The enhancement of electroanalytical data by on-line fast Fourier

Instrumentation

Donald E. Smith Department of Chemistry Northwestern University Evanston, 111. 60201

mechanistic conclusions can be deduced (3-5), is potentially a highly significant development for electrochemical analysis. I t could lead to a new dimension in electroanalysis, since existing electrochemical assay procedures utilize only a small portion of the data base inherent in the cell response waveform. Essentially all conventional electrochemical assay procedures are based on the observation of the cell’s response, either a t a single frequency (fundamental and second harmonic ac polarography), a single point in time (pulse polarography, dc polarographic response a t end of drop life), or some form of time-integral response (dc polarographic average currents, chronocoulometry). Furthermore, usually the detailed character of the analytical response vs. potential profile is ignored, an assay being based on the response a t a single potential (peak potential), or in a potential region where the response is potential-independent (plateau currents). Consequently, except for cursory qualitative inspection to be sure that the wave shape appears “normal” and is properly located on the dc potential axis, a typical electrochemical assay ignores the usually recorded response-potential profile, and the response-time or response-frequency profiles usually are unavailable as part of the instrument readout. In other words, a careful kineticmechanistic type examination of an electrode response is seldom, if ever, a part of a routine assay procedure. The main reason for this state-of-affairs is quite obvious. With conventional in-

The Enhancement of Electroanalytical Data by On-Line Fast Fourier Transform

Data Processing in Electrochemistry The on-line fast Fourier transform

(FFT) procedure, which enables rapid, precise, and convenient measurement of the small amplitude frequency domain response spectrum of an electrochemical cell, has obvious and important advantages for kinetic-mechanistic applications in electrochemistry. This kind of objective has been the primary stimulus and focus of most literature reports dealing with on-line F F T electrochemical relaxation measurements (ERM) (see ref. l and its references). Consequently, the electrochemical literature provides a t least a de facto suggestion that the frequency domain-kinetic-mechanistic triumvirate represents the terms with which F F T electrochemical applications should be associated. However, the possible application scope of on-line FFT procedures in electrochemistry actually is much broader than this, encompassing unique analysis procedures and ERM approaches which focus on time domain or time domainrelated observables. This article is designed to survey some of this latter class of applications. It will begin by considering some suggestions for electrochemical analysis which follow directly from the new-found ability t o rapidly acquire small amplitude ERM response spectra via the F F T ( I ) . This will be followed by consideration of some more general applications of F F T procedures which effect convolution, deconvolution, and correlation of data arrays. All concepts discussed effect data enhancement by providing for greater information content, minimized noise, and/or more desirable data presentation formats. When evaluating the concepts discussed below, the reader should be aware of the state of minicomputer technology regarding execution times for the F F T and related operations.

We have mentioned ( I ) that a software-implemented transfer function calculation requires approximately 2 s with 512-point time domain arrays using fixed-point arithmetic and a Raytheon 704 minicomputer. This can be improved substantially with hardware F F T processors, where one device (2) can perform the same computation in only about 60 ms, using floating point arithmetic. These are simply examples, not the state-of-theart. Analytical Applications of Capability for Real-Time KineticMechanistic Characterization of Electrode Reaction The fact that minicomputerized FFT data processing enables rapid assessment of the small amplitude faradic admittance’s frequency domain response, from which detailed kinetic-

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Figure 1. Relationships between time and frequency domain operations ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976

517A

strumentation, quantitative kineticmechanistic characterization of an electrode response has been simply too time-consuming to be analytically palatable. However, the minicomputer-FFT combination changes the picture by making available on an analytically acceptable time scale phenomenological kinetic-mechanistic insights which normally are sought only in fundamental investigations. Indeed, from the measurement and computation time scales indicated above, it is evident that the additional time investment associated with computerized extraction of the kinetic-mechanistic information is small, relative to the time devoted to an electroanalytical assay using conventional equipment and procedures. Thus, using as a frame of reference the electrochemical analyst’s normal experience, one can view the phenomenological kineticmechanistic information provided by F F T strategies as being available in real-time. Recognizing that, given a minicomputer, an electrode reaction’s kinetic-mechanistic status can be monitored and transmitted to the analyst in the course of an assay run, one can readily deduce some potentially useful applications of this new-found capability. Monitoring Kinetic Status of Electrode Process and Compensating for Undesirable Rate Process Interferences. In the context of ERM, the so-called reversible or diffusion-controlled (3-5) electrochemical response normally is considered the most desirable for analytical purposes, particularly with highly sensitive relaxation methods, such as differential pulse polarography, second harmonic ac polarography and phase-selective ac polarography. There are exceptions, such as dc polarographic assays based on the highly sensitive catalytic wave (6, 7), but these are rare. The bias toward diffusion-controlled processes is well justified because the response-controlling diffusion rate is relatively insensitive to sample-induced variations in solution or interfacial conditions encountered in typical assay procedures. The same is not true for the other rate processes which often influence the electroanalytical response, such as heterogeneous charge transfer, coupled homogeneous chemical processes, adsorption, and other surface reactions (3-5). Background constituents in the sample, such as surfactants and electrode surface oxidants, may alter in situ the electrode surface state regardless of solid electrode pretreatment, or whether the assay uses the often reproducible liquid mercury surface (8-1 1 ) . If these background substances fluctuate with time (continuous monitoring) or from sample-to518A

