow cathode lamp, and the transmission of polystyrene were used. The effect of a missed point was simulated in an interferogram that resulted from the measurement of a He-Ne laser source. The interferogram was double sided, 1710 points long, and was triangularly apodized before the dropped points were simulated. A single point was dropped out at three different locations frcm the center of the interferogram and then the rest of the points were appropriately indexed. For example, if point 1000 was dropped, the values of points 1001 to 1710 would be reindexed as 1000 to 1709. This then results in an abrupt phase shift at point 1000 and the
A
B
Figure 8. Effect of an extra point on the spectrum of an iron hollow cathode lamp 1366
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
Figure 9. Effect of an extra point on a polystyrene spectrum Fourier summation will not proceed properly. This would tend to be more serious the closer the dropped point is to the center of the interferogram. The effect this has on the resulting spectrum is shown in Figure 7A. Peak 1 is the normal case. For peak 2 the missed point occurred at point 1000, for Peak 3 at 1300, and for Peak 4 at 1600. Note that Peak 2 is severely distorted and as the missed point is moved further out from the center of the interferogram (Point 855) the distortion becomes less severe as predicted above. Since the source is a laser line, these figures approximate the resolution function of the spectrometer if a data point is missed. It is obvious that this type of error is very serious and must be completely avoided if accurate spectra are to result. In a similar manner, the effect of an extra point was simulated on the same interferogram. This time a point was added between two present points at three different locations from the center of the interferogram and the series of data points were appropriately reindexed. For example, if a point was to be added between points 1000 and 999, it would be indexed as point 1000 and its value set as the average value of the original point 1000 and point 999. The values of the original points 1000 to 1710 would be reindexed as 1001 to 1711, point 1711 having a value of zero. This
would result in a discontinuous phase shift similar to that caused by the missed point and a similar effect would be expected. This is shown in Figure 7B. Peak 1 is again the normal case. For Peak 2, the extra point was set at point 1000, for Peak 3 at 1300, and for Peak 4 at 1600. Again, for the point closest to the center of the interferogram, the peak is severely distorted. Thus, an extra point is also a very serious error and must be completely avoided. The effects that an extra point has on line and continuous spectra are shown in Figure 8 and Figure 9, respectively Figure 8 A is the normal iron hollow cathode spectrum and Figure 8B shows the distortion that results when an extra point is added at point 1100. The interferogram is 1540 points long, and the center is at point 800. Figure 9A is a normal low resolution polystyrene single beam absorption spectrum, and Figure 9B is the resulting spectrum when an extra point is added at point 270. The interferogram is 490 points long and the center is at 235. Triangular apodization was used in all cases. The overall effects are a distortion of normal bands and lines and an overall reduction in resolution, both of which are very hard to quantitate. The main point to realize is that both errors are very serious and must be avoided at all costs. This computer simulation study proved effective
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
1367
C
A
I ; B
I
G
E
Figure 10. Effect of a bad point on a spectrum in illustrating their severity and also indicated the types of’ effects these errors have on a spectrum and thus allows for their recognition and, hence, detection. EFFECT OF A BAD SAMPLING POINT A complication that can arise with analog-to-digital converters and readout systems is that a so-called bad point can sometimes occur. In this case a n erroneous value is assigned to a sampled point, often orders of magnitude different than what it should have been. This type of error has been simulated in a n interferogram and the results are shown i n Figure 10. This type of error is completely intolerable in Fourier transform spectrometry as can be seen from the spectra resulting from the transformation of such interferograms. Even the small bad point in interferogram C causes visible fringing in spectrum D. A comparison of spectra F and H calculated from interferograms E and G shows that the frequency of the fringes produced in the spectrum depends on the position of the bad point. Higher frequency fringes are produced by points further from the central fringe. This type of error could also be caused by a noise spike on the wing of a n interferogram. This error is most serious when the interferogram is measured in one scan. If several scans are time averaged, this type of error may be minimized if it occurs in the measurement step rather than during the analysis of the interferogram. 1368
Some of the advantages of rapid and repetitive scanning of the interferogram are discussed by Mertz (11). RESOLUTION OF THE ANALOG-TO-DIGITALCONVERTER An interferogram has a wide dynamic range, especially when a broadband source is observed, and the small fluctuations on the wings are important. This dynamic range, in general, is greater in the interferogram than in the spectrum (12). This can be seen intuitively because the central fringe intensity represents the summation of all the frequencies in the spectrum when the interferometer is fully compensated. Also the process of Fourier summation is capable of detecting small sinusoidal fluctuations buried in the noise on the wings of the interferogram. Thus the analog-to-digital converter must have a high resolution, on the order of 12 to 14 bits in order to avoid introducing digitization noise when high resolution spectra are to be calculated from interferograms. Twelve-bit resolution in a n analog-to-digital converter means that the peak input voltage can be divided into a (11) L. N. Mertz, J . Phys. (Paris), Suppl., 28, C2-87 (1967). (12) I. Coleman and L. N. Mertz, “Experimental Study Program to Investigate Limitations in Fourier Spectroscopy,” Block Engineering, Inc., Cambridge, Mass. 02139. This publication is available from the Clearing House for Federal Scientific and Technical Information, Springfield, Va., 22151. The order number is Ad 665,890 (1968).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
I
c
A
E
G
A
Figure 11. Effect of insufficient resolution in the analog-to-digital converter maximum of 4096 levels (212). Thus a signal with a dynamic range of up to 4000 to 1 can be accurately digitized. The effect of not having enough resolution in the analogto-digital conversion step was simulated o n a computer using a low resolution polystyrene interferogram. The decimal digital value of each point was converted to a thirteen-bit binary number plus a sign bit using a successive approximation method. The most significant bit had a value of 4.096 volts and the least significant bit a value of 0,001 volt. The decimal digital values of the interferogram points could be easily regenerated from the binary numbers by a simple weighted summation on the computer. It is a simple matter to set one or several of the least significant bits equal to zero before regenerating the decimal digital values of the interferogram samples. Thus if the last four least singificant bits are set equal to zero, the resolution is only 9 bits and the value of the least significant bit would now be 0.016 volt rather than 0.001 volt. The interferograms and spectra resulting from a series of such truncations are shown in Figure 11. Spectrum B resulted from the transformation of 13-bit interferogram A , spectrum D from 8-bit interferogram C , spectrum F from 6-bit interferogram E, and spectrum H from 4-bit interferogram G. The main effect is a loss in resolution as would be expected when the small
fringes o n the wings of the interferogram are not properly digitized. It is interesting to note that the resolution of an analogto-digital converter can be improved by the addition of random noise (13, 14). If the amplitudes of the signal modulation frequencies are smaller than the size of the least significant bit, as they might be on the wings of the interferogram, they will not be digitized accurately. However, a random noise signal can be added to the interferogram signal. This raises the instantaneous signal level so that it is digitized accurately. A number of scans must be made and averaged in order to reduce the noise level to its original level, or in the case of Fourier transform spectrometry, band limited random noise can be added such that its frequency spectrum does not overlap with that of the signal frequencies. RECEIVED for review May 13, 1970. Acception July 27, 1970. Taken in part from the Ph.D. Thesis of Gary Horlick, University of Illinois, 1970. This work was supported, in part, by a Division Summer Fellowship awarded to one of the authors (G.H.) by the Analytical Division of the American Chemical Society. (13) J. Butterworth, P. E. MacLaughlin, and B. C. Moss, J. Sci. Instrum., 44, 1029 (1967). (14) I. Coleman, J . Opt. SOC.Amer., 56 (8), IV (1966).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 12, OCTOBER 1970
1369
