Anal. Chem. 1980, 52, 1335-1344
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Linear Parameter Estimation of Fused Peak Systems in the Spatial Frequency Domain David P. Binkley' and Raymond E. Dessy" Chemistry Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 2406 1
The use of fast Fourier transforms are introduced as a new and unique approach to performing linear least-squares parameter estimation in the spatial frequency domain. A new approach to determine the proper cut-off frequency is suggested and evaluated. Theoretical discussions and examples are given to explain how parameter estimation analysis can be performed In the spatial frequency domaln to resolve Pb(I1) and TI( I ) in HNO, uslng square-wave voltammetry. Experimental verification is given by application of the method to several mixtures of Pb(1I) to TI(1) in 0.9 M ULTREX FINOB. The algorithm is general and readily applicable to other methods of analysls.
Fast sweep voltammetry is a particularly useful technique for performing multielement analysis in a short period of time. T h e analysis becomes more complicated when faced with determining two electroactive species with similar half-wave potentials. As half-wave potentials approach one another, it is actually possible for the peaks of interest to coalesce or fuse, leading t o a very difficult or impossible analysis. With the development of pulse polarographic and pulse voltammetric methods of analysis, resolution of nearby peaks has become less of a problem. However, even with these newer techniques, it is possible to have peaks interfere with the analysis under investigation. Several investigators have addressed themselves to this problem through the use of computers. The approaches have taken two extremes. The first is a rather complex numerical deconvolution approach taken by Perone ( I ) . Basically, Perone developed an empirical equation which fitted a large variety of stationary electrode polarograms (voltammograms) by adjustment of appropriate constants. This function was first fit to a number of different standard voltammograms and the constants of the functions for each species of interest were stored in a library in the computer. When an unknown mixture was t o be analyzed, the constants were used t o regenerate the standard curves, a composite of which was then fitted to the unknown signal. The approach was totally mathematical in nature and required rather large programs t o generate synthetic curves. Good results were reported for widely separated systems such as a 10/1 mixture of Pb(I1) to Cd(I1) in 1.0 M KC1 (peak potential separation of approximately 200 mV). Perone also reported results for a closely spaced voltammetric system. This was a 1/1 mixture of In(II1) and Cd(I1) in 1.0 M HC1 in which the separation of peak potentials is 38-40 mV. The other extreme is the approach taken by Bond ( 2 ) . His approach was completely experimental. A Princeton Applied Research Model 374 microprocessor controlled polarograph was used for the analysis. First a differential pulse polarogram of the solution to be determined was recorded and stored in memory. It is assumed for this discussion that the unknown 'Present address: Perkin-Elmer MS 124, Main Avenue, Nonvalk, Conn. 06856. 0003-2700/80/0352-1335$01 .OO/O
contained a mixture of two electroactive species, A and B, with similar peak potentials. After the unknown had been recorded, a blank solution containing neither A nor B was placed in the polarographic cell. If A was to be determined, then increasing concentrations of B were titrated into the cell and differential pulse polarograms of the blank plus B were recorded. After each addition of B, the current-potential curve for t h a t solution was automatically subtracted from t h a t for the polarogram of the mixture already stored in memory. The resultant current-potential curve was then examined visually. After addition to the blank of the approximate amount of B present in the sample solution to be determined, the resultant differential pulse polarogram obtained by the subtraction procedures would contain the peak height of A void of interference from electrode process R. A could then be determined in the normal way by reference to a standard calibration curve. This procedure was completely general and did not depend on knowing the mathematical formulation of either electrode process. Needless to say, a t 8 to 10 min per differential pulse polarogram, the analysis was time consuming. The entire procedure would have to be repeated if B were to be determined, this time adding successive aliquots of A t o a new blank. Bond reported the analysis of trace amounts of lead(I1) and thallium(1) in 1 M ZnS04 acidified to p H 2 with H2S04. In this supporting electrolyte, the peak potentials were separated by 60 mV. Synthetic curves indicated that thallium could be determined from completely resolved curves in the presence of a t least a 50-fold concentration excess of lead (n = 2) and lead from completely resolved curves in the presence of a 100-fold excess of thallium (n = 1). Most recently, Perone introduced a multilinear least squares regression method for the quantitative resolution of peaks separated by as small as 30 mV. The technique used an arbitrary fitting function to fit square-wave voltammograms with an acceptable precision based on minimizing the sum of the squares of the residuals using the gradient expansion algorithm of Marquardt ( 3 ) . Linear least-squares parameter estimation is becoming more common in laboratory data processing systems as computer networks are becoming more available to the average chemist. In fact, a recent review article by Tracey and Brubaker has indicated the potential advantages of this data handling tool ( 4 ) . The technique has been used in such areas as controlled potential coulometry, multiple wavelength spectroscopy, and for the determination of time constant estimates in ion-selective electrodes. Linear least-squares parameter estimation is capable of handling such fused peak voltammetric systems as the closely spaced lead(I1)-thallium(1) system. As applied conventionally, i t represents a very reasonable compromise between the extremes offered by Perone and Bond. In the introduction to his book on transform techniques in chemistry, Peter E. Griffiths (5) notes that the research papers which have appeared in ANALYTICAL CHEMISTRY involving transform techniques can be grouped into two major cate0 1980 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JULY 1980
