I AIDS FOR ANALYTICAL CHEMISTS Fourier Domain Interpolation of Sampled Spectral Signals G. Horlick* and W. K. Yuen Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada, T6G 2E1
There is increasing utilization of electronic image sensors as spectrochemical detectors ( I ) . In most spectrochemical applications, the image to be measured with the electronic image sensor is a spectrum as formed in the exit focal plane of an instrument such as a monochromator or a spectrograph. The image sensing and charge storage elements of some sensors such as photodiode arrays and charge transfer (coupled) devices are discrete. Thus, the spectral image is sampled by the discrete array of sensors. As with any sampling operation, it is important to ensure that the sampled set of image points is an accurate representation of the original image, i.e., to ensure that aliasing does not occur. In a study of digitization, Kelly and Horlick (2) derived the sampling interval necessary in order to sample several common peak shapes with a given accuracy. Their criteria expressed as “number of samples per full-width a t half-height” are given in Table I. These criteria were developed under the constraints that no prior knowledge existed about the shape of the peak and that the accuracy of digitization would be determined by the accuracy with which the digital data could be converted back to an analog signal that could be exactly superimposed on the original signal. The per cent error tabulated indicates how much the regenerated peak height could deviate from that of the original peak. In our work utilizing photodiode arrays as spectrochemical detectors (3, 4 ) , it was necessary to select an entrance slit width and, hence, spectral line width that was compatible with the sampling interval capability of the sensor. The individual photodiodes are on 25.4-pm (0.001-in.)spacing. Coupling with our monochromator, which has a dispersion of about 20 &mm, results in an Angstrom/diode value of 0.5. Based on the criteria presented in Table I, it was felt that about 5 samples should be taken across the full-width a t half-height of the spectral line in order to avoid serious sampling errors (assuming a predominately Gaussian shape) and to provide a relatively smooth peak profile. Thus, the spectral lines in the focal plane should be about 2.5 A wide. This was approximately achieved with an entrance slit width of 100 pm. Discrete array sensors were recently criticized for requiring a somewhat large number of sensors to sample a spectral line in order to avoid aliasing (1).The above criterion (which results in about 11or more sensors across a complete line) does seem, in an intuitive sense, to require too many sensors to measure too few spectral resolution elements. However, it was not established solely on the basis of avoiding aliasing. In sampling peak type signals, it is often desirable to oversample in order to generate a well defined peak shape. This facilitates measurement of peak parameters such as position, height, and area, but is inefficient with respect to the number of data points that must be acquired and processed. Using interpolation techniques, it is possible to optimize and, hence, minimize the number of samples that must be acquired without loss of precise peak parameter information. A simple and versatile interpolation method for peak type spectral signals can be based on a zero filling technique utilized in Fourier transform spectroscopy.
In Fourier transform spectroscopy, poor photometric accuracy and inaccurate band shapes can result for lines of width less than or equal to the resolution at which the measurement is made (5).The basic problem results from the fact that strict application of Nyquist’s sampling theory to interferograms or any conventional spectral signal results in the calculation or measurement of only one output data point per resolution element. For narrow lines, this results in a rather jagged display of the band shape even though the signal has been rigorously sampled. Griffiths indicates that “zero filling” (extending the interferogram with zeros) is an effective way of overcoming this problem ( 5 ) .Zero filling in the Fourier domain amounts to interpolation in the spectral domain by an interpolation function equivalent to the Fourier transform of the apodizing function used on the interferogram. This Fourier domain interpolation provides a solution to the apparent oversampling necessary with discrete array sensors. The spectrum of the Mn triplet at 403 nm as measured with the photodiode array spectrometer is shown in Figure la. The entrance slit was 100 pm and clearly the triplet (403.449, 403.307 and 403.076 nm) is not resolved. Based on our suspicion that we are oversampling this spectral peak, it should be possible to reduce the spectral slit width and perhaps resolve the triplet. This is true, as can be seen by the spectrum measured with a 15-pm entrance slit width (Figure Ib). However, as with the case of Fourier transform spectroscopy discussed above, the peaks are somewhat jagged and, for example, it is difficult to measure accurate peak positions and intensities from this spectrum. The spectra measured for Figures l a and l b were 128 points long, although only that portion in the vicinity of the Mn triplet is shown. The peak maxima in Figure l b are located at points 95,97, and 102. Using the two outside peaks as reference lines, there are 0.0533 nm per point. The wavelength of the central peak can be calculated from this value and it is 403.342 nm as compared to the listed value of 403.307 nm. Using Fourier domain interpolation, more accurate results can be obtained. The Fourier transform of the spectrum shown in Figure 1b was taken and the result is shown in Figure 2. It was zero filled to 512 points and retransformed. The resulting spectrum is shown in Figure IC.Again only that portion in the vicinity of the Mn triplet is shown; however, because ~
Table I. Number of Samples for Common Peak Shapes Maximum error, %
of peak
height 10 1 0.1
0.01 0.001
Samples per full width a t half-height Triangle 5.5 40 357
... ...
