Fourier Method for Digital Data Smoothing in Circular Dichroism Spectrometry C. Allen Bush Department of Chemistry. lllinois lnstitute of Technology. Chicago, 111. 60676
A digital smoothing method for circular dichroism spectra utilizing the Fast Fourier Transform algorithm and its truncated inverse is described. We introduce a new method for selecting the point of truncation of the Fourier coefficients involving computation of the standard deviation of the short period coefficients from zero. The data acquisition method for the circular dichroism spectrometer involves digitization of the modulated (ac) signal, the total (dc) signal and a voltage proportional to the wavelength. Data are recorded at medium speed on a digital magnetic tape which may be read by a general purpose computer. All mathematical manipulation, including filtering, base-line subtraction, signal averaging and plotting of the spectra is done on a large computer programmed in FORTRAN. For an idealized case, signals may be recovered from noise as large as twice the signal and for a practical case of nucleoside circular dichroism, negligible curve distortion is found for circular dichroism spectra with a signal/noise ratio of five.
Circular dichroism (CD) spectra, like many other types of spectra of interest to chemists, often suffer from a poor signal-to-noise ratio. In the case of electrically allowed absorption bands, the ratio of CD to total absorption is often as small as 10- 4. Although modern CD machines can measure these effects, the signal-to-noise ratio is often rather poor and any long period fluctuations in the light source make it necessary to repeat an experiment several times before the experimentalist is convinced of the reliability of his measurement. Published CD spectra are often deceptively smooth in appearance as a result of considerable smoothing and averaging which is usually done by techniques which are intuitive rather than mathematically precise. Such smoothing is possible if one has the forehand knowledge that no sharp features exist in the CD spectrum. For the case of electrically allowed transitions of the T - T * type in organic chromophores, the absorption spectrum is usually a slowly varying function of the wavelength and vibrational structure is small or absent. Therefore, one expects the CD curve also to be smooth and without any sharp features. It is this fact which allows one to draw a smooth line through the noisy spectrum of such compounds leading to the commonly published spectra. The smoothing procedure can be mathematically formalized using digital procedures if one has a method for digital recording of the actual CD spectrum. A smoothing procedure for CD spectra utilizing a piecewise sliding polynomial fit has been previously described by Tinoco and Cantor ( I ) . This technique is based on the method of Savitsky and Golay ( 2 ) . In the pages which follow, I will describe a digital smoothing scheme for CD curves which is based on Fourier methods. Since the general features of Fourier digital (1) I. Tinoco and C. R . Cantor, Methods Biochem. Anal.. 18, 81 (1970). (2) A . Savitskyand M. J. E. Golay, Anal. Chem., 36, 1627 (1964).
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filtering have been extensively discussed in the literature, I will give only a brief outline of the principles (3, 4 ) . The CD spectrum is measured in the usual way and the signal plus accompanying noise is sampled and digitized. The Fourier expansion coefficients of the CD spectrum are then computed using the fast Fourier transform digital algorithm. Since the true spectrum is anticipated to be a smooth function of the wavelength, the Fourier coefficients corresponding to short period variations can be assumed to describe primarily the noise. One can set these Fourier coefficients equal to zero and compute the inverse Fourier transform retaining only the long period coefficients which describe mainly the signal. The resulting spectrum is a smooth curve which is a faithful representation of the true CD signal. The present work describes a particularly simple and relatively inexpensive method for digital sampling of the spectrum. We record the digitized data on a magnetic tape which is capable of medium speed data rates. All the mathematical manipulation necessary for digital filtering, base-line subtraction, scaling, signal averaging, and plotting of the resulting curve is accomplished on a large general purpose digital computer. This method avoids the difficulties inherent in programming a mini-computer while retaining reasonably fast data acquisition rates of 1 to 20 data points per second. I will also propose a scheme for choosing the optimum cutoff in the Fourier coefficients so as to separate signal from short period noise. The technique is sensitive both to the noise level and to the signal complexity so it should be applicable not only to slowly varying functions of the wavelength but also to spectra having some structure. Thus, this filtering method may find application