sample (batch assays), a corresponding fluctuation in the heterogeneous rate is likely, which will lead to a nonreproducible electroanalytical response i f the latter is influenced by the heterogeneous rate (e.g., as with a quasi-reversible process). A similar conclusion is reached for cases where a coupled homogeneous chemical reaction’s kinetic status influences the electroanalytical response, and that kinetic status is sensitive to normally encountered variations in sample makeup. Such fluctuating kinetic phenomena undoubtedly represent a predominant origin of the frequently greater analytical irreproducibility associated with “kinetically complicated” (i.e., nonreversible) electroanalytical responses, relative to the diffusioncontrolled “ideal”. This and the fact that the diffusion-controlled response usually is larger than the kinetically complicated counterparts make obvious the reasons for the preference accorded the diffusion-controlled process in electroanalysis. Unfortunately, nature does not always cooperate, and many analytically promising electrode processes do not achieve the diffusioncontrolled ideal under accessible assay conditions. This is particularly true when one is using a technique with a relatively short effective observation time, such as pulse polarography, fundamental and second harmonic ac polarography, and coulostatic analysis; and when one is using a solid electrode where heterogeneous charge transfer rates often are notably smaller than a t mercury ( 1 1 ) . These “kinetic fluctuations” attending ERM-based assay procedures hardly can be considered devastating for this approach to chemical analysis in view of the monotonically increasing popularity of these methods (12, 13). It is evident that equally annoying sources of quantitative data fluctuations harass users of nonelectrochemical procedures. Nevertheless, it should be apparent that the scope of electroanalytical measurements, not to mention operator satisfaction, will be greatly enhanced if the magnitude of these kinetic fluctuations can be monitored and appropriate action taken to minimize their effects on the analytical procedure. This possibility is precisely what the on-line F F T admittance response spectral measurement provides. One can program the minicomputer to extract from the admittance spectrum the essential kinetic-mechanistic data about the electrode process and use these data to either compensate the observed response mathematically for the undesirable kinetic effects (the preferred strategy) or take other appropriate action, such as providing a warning message via teleprinter or scope display informing the operator

ANALYTICAL CHEMISTRY, VOL. 48, NO. 6, MAY 1976

of the occurrence of an intractable kinetic situation which invalidates the assay procedure for the sample in question (last-ditch strategy). We are talking about a relatively novel procedure in which the status of an analytical probe’s response characteristic is monitored and useful action is taken if that characteristic changes. For clarification purposes, we will give a detailed example of how this type of procedure can be applied to a quasi-reversible ERM admittance response (response controlled by diffusion and heterogeneous charge transfer kinetics) of a single-step electrode reaction represented by:

O+ne+R

(R1)

where species 0 is initially present in the solution phase. For this case, the rate law for the small amplitude frequency domain response of the faradic admittance may be written (expanding plane model) ( 3 , 5 )

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effects of heterogeneous charge transfer kinetics on the dc and ac time scales, respectively (3,5). F ( t ) and G ( w ) are unity in absence of heterogeneous charge transfer kinetic effects. I t is evident that, in principle, the diffusion-controlled (reversible) response can be extracted from an observed quasi-reversible current amplitude if F ( t ) and G ( w ) are known, simply by using the rearranged form of Equation 12,