gories. The first category describes those techniques in which measurements are made using an instrument whose output is the transform of the conventional spectrum. The techniques which fall into this category are possible because of some multiplex advantage inherent in the instrument design. An excellent example of a n analytical technique which falls into this category (besides the more common FT-IR and FT-NMR experiments) is given by the work performed by Smith (6-10). Smith’s work illustrates how Fourier transforms and their associated multiplex advantage can be applied to the electrochemical technique of ac polarography. I n this paper, we introduce a totally new and unique approach to linear least-squares parameter estimation through t h e use of Fourier transforms; a n application of Fourier transforms t h a t falls into the second major category. This second category describes analytical applications in which data measured by the conventional approach are enhanced or made more meaningful through the use of a Fourier transform algorithm. By performing the linear least-squares parameter estimation calculation in t h e spatial frequency domain, we introduce a data handling application which provides significantly improved results with a substantial savings in computation time relative to t h e traditional approach. I t should be stressed t h a t the data handling technique introduced here is not discipline-specific as are those applications which fall into category one, but is quite general and could easily be applied to other areas of analytical chemistry which suffer from fused peak systems.
EXPERIMENTAL Cell a n d Electrodes. Metrohm titration vessels were used for all analyses. The cells were cleaned by first washing thoroughly with soap and water and rinsing with distilled deionized water. This was followed by several 10% nitric acid rinses. The cells were then soaked in 6 M nitric acid for a minimum of 1 2 h and finally rinsed with distilled deionized water. A Metrohm EA404 standard reference electrode with calomel electrode, asbestos diaphragm, and ground joint for use with the titration vessel assembly was used. The counter electrode was a Metrohm EA202 platinum wire electrode, wire diameter approximately 0.8 mm, with ground joint. The working electrode was a Metrohm E410 hanging mercury drop electrode which is essentially a microburet complete with micrometer for reproducing the mercury drop size. Metrohm EA842-120 mercury capillaries were used with the HMDE assembly. Solutions. AU solutions were prepared from distilled deionized water stored in a Teflon jug which had been thoroughly washed and rinsed. Hydrochloric acid (Fisher reagent grade) was used as the supporting electrolyte for the preliminary lead(I1) analyses. The HC1 was purified by controlled distillation in a Teflon desiccator containing a beaker of stock HC1 and a Teflon beaker of distilled deionized water; 1.0 and 1.5 M HCl could be obtained over a 24-h period. The distilled HC1 was collected and stored in polyethylene containers. All lead solutions were run in 0.771 M distilled HC1 solutions. ULTREX nitric acid was used as the supporting electrolyte for the thallium(1) investigations. Nitric acid solutions were 0.9 M. Stock solutions of 0.1 M lead and 0.025 M thallium were used. The supporting electrolyte solutions were spiked with the stock metal ion solutions using a Rainin Pipetman continuously adjustable digital microliter pipet. All solutions were degassed for 15 min with nitrogen gas which had been purified by passage through vanadous chloride solutions. Instrumentation. All experiments were performed using a homemade three-electrode potentiostat. A computer network consisting of ten Digital Equipment Corporation LSI-11 minicomputers was used for program development, debugging, and execution. All programs to perform square-wave voltammetric experiments were written on the host computer in MACRO-11 assembly language under an RT-11 operating system. The programs were then down-line loaded into a satellite computer where the
actual experiments were performed. The computer network was controlled by a multiprocessor program supplied by Digital Equipment Corporation called Remote-11. The plotting package was written in Fortran IV and drives the Benson-Lehner digital incremental plotter or a Tektronix 611 storage scope, both of which are attached to the host computer. The ADAC Corporation model 600-LSI-11 data acquisition and control system was used to sample the current, monitor the reference electrode potential, and apply the square-wave potential waveform. The device contains two 12-bit, 110 V digital-to-analog converters. The board also contains a multiplexer of up to 64 analog input channels, a programmable gain amplifier with automatic zeroing, and a high speed 12-bit analog t~ digital converter.