in other kinds of spectrometry such as Raman or infrared. METHODS D a t a Acquisition Hardware. For reasons outlined above, we chose to make the data acquisition hardware as simple as possible consistent with reasonable data rates. This latter requirement rules out the use of a Teletype paper tape punch which is certainly the least expensive digital recorder. We chose instead an inexpensive 7-track incremental magnetic tape recorder. The tapes written by such a recorder can be read by almost any general purpose digital computer. While the data recording rates are not as fast as those obtainable by direct recording in the memory of a dedicated computer, such a tape drive can be operated at 200 to 300 characters per second and the storage capacity is essentially unlimited. In order to explain exactly which voltages must be recorded, it will be necessary to briefly describe the measurement of circular dichroism. Most CD machines use a light modulator driven by a fixed frequency alternating current. The modulator produces alternatively left and right handed circularly polarized light. Most popular commercial instruments including the Cary 6003, the (3) G Horlick Anal Chem 44,943 (1972) (4) J W Hayes D E Glover D E Smith and M W Overton Anal C h e m , 45,277 (1973)
Jasco and the Jouan use a Pockels cell driven by an ac voltage having a frequency in the audio range. The modulated beam passes through the circularly dichroic sample and is detected by a photomultiplier. The detected voltage is composed of a small ac component which is in phase with the modulator and a large dc component which is proportional to the total transmitted light intensity, IO. The ac signal is proportional to ZI - I,, the difference in intensity of left and right handed circularly polarized light transmitted by the sample. The CD may be shown to be proportional to the ratio of the ac signal to the dc signal ( 5 ) .In the commercial instrument, this ratio is computed by an analog slide-wire recorder. In our scheme, we chose to record these two voltages separately and to compute the ratio digitally. Our recording speed is fast enough so that the time necessary to record an extra voltage is not a serious problem. The CD machine in our laboratory is a Cary 60 with model 6003 CD accessory. In this instrument, the ac signal is available a t TF'-4 of the upper chassis and the dc voltage may be taken from terminal 1 of TS-600. The voltage a t TP-4 is only 30 to 40 mV for weak CD signals so we included an operational amplifier to amplify this voltage by a factor of 10. We also included an adjustable offset voltage to facilitate convenient setting of the base line. The dc voltage is maintained relatively constant a t about 1.26 volts by the Cary 6003 circuit which adjusts the photomultiplier dynode voltage in response to fluctuations in light intensity. Our simple data acquisition system makes no provision for synchronizing the recording of signal voltages with the monochromator wavelength. Therefore, we must also record an indication of the wavelength along with each signal recording. Although the wavelength could have been digitized with a shaft encoder, we chose to attach a ten-turn precision heliopot to the wavelength drive and use it as a voltage divider. This potentiometer is attached to the wavelength shaft which protrudes from the voltage programmer of the CD modulator on the left side of the Cary 60. T o summarize, there are three voltages which go together to make up one data point; the ac voltage, the dc voltage, and a voltage proportional to the wavelength. In our system, these three voltages are sampled by an analog multiplexer and digitized in the range of &2 volts with a resolution of 1 mV. The multiplexer-DVM is a Datos 901 commercially available from Data Graphics Corporation, San Antonio, Texas. The sampling rate may be adjusted in the range of 4 to 24 voltage readings per second, A voltage recording consists of the four digits of the DVM, a sign and a channel number indicating which voltage is being recorded. These six characters are serialized by a Data Graphics model DGC-300-M digital multiplexer and written in a BCD code on the 7-track tape unit, (Cipher Data Products, San Diego, Calif., Model 70). Three such voltage recordings make up a single data point in our system. The tape controller may be set to record a selected number of data points followed by an inter-record gap. After writing a chosen number of records, the system stops while the Cary 60 is readied for another scan. Figure 1 shows a block diagram of the data acquisition system. Software for Data Treatment. Neither the scan rate of the Cary 60 nor the data recording rate is exactly reproducible. Moreover, the data recording is interrupted from time to time during writing of inter record gaps on the magnetic tape. Therefore, the wavelength record is asynchronous. Nevertheless, a voltage proportional to the ( 5 ) L. Velluz, M. Legrand, and M . Grosjean, "Optical Circular Di chroism," Academic Press, New Y o r k . N.Y.. (1965).