Knowledge of F ( t ) and G(w) is conveniently accessible if cot 4 is measured (Equations 3 and 4). Thus, F ( t ) and G ( w ) can be computed simply from the observed value of cot 4 and the values of the experimentally controlled parameters, j , t , and w . In cases where the dc process is diffusion-controlled, F ( t ) = 1,and only the cot 4 value is required [to calculate G ( w ) ] . Regardless, the quasi-reversible kinetic correction represented by Equation 13 is implementable without requiring that one carry the kinetic characterization of the system to the point of explicitly evaluating k , and cy. The required phenomenological kinetic information amounts to nothing more than one additional observable beyond what is conventionally recorded in an assay run involving an ac response. In principle, these conclusions are applicable to observations performed a t a single frequency. However, invoking Equation 13 on the basis of single frequency information entails the assumption that the electrode process in question is indeed quasi-reversible, which is somewhat unsafe and unnecessary. Using a detailed frequency domain spectrum revealed by FFT processing of the response to a multiple frequency input ( I ) , as we are advocating here, one has the basis for establishing with reasonable certainty that a quasi-reversible response is or is not present, and proceeding accordingly. A simple numerical analysis (14) shows that the kinetic correction embodied in Equation 13 will become inaccurate when the electrode process nears irreversibility, which is a minor problem because in this irreversible realm the response magnitude becomes insensitive to fluctuations in k , (15)and the correction is less important. The concept in question works best under quasi-reversible conditions, where it is most needed. Its validity has been established in our laboratory ( 1 4 ) with the aid of some offline computations. A program for totally automated, on-line implementation of the kinetic correction concept remains to be developed, but this is a routine matter. 520A

Although the computations may be somewhat more involved, our investigations have shown (14) that most other sources of kinetic fluctuations of the ERM response, such as coupled homogeneous chemical reactions, can be handled in a manner similar to that described above for the quasi-reversible case. Of course, an overly complicated mechanistic pattern involving numerous competing rate processes may cause difficulties, but such situations usually are avoidable in analysis work. Assays Based on Direct Observation of Electrode Reaction Kinetics. The foregoing section considered compensation for heterogeneous and homogeneous reaction rate constant variations associated with sample background components whose concentrations are not controlled from assay to assay. These same influences on an electrode reaction’s kineticmechanistic status can be applied as the basis for assaying these rate-influencing “background constituents”. Thus, with the real-time kinetic information provided by the F F T admittance spectral measurement, one can envision applications such as analyzing: the water content in a nonaqueous solvent based on its effect on the rate of protonation of an organic or organometallic electrode reaction product; chloride ion from its effect on the Bi+3/Bi(Hg) heterogeneous charge transfer rate ( 1 1 ) ;a surfactant’s concentration from its effect on the Cd+*/Cd(Hg)heterogeneous rate (11), etc. This type of assay procedure based on electrochemical kinetics is relatively unused, particularly where k,qmeasurements are the basis for the assay. Nevertheless, the potentialjties are nontrivial. The method does not require electroactivity in the component which is the assay’s objective; therefore, a rather broad range of possibilities exists. The homogeneous analog of this procedure represents a well-known, rapidly expanding area of analytical chemistry (16-18). Perhaps the computerized F F T electroanalytical approach will help electrochemical methods contribute more noticeably to this area. FFT-Aided Convolution, Deconvolution, and Correlation in Electrochemical Measurements In addition to providing the means for revealing a signal’s spectrum for purposes of cell admittance measurements, the F F T is a powerful mathematical tool for implementation of a variety of other data processing strategies which have a broad spectrum of applications in ERM. Most of these are forms of convolution, correlation, and the inverse operations, deconvolution and decorrelation. These mathe-