THEORY A theoretical approach was chosen as the preferred route to investigate the performance of Fourier transform analysis on square-wave voltammograms. Fourier transforms are sensitive to changes in t h e normally used peak parameters including peak height, peak position, and general peak shape. A change in any of these peak parameters will be reflected in a change in the real and imaginary spatial frequencies. A major goal of our work was to be able to obtain squarewave voltammograms with appreciable signal-to-noise ratios in a single scan of a hanging mercury drop electrode. In order to do this, i t was necessary t o determine t h a t point in t h e spatial frequency domain a t which the spatial frequencies no longer contribute significantly to the useful information contained in the voltammogram. Osteryoung has shown that the theoretical square-wave voltammograms agree quite well with experimental ( 1 I ) . Consequently, the Fourier analysis was first examined via computer simulation of model data; then t h e concept was applied to real data. Similar approaches have been taken in other areas of analytical chemistry as evidenced by Cram’s investigation into t h e effect of Savitzky-Golay weighted digital filters on chromatographic signals (12). I t is much more practical to first evaluate new data treatments on simulated data than with actual experimentation and data acquisition, which introduce uncontrolled variables. Noise in its various forms can be considered as a random error that directly affects the precision and useable range of analytical measurements. Consequently, i t is desirable t o remove as much noise as possible without distortion of t h e signal beyond acceptable limits. In the technique of square-wave voltammetry, the forward current measured can be considered as a linear sum of t h e current due to driving the electrode reaction in the cathodic direction and any noise inherent in t h e system used for t h e analysis. The reverse current measured is a linear sum of the current due to driving the electrode reaction in the anodic direction and any noise present. Since noise is a random error superimposed on the signal of interest, it can be measured statistically. One such measure is the variance of the signal. T h e normal procedure for calculating a n experimental square-wave voltammogram is to subtract the reverse current from the forward current to obtain a forward difference current. If the forward current has a noise level or variance VI and the reverse current has a noise level or variance V2 then the difference current will have a noise level or variance of ( V1 V2). Thus, any approach which has been taken in the past, such as ensemble averaging of difference currents, is already hampered by having a noise level or variance increased from the actual experimental measurement. Combining this with the fact that many ensemble averages are necessary, with resultant loss in analysis time, this approach can be considered as less than optimal. A very useful property of Fourier transforms is linearity. If x ( t ) and y ( t ) have the Fourier transforms X ( f , a n d Y o , respectively, then the sum x ( t ) + y ( t ) has the Fourier trans-
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ANALYTICAL CHEMISTRY, VOL. 52, NO. 8, JULY 1980
+
form X ( f ) Y o . This can also be expressed via the Fourier transform pair notation: ritl
+
y h
0
x/r/ + Y i i l
T h e technique of Fast Fourier Transform square-wave voltammetry makes use of this important property. The forward difference current can be determined by carrying out the current subtraction in the spatial frequency domain. The forward and reverse currents are sampled and stored in memory as in the conventional mode. At the completion of the scan of the desired potential range, the forward and reverse current arrays are each submitted to a fast Fourier transform algorithm. The resultant real and imaginary spatial frequency arrays are truncated at a previously determined point and the subtraction is carried out only over those spatial frequencies which contribute to signal due to electrode reactions and the noise has been eliminated. The real and imaginary spatial frequency arrays obtained from the subtraction procedure may be inverse Fourier transformed to obtain a current-potential voltammogram. The obvious advantage of this technique is t h a t any noise present is eliminated before the subtraction procedure producing a much better representation of the true signal. Signal-to-noise ratios should be appreciably enhanced. In this article such fast Fourier transform square wave voltammetry will be compared to conventional square-wave