CIPHER
DATOS-901
MULTIPLEXER
DIGITAL SERIALIZER TAPE CONTROLLER
-
WAVELENGTH POTENTIOMETER
DG C -300 M
+
Block diagram of the data acquisition system available from Data Graphics Corporation, San Antonio, Texas Figure 1.
wavelength is recorded so that the wavelength for each data point may be readily computed. Next, the ratio of the ac voltage to the dc voltage is calculated, yielding a number which is proportional to the ellipticity. In order to determine the proportionality constant, I have used the calibration data of Cassim and Yang (6) who report the molar ellipticity of d-10-camphor sulfonic acid a t 290.5 nm to be [e] = 7260. The fast Fourier transform, which will be used in the filter, requires that the data points be a t equally spaced intervals. Moreover, standard wavelengths will also become useful later in the program when base-line subtraction and signal averaging are done. Therefore, the data points are first reduced to a standard interval by linear interpolation. There is a second problem which must be addressed before the Fourier coefficients may be calculated. Since one cannot expect a spectral scan to give equal CD a t both ends of the spectral range, the periodic nature of the Fourier transform may recognize a discontinuity at the ends of the range. This problem has been discussed in connection with Fourier filtering of polarograms and may be remedied by linear detrending ( 4 ) . The detrended CD, e d ( x ) is defined as Bd(X) = O(X)
- a - b(X - A,)
(1)
where XO is the short wavelength limit of the curve. In order to select the linear detrending parameters, a and b, we calculate the average of several points a t each end of the spectrum. a and b are selected to make these averages zero in o d . The two detrending parameters are saved so that the trend may be added back to the filtered data before output. This procedure greatly reduces discontinuities a t the ends of the range and yields a substantial improvement in the behavior of the filter ( 4 ) . Once the Fourier expansion coefficients have been computed, the next problem is selection of the cutoff for the filter. In fact, the Fourier coefficients themselves can be employed for this task. It is well known that a plot of the amplitudes of the Fourier coefficients describing a spectrum having a Gaussian band shape is itself a Gaussian curve peaked about infinite period. A plot of the Fourier amplitudes for a typical Gaussian type spectrum has been given in Figure 1 of the paper of Horlick ( 3 ) .It indeed has the expected shape with the amplitudes dropping off to small values for the short period coefficients. The short period coefficients describe the noise in the spectrum and they oscillate about zero. If we compute the standard deviation from zero for these short period coefficients, it will be approximately independent of the number of coefficients included in the calculation of the standard devia(6) J. Y. Cassim and J. T. Yang, Biochemistry8, 1947 (1969)
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Table I. Standard Deviation of 128-L Shortest Period Complex Fourier Coefficients for Several Simulated Spectra Sampled at 256 Points S/Nb
0.25 0.25 0.125
2.0
0.1
0.5
0.4 0.2
1.0
0.125 a 3
5.0
Standard deviation = o(128-L)
rms noise
Aa
L
0.04
=
L = 7
32
...
S/N
0.25 0.25 0.125 0.125
2.0 0.5 5.0
a
1.0
Amplitude ofmaximum
1.005 1.029 1.058 0.96
L = 3
cutoff
0.1025 0.4093 0.2106
0.1056 0.4118 0.2237
0.1168 0.4131
... 0.4344
4 3 5
L
L = 10
L = 9
L = S
L = 7
L = 6
0.0417
0.0417
0.0466
...
...
=
32
- 1 to
+l. S/N is ratio of signal to the peak-to-peak noise.
DisplaceNumber of menta of maximum Maximum complex (in data overshoot, coefficients retained points) 5%
+1
+5 $1 $1
6
6 22 3
10
11
7
5
Points displacement out of 256 total data points.
tion. If, however, any coefficients describing signal are included in the calculation, the computed standard deviation will rise above the value characteristic of the noise. Our procedure for deciding how many Fourier coefficients are needed to describe the signal is as follows. We select a group of short period Fourier coefficients certain not to contain any signal (the upper 15/~6of the coefficients in our case). We compute the standard deviation and then repeat the calculation including longer period coefficients one a t a time until a significant rise (5-40%) in the standard deviation is noted. It will become obvious from our results below that in this method of determining the cutoff, both the noise level in the measured spectrum and the sharpness of the measured CD bands influence the cutoff of the filter. Therefore, although the CD bands we treat in this paper are all slowly varying functions of the wavelength, we have also had success using this method on the CD of phenyl alanine containing peptides which have sharp vibronic features in the near ultraviolet. Once we have determined the position of the cutoff, we set all coefficients having periods shorter than the cutoff equal to zero. We then compute the inverse Fourier transform and restore the linear trend. This filtering procedure is applied to all CD scans, both sample scans and base lines. We then subtract a selected base line from a sample scan and scale the result to units of molar ellipticity. Any number of these spectra can then be signal averaged. The smoothed, averaged spectra are then plotted on the digital plotter. In our installation, this is the University Computing Corporation model MTD-345 which is similar to the popular Cal Comp plotter. RESULTS Tests of the Filter on Gaussian Bands with Noise Added. In order to test the method outlined above, I constructed Gaussian curves which are similar to a typical CD band. These bands have a maximum amplitude of 1 a t the center of a range from -1 to +1. The widths of the bands are characterized by A , half width a t l / e of maximum. White noise was simulated by generating random numbers between -x/2 and +x/2 and adding them to the 892
L=4
0.4112 0.2118
Table 11. Comparison of Filtered Curves to Original Gaussians
A
L=5
0.1030 0.4131 0.2068 0.0413
is half width a t l / e of max., range is
L = 6
A N A L Y T I C A L CHEMISTRY, VOL. 46, NO. 7, JUNE 1974
...