matical operations can be achieved by a variety of algorithmic strategies (19-21). However, the speed and efficiency of the F F T make its use a particularly appealing approach, despite the fact that seemingly indirect computational pathways are involved. The relationships between the time domain versions of convolution, correlation and their inverses, and the Fourier transform operation are given by certain mathematical theorems (1922) whose consequences are indicated schematically in Figure 1. Figure 1 indicates, for example, that time domain convolution of two waveforms, A ( t ) and B ( t ) ,may be effected either by some direct time domain algorithm or by Fourier transforming A ( t) and B ( t ) ,m u l t i p l y i n g their spectra, and then inverse Fourier transforming. Similarly, deconvolution may be effected by some “direct” algorithm or by dividing the spectrum of the convoluted waveform by the spectrum of the function to be deconvoluted therefrom, and inverse Fourier transforming. Use of the F F T in this context is accompanied by certain subtleties which have the same origin as the “leakage” effects discussed in the first article ( 1 ) and can give rise to significant computational error (19-21, 23) unless appropriate steps are taken. The “appropriate steps” involve minor time domain data array manipulations in most instances (19-21,23) which are not explicitly indicated in Figure 1 because sufficient space is not available for discussion of these subtleties in the F F T approach. They are well described in the literature (19-21,23). The primary messages to be extracted from Figure 1 are that simple multiplication and division in the frequency domain imply convolution and deconvolution, respectively, in the time domain, and that frequency domain multiplication and division using the complex conjugate of one spectrum are equivalent to time domain correlation and decorrelation, respectively. Deconvolution Aspect of Cell Admittance Computation and Kinetic Correction. From the latter observations, one may immediately recognize that the master equation for computation of the cell admittance from the Fourier spectra of the applied potential and current response ( I ) ,

[ A ( w )I, ( w ) , E ( w ) ,and E * ( o )represent the spectra of cell admittance, the cell current response, the applied potential, and its complex conjugate, respectively], is equivalent to deconvoluting the time domain applied potential waveform (with effects of poten-

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521 A

tiostat nonideality) from the observed time domain current response to obtain the time domain impulse response of the cell. If the latter waveform were desired for data analysis, it is obtainable from A ( w ) simply by inverse Fourier transformation. Similarly, the frequency domain kinetic correction represented in Equation 13 is equivalent to deconvoluting from the cell’s time domain impulse response the effects of slow heterogeneous kinetics. Consequently, FFT-assisted operations we have already discussed implicitly contain examples where all but the final step of time domain deconvolution via the FFT is implemented. Semiintegral and Semiderivative Voltammetry (Convolution and Deconvolution Voltammetry) and Related Operations. Oldham’s semiintegral voltammetry (24-27) is identical to convolution of the observed current with the t-lI2 function, as Saveant and coworkers explicitly acknowledge in their convolution voltammetry (28-30). Applications of these forms of convolution to processing of chronoamperometric data derived from linear sweep and constant potential experiments in electrochemistry are well documented. Greatest success and interest have been generated in the context of stationary electrode linear sweep and cyclic voltammetry where the undesirable asymmetric peak-shaped voltammogram is converted to the sigmoidal-shaped response characteristic of the dc polarogram (24-30). Figure 2 depicts this operation for a voltammogram obtained with a partially resolved two-component reversible system by the transformation from Figure 2A to 2B. Convolution (or semiintegral) voltammetry currently represents the most visible example, other than admittance measurements, where the FFT can assist effectively in electrochemical data processing. Of particular interest in this writer’s laboratory are the possibilities of invoking F F T deconvolution procedures to ERM data to effect narrowing of the response-potential profiles (greater resolution) and other useful data enhancement goals. The broadening function to be deconvoluted from initial data can be defined in a variety of ways, depending on the phenomena one wishes to define as “broadening”. The “kinetic correction” discussed above is one example. There, the initial admittance polarogram is broadened by the heterogeneous or homogeneous kinetic contributions and is narrowed by mathematically deconvoluting the kinetic effects in the manner described. The stationary electrode linear sweep or cyclic voltammogram once again provides a promising situa522A

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tion. Here, the diffusion-controlled voltammogram can be considered to be distorted by a broadening function originating in the diffusion process which has the form t-”* with planar diffusion (31-34) and other well-defined forms for other diffusional geometries. Deconvolution of t-’I2 from a reversible linear sweep voltammogram (semidifferentiation) produces for each component the much sharper and symmetrical l/cosh2 Cj/2) shape function, where j is given by Equation 7 (31-34). The effect of this operation is illustrated by the transformation from Figure 2A to 2C. Its advantages relative to the inverse operation of convolution, as far as peak resolution is concerned, are apparent in Figure 2. Experimental demonstration of this operation has been reported ( 3 4 ) .One may consider the remaining l/cosh2 G/2) shape (Figure 2C) as due to “nernstian broadening” of an impulse response which is characteristic of each component in the system. In principle, it is possible to deconvolve the nernstian broadening function from the profiles of Figure 2C to obtain these impulse functions, which will be located at each electrode reaction’s reversible half-wave potential with a concentration-proportional magnitude, as shown by the transformation from Figure 2C to 2D ( 3 2 , 3 3 ) .