Exponential Lorentz 5.9 50
454
... ...
Gaussian
1.8
1.5
3.6 4.8 6.3 8.3
2.2 2.6
ANALYTICAL CHEMISTRY, VOL. 48, NO. 11, SEPTEMBER 1976
3.0 3.3
1643
03.076
403 307
128 POINTS
b l L
I
ZERO FILLED TO 512
-
Figure 2. Fourier transform of the spectrum shown in Figure l b illustrating Fourier domain interpolation (zero filling)
L
Figure 1. Spectra of the Mn 403 nm triplet: ( a ) 100-Mrn slit width, ( b ) 15-pm slit width, and ( c ) 15-prn slit width after Fourier domain interpolation. Wavelengths are given in nm
of the zero filling (Le. interpolation), there are three times as many points. The peak maxima in Figure ICare now located at points 375, 386, and 404. Using the same method as before, there are now 0.0129 nm per point and the calculated wavelength of the central peak is 403.308 nm which is in excellent agreement with the listed value of 403.307 nm. No apodizing function was applied before zero filling and retransformation. This results in interpolation of the spectrum with the monochromator line shape function. If desired, apodization techniques could be used to control the nature of the interpolation function. This is necessary in some cases to avoid generation of excessive side lobes ( 5 ) . The spectra shown in Figure 1indicate that Fourier domain
interpolation is quite useful in optimizing the sampling capability of discrete array image sensors. The technique is general and, hence, applicable to any sampled spectral signal. As a final comment, interpolation does not mean that we can successfully violate the sampling criteria presented in Table I. Interpolation by its very nature means assumption of some type of line shape function with which to interpolate; either the instrumental line shape function or, via apodization, some imposed interpolation function. The criteria presented in Table I are based on the premise that there is no prior knowledge of the signal shape and, if it is desired to ascertain within a given accuracy that a peak signal is a particular shape, then these sampling criteria must still be followed.
LITERATURE CITED (1) (2) (3) (4) (5)
Yair Talmi, Anal. Chem., 47, 658A (1975). P. C. Kelly and G. Horlick, Anal. Chem., 45, 518 (1973). G. Horlick and E. G. Codding, Anal. Chem., 45, 1490 (1973) G. Horlick, Appl. Spectrosc., 30, 113 (1976). P. R . Griffiths, Appl. Spectrosc., 29, 11 (1975).
RECEIVEDfor review December 29,1975. Accepted April 30, 1976.
Method to Reduce Noise in Silver Nitrate-Benzyl Cyanide Columns Fred B. Wampler University of California, Los Alamos Scientific Laboratory, Los Alamos, N.M. 87545
A recent photochemical kinetic study in this laboratory required the determination by gas chromatography of minute mol, of cis-2-pentene in the presence of quantities, 6 X 1.2 X mol trans-2-pentene. The literature available for the separation of the isomeric 2-pentenes (1-3) suggested a column prepared by dissolving silver nitrate in some nonvolatile polar solvent. Ethylene glycol, propylene glycol, and benzyl cyanide are most commonly used. Columns prepared using benzyl cyanide as a solvent were found to be superior to the other suggested solvents. A 12-ft column, %-inch o.d., consisting of 10%by weight of silver nitrate on 60/80 mesh Firebrick, operated at 22 O C and at a He flow rate of 30 cm3/min, was found to give a very good separation of 30 min between the two isomers when large quantities, mol, of each isomer were present. However, even when matching reference and sensing columns were used in the Perkin-Elmer Model 810 FID gas chromatograph, a noise level from column bleed hindered the operation of the gas chromatograph at the sensitivity level necessary to detect 6 X mol of the isomers. The signal from the eluting solvent, benzyl cyanide, caused the recorder to go off scale at the high 1644
ANALYTICAL CHEMISTRY, VOL. 48,
NO. 11,
sensitivity levels required. Conditioning of AgN03 columns at elevated temperatures cannot be done very well since solutions of AgN03 decompose above 50 T.Also, i t was desirable to have the benzyl cyanide present on the column support since columns prepared from it were far superior to those using propylene or ethylene glycol in resolving the two isomers. It was found that by inserting a 12-ftlength of a 8%DC-550 silicone oil column between the detector and the AgN03 column that the benzyl cyanide was completely retarded in the silicone oil column, but that the separation achieved for the isomers in the AgN03 column was not diminished since the order of elution, trans first, was the same for both columns. However, the use of this length of silicone oil column by itself could not resolve the isomers. Analyses performed with and without the silicone oil column indicated that the resolution of isomers was the same in both cases but that, for high sensitivity runs, the silicone DC-550 column was necessary to retard the benzyl cyanide from entering the detector. Under the operating conditions, the benzyl cyanide took about 4 weeks to start eluting from the silicone oil and a t this time a replacement silicone oil column was inserted. This technique
SEPTEMBER 1976