...
8
Cutoff determined by 5% rise in standard deviation.
Gaussian. Since the maximum signal amplitude is 1, we can define the ratio of the signal to the peak-to-peak noise as 1/x. The peak-to-peak noise, which may be easily estimated from an experimental spectrum, is approximately five times the rms noise. These noisy curves are sampled a t 256 equally spaced intervals and the Fourier coefficients calculated. There are 128 linearly independent complex Fourier coefficients which correspond to 128 sines and 128 cosines. In order to test the standard deviation method outlined above for selecting a cutoff, we computed the standard deviation of 128-L shortest period coefficients. Our results for various values of L for several simulated spectra are given in Table I. The general features of the standard deviation method can be recognized in the data of Table I. First, the noise level in the spectrum is proportional to the standard deviation of the short period coefficents. In fact, if the Fourier coefficients are appropriately normalized (by l/vm, the standard deviation is essentially the same as the rms noise in the original signal, as may be seen in Table I. Also, the standard deviation is approximately constant unless Fourier coefficients which describe signal are included in the calculation. Second, the number of coefficients necessary to describe the signal increases as the band width of the signal decreases. Thus. the selection of the cutoff is sensitive both to the signal band width and to the noise level. For this simulated case and for a fairly broad curve (A = 0.25), a signal can be recognized in the presence of substantial noise (S/N = 0.5). Whether such performance could be expected from any real spectra will depend on whether the noise is truly “white” noise as it is in this idealized example. While we have found true Cary 6003 spectra to be somewhat more difficult to filter than these idealized examples, in the absence of severe fluctuations in the light source intensity, the real spectra behave similarly to the data in Table I. The inverse transforms of the truncated Fourier expansions for the examples shown in Table I were calculated and compared to the original noiseless Gaussian signals in order to test the fidelity of the filtering method. I performed the inverse transform keeping two coefficients above the cutoff indicated in the right-most column of Table 1. Some observation on the result are tabulated in Table 11. Comparison of the filtered scan to the original Gaussian reveals that for even the worst case (S/N = 0.5) there was only a modest displacement of the position of the maximum ( 5 parts in 256). Moreover, there is only a small error in the amplitude of the maximum (3-570). There is, however, some distortion of the curve shape in the cases of lowest signal to noise ratios. This distortion takes the form of “ringing” or overshoot of the base line. We can see in Table I1 that the amount of overshoot can be substantial (22%) for bands with very high noise levels.
I/
'
I
F i g u r e 2.