voltammetry with respect to mean noise levels, signal-to-noise ratios and reproducibility for Pb(I1) in HCl and Tl(1) in ULTREX "03. In order to develop a square-wave voltammetric data processing system that handles data completely in the spatial frequency domain, it was necessary to generate theoretical square-wave voltammograms so that variations of peak parameters in the time domain could be studied in the spatial frequency domain. A program to calculate theoretical square-wave voltammograms was obtained from Osteryoung. This program was originally written in PDP-12 Fortran and suitable modification was done to execute it in PDP-11 Fortran IV. The program was then modified to allow the user to specify the number of electrons in the electrode reaction and the half-wave potential. T h e desired potential range was always adjusted to generate a final data file with a power of 2 number data points. In all cases reported here, 128-point voltammograms were generated. A Fortran IV program was written to perform fast Fourier transform analysis of the square-wave voltammograms generated. T h e fast Fourier transform subroutine used in this work was obtained from the Digital Equipment Corporation Laboratory Applications-11 Software package. The subroutine executes a Cooley-Tukey algorithm and the data is handled as signed binary fractions. A sine look-up table is used so that the trigonometric functions need not be calculated during execution. The subroutine will perform a 1024-point transform in approximately 0.5 s. Data files are read in from disk, preprocessed, and Fourier transformed. Amplitude and phase spectra are calculated from the returned real and imaginary spatial frequencies. The user may specify a CRT or printer listing of the real/imaginary amplitude or phase spectra. The user may also specify a display of all four spectra on either a Tektronix 611 storage scope or a digital incremental plotter. There is some point in the spatial frequency domain a t which spatial frequencies no longer contribute significantly t o the voltammogram peak but are essentially due to noise. This point is termed the cut-off frequency and is designated Fco. The selection of the proper cut-off frequency has always been a problem associated with Fourier transform data analysis. Several approaches have been investigated and reported (13-15). Of the approaches investigated, none are definitive in the selection of the cut-off frequency. The most
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Table I. Pb(I1) Theoretical Cut-Off Determination Data cut off 6 7 8 9 10 11
12 13 14
15 16
normalized sigma
peak difference
peak error, %
12.7 7.51 4.09
-1655
-11.5 -5.71 -2.14 -0.174 0.740
2.10
1.18 0.917 0.803 0.672 0.516 0.371 0.247
-819
-:307 -- 25 +:LO6
+I48 + 142 +I16 i- 85 -I- 57 i- 35
1.03
0.991 0.810 0.593
0.340 0.244
important point in determining a cut-off frequency should be the objective of the truncation operation and the amount of error acceptable in the truncated data with respect to the desired peak parameters. These factors will vary from experiment to experiment as peak shapes change. A method was developed to determine statistically valid cut-off frequencies for voltammetric data. The technique is general and is useful for other types of analytical data. Basically, the method optimizes the number of spatial frequencies retained with respect to the important peak parameter. In voltammetry, peak heights are proportional to concentration, and the method developed here examines the effect of sequentially retaining more spatial frequencies on peak height of data regenerated via inverse Fourier transformation. If chromatographic data were being studied, the cutoff frequency could be optimized with respect to peak area. A program was written in Fortran IV to read a specified theoretical square-wave voltammogram on disk into memory, preprocess the data for Fourier transformation, submit the data to the forward FFT algorithm. and calculate an amplitude spectrum based on the returned real and imaginary spatial frequencies. The first amplitude spatial frequency which contributes less than 0.1% to the amplitude array is determined and reported to the user. The user then specifies a cut-off determination tolerance factor. The tolerance factor controls t h a t region of spatial frequencies over which the truncation procedure is analyzed. For example, if the first amplitude spatial frequency to contribute less than 0.1 7" to the amplitude array was the tenth spatial frequency and a tolerance factor of five was specified, then t h e data would be truncated with cut-off frequencies of five to fifteen (first spatial frequency