Single scan CD s p e c t r u m of adenosine in 10-3Mp h o s p h a t e buffer
at p H 7 9
The noisy spectrum is the difference between a single sample scan and a single base line The smoothed spectrum is Fourier filtered keeping 5 of 256 linearly independent complex Fourier coefficients The wavelength is in nanometers and the CD I S in units of molar ellipticity X
The error we find here due to truncation of the Fourier expansion is similar to the sampling error discussed by Kelly and Horlick (7). The error illustrated in Figure 4 of their paper is essentially the same as the ringing effect discussed above. In fact, truncation of the Fourier expansion to N sines and N cosines is equivalent to sampling a noiseless signal at 2 A' points. Therefore, we expect it should be possible to compare the error due to truncation to the error due to sampling indicated in Table I1 of ref. 7. For a fixed ratio of Gaussian width to sample interval, one should find constant error a t a level indicated in Table I1 of ref. 7 . This is true for the noiseless Gaussian functions studied by Kelly and Horlick but is not true for our data shown in Table 11. The error introduced by truncation of the Fourier representation of a noisy Gaussian in somewhat larger than that predicted by Kelly and Horlick (7). The problem of curve distortion can be alleviated by use of a shaped "window" rather than a simple square cutoff filter as we have used here (8, 9). Although we have not yet tested any filter window other than the simple square cutoff, one could certainly improve the fidelity in curve shape for difficult cases after some experimentation with the window choice. Application of the Method to Actual CD Data. For gathering some actual CD data on which to test the filtering method, we operated the data acquisition system as follows. The DVM recorded data a t the rate of 4 voltage readings per second or 1 complete data point every 0.75 second. Each scan of the spectrometer was composed of 14 records of 30 data points each for a total of 420 data points per spectral scan. The Cary 60 was operated at a ( 7 ) P. C. Kelly and G. Horlick, Ana/. Chem.. 45, 518 ( 1 9 7 3 ) . (8) J. A. Blackburn, Ed, "Spectral Analysis," Marcel Dekker, New York, N.Y , ( 1 9 7 0 ) . (9) A. J. Starshak, Thesis "Spectral Analysis of Liquid X-Ray Diffraction Data" Illinois Institute of Technology, Chicago, Ill., ( 1 9 7 0 ) .
scan speed such that about 80 nm was covered in 5 minutes. The time constant of the Cary 6003 was set a t 1 second. As stated above, cl-10-camphor sulfonic acid in water was run for the purpose of calibration. For this case, the signal is so strong that the filter has almost no effect. The performance of the filtering procedure on spectra which give relatively weak signals may be seen in the data of Figure 2 which is a machine-generated plot of both raw and filtered data for a single scan spectrum of the nucleoside adenosine. The noisy spectrum in Figure 2 is the difference between an unfiltered sample scan and an unfiltered base line. The smoothed curve is the difference between the same sample scan and base line, each of which had been filtered so as to retain 5 complex Fourier coefficients. The asymmetry ratio (Ac/c) for adenosine a t 260 nm is 5 x placing this compound in the category of difficult samples. Since the spectral scan time is only 5 minutes, it is practical to take several scans and average them. Signal averaging may be used to suppress noise whose period is longer than or comparable to that of the signal itself. Figure 3 shows an average of 3 separate spectra of adenosine. The suggestion of a shoulder in the 254-nm region in Figure 2 is confirmed by its persistence in the averaged spectrum. The possibility of resolution of the CD due to the adenine chromophore in the 260-nm region into several contributing components has been discussed previously (10, 11). The spectrum of Figure 3 is quite consistent with available data on this compound. T o counter the suggestion that the Fourier method always introduces false features into the filtered spectrum, the data of Figure 4 are offered. This curve is the average ( l o ) C A Bush J Amer Chem SOC 9 5 , 2 1 4 ( 1 9 7 3 ) ( 1 1 ) J S 1ngwall.J Amer Chem Soc 94, 5487 (1972) A N A L Y T I C A L CHEMISTRY, VOL. 46, N O . 7 , JUNE 1974
893
0 0
,
2:o.i:’
:2:.oc
23:.:c
.rc.so
.
I~
I~
27P.:?
.._I~-
?a:.&;
:32.:?
‘cc.::
E,:-,,
L-’,-
Figure 3. Average of three separate filtered s p e c t r a of adenosine in
..
~
?B?.c:
94:.:.‘
10-3Mp h o s p h a t e buffer at p H
7.9
Wavelength is in nanometers and the CD is in units of molar ellipticity X
220.02
230
.:o
1
213.11:
250
:;z.::
.:o
n;i\
~
~
. . ~ -. . ~ ~ 2 8 . .:o 23: ~~~
2’”
.:2
.::
FLEI.7-7
Figure 4. Average of three separate filtered s p e c t r a of 3’,5‘-cyclic a d e n o s i n e monophosphate in 1 0 - 2 M phosphate buffer at p H 7.9 Wavelength is in nanometers and the CD is in units of molar ellipticity X
of 3 spectra of 3’,5’-cyclic adenosine monophosphate which shows no obvious shoulders in the same wavelength region. The absence of obvious shoulders in the spectrum of this compound has been commented upon in the literature (12). (12) C . A. Bush and H . A. Scheraga, Biopolymers, 7, 395 (1969).
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DISCUSSION Since the Fourier filtering method essentially guarantees that one can draw a smooth line through almost any amount of noise, one may question the wisdom of applying it to noisy spectra. In fact, we have designed this
method for studying qetailed spectral features of compounds which give noisy spectra. The question of whether the features thus detected represent true physical properties of the compounds under study can be answered only by the test of experience. If the spectra are accurately reproducible and can be related to molecular properties, then the Fourier method can be considered useful. One should obviously be aware of the possibility of the ringing or overshoot artifact we have described above. This effect would tend to accentuate spectral shoulders in some cases. The fact that a smoothed CD curve can be represented by only 5 complex Fourier coefficients reveals the comparatively low information content of the spectra. The Fourier representation of spectra with many sharp lines such as NMR or mass spectra would require many more terms. It is just the smoothness of the CD spectrum that makes digital filtering so effective in our case. In fact, our results suggest that an improved method for reporting smoothed spectra in the literature might be to simply tabulate the Fourier coefficients rather than to reproduce the full spectrum. It is true that digital filtering has an effect similar to that of analog filtering devices placed at the output of the spectrometer. However, digital filters enjoy several advantages over analog methods. First, the shape and cutoff point of the filter may be decided after the data have been recorded. In fact, several different filters may be tested and the best one selected. Second, the digital filter considers all the data both to the left and to the right of any data point under consideration. Analog filtering can take into account only data which were recorded prior to the data point under consideration. A second question is whether the Fourier method of digital filtering has any major advantages over the Savitsky and Golay ( 2 ) method of sliding polynomial curve fitting. While the choice between the two methods is not obvious, we can point out some differences in the two techniques. The polynomial method weighs the local data more heavily, while the Fourier method weighs the entire spectrum equally. A minor problem in the polynomial method is the choice of the polynomial itself. This choice depends somewhat on the nature of the curve shape; a maximum or a minimum is best represented by a polynomial of even degree while a shoulder or inflection is better represented by a polynomial of odd degree. For any one CD curve, one can choose polynomials and intervals so as to get good smoothing. However, the Fourier method has the advantage of being easier to handle mathematically. The basis functions of the Fourier method are, unlike polynomials, orthogonal. The resulting uniqueness of the Fourier fit makes design of computer algorithms much easier. Also, statistical analysis of the Fourier filter is much more
straightforward than for the case of the sliding polynomial method. The rms noise level of the signal can be directly estimated from the standard deviation of the short period Fourier coefficients (See Table I). The noise suppression is given by the ratio of the number of coefficients kept to the number of data samples. In closing, I wish to comment on our choice of the simple data logging method with all computation done on a general purpose computer. The advantages of simple logging are fairly obvious. The hardware is less expensive than that for even a small mini-computer system. The commercial apparatus described above was purchased ready-made for about $7500. The ease of programming the large general purpose machine makes it possible to test sophisticated methods of data treatment without a disproportionate expense in programming. Moreover, the programs, which are all in FORTRAN, are exportable. Only the magnetic tape reading and the MTD-345 plot routines are unique to the equipment of the local installation. The use of a magnetic tape recorder avoids the problems of very low data recording rates of card and paper tape punches. The obvious disadvantage in using an off-site computer is the loss of immediacy of computed results. Typically, we record several hours of spectra before taking the magnetic tape to the computer center for calculation of the filters. With a dedicated computer at the spectrometer site, one could examine the filtered results scan by scan. One possible method for regaining some of the immediacy lost in our method would be the use of a medium speed data transmission over a dial-up telephone line to the large general purpose computer. The expense of a medium speed (2000 to. 4800 bits per second) terminal would sacrifice economic advantages but retain the ease of programming characteristic of the large general purpose computer. It would require 15 to 30 seconds to transmit a spectrum of the type described above over such a medium speed line. If a reasonably short turn-around time could be expected, the Fourier filter calculations could be economically performed in a remote batch mode and returned to the spectrometer site in a few minutes. Such a procedure might provide a reasonable compromise between the immediate results of a dedicated mini-computer and the flexibility of a system of the type we have described above. ACKNOWLEDGMENT I wish to thank Steve Leasure and Phillip Coduti for assistance in the early stages of this project. Received for review August 27, 1973. Accepted January 22, 1974. This research was supported by NSF Grant GP20053.
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