This final step is somewhat unrealistic in practice because it amounts to invoking very high-order derivatives (35) and will thus be thwarted by effects of noise, computer round-off and quantization errors, etc. However, one can envision deconvolution steps which will provide a compromise between the ideal of Figure 2D and the nernstian-broadened peak of Figure 2C and provide a significant resolution improvement (14). Except for the specific nature of the functions employed, the peak sharpening deconvolution strategies described here are hardly new or unique. Such data enhancement concepts have been employed in spectroscopy, in particular, for many years (36, 37). These earlier studies have made it clear that the degree of success realized with such procedures is intimately related to measurement noise levels. Electrochemical data tend to be less noisy than their spectroscopic counterparts; therefore, it is possible that electroanalytical procedures will benefit more from deconvolution procedures than perusal of the relevant spectroscopic literature might suggest. The crucial experiments remain to be done. Digital Filtering. The issue of how noise can limit the benefits of the foregoing types of data enhancement

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Figure 3. Fourier transform smoothing of fundamental harmonic ac polarographic data A, original polarogram: 6,Fourier spectrum: C. smoothing function: D. filtered Fourier spectrum: E, smoothed polarogram. B and D: 0 = real spectral components, 0 = imaginary components or both components when they are equal. System: 1.0 X M Cd+* in 1.0 M KN03, 0.1 M sodium acetate, 0.040 M nitrilotriacetic acid, at 25 OC, pH 4.95, second wave. Applied: 10.0-mV peak-to-peak 128-Hz sine wave, computer-controlled incremental scan. Measured: in-phase 128-H2 alternating current at end of mechanically controlled drop life (38)

schemes leads directly to the subject of noise minimization via digital filtering. I t already has been made apparent in the previous report ( 1 ) that superior data precision is provided in FFT admittance spectral measurements by the unusual speed of the measurement operation which makes extensive ensemble averaging a palatable option. In addition to the ensemble averaging strategy, one can suppress noise quite effectively with the aid of digital filtering schemes which employ the F F T (19,20,38-40), and successful applications to electrochemistry have been reported (38,41). F F T digital filtering is routinely invoked in our laboratory. The simplest example involves Fourier transforming the data array to be filtered, multiplying the array's frequency

spectrum by a frequency domain filter function, and inverse Fourier transforming. Of course, this is equivalent to time domain convolution of the original data array with the filter function's time domain representation. An illustration of this particular F F T digital filtering sequence is shown in Figure 3. Its effect on a rather noisy second harmonic polarogram is shown in Figure 4. The alreadymentioned subtleties of the F F T as a convolution and correlation strategy yield problems whenever the sequence illustrated in Figure 3 is applied to data arrays with nonzero initiation and termination points (38,40).Once again, these problems can be substantially eliminated by minor modifications of the time domain arrays (38, 40). Advantages of FFT digital filter-

ing relative to other digital filtering approaches have been enumerated (20,21, 38-40). Correlation Procedures in Electrochemistry. Inspection of Figure 1 and Equation 14 confirms that the admittance calculation also contains implicitly, in addition to deconvolution, the frequency domain equivalent of time domain correlation. The cross, the power spectrum, I ( o ) E * ( w )is F F T of the time domain cross-correlation between the applied potential and current response waveforms. Computing this quantity by itself provides the digital equivalent of analog lock-in amplification at each frequency and yields the phase relations between the potential and current. The magnitude of the cross-power spectrum is "colored" by the applied potential spectrum, but this effect is removed by dividing by the quantity, E ( w ) E * ( w )which , is the applied potential autopower spectrum whose inverse F F T is the potential waveform's autocorrelation function. These are not the only possible modes whereby correlation operations are invoked. A final link in the framework of FFTbased data processing procedures is the use of cross-correlation of unknown and standard responses for purposes of qualitative and quantitative assays. Merits of the usual correlation pattern and its frequency domain counterpart (cross-power spectrum) are many (19-22,42). One can envision the correlation of standard voltammograms of various types (response vs. potential correlations), analogous to what is done in spectroscopy (response vs. wavelength correlations) (42), to enhance the selectivity of electroanalytical methods. Of particular interest to us is the use of multidimensional cross-correlations involving various observables (dc, fundamental and second harmonic response, linear sweep voltammograms, spectroelectrochemical responses, etc.) and experimentally controlled variables (time, potential, frequency). Unusually complex multicomponent

C

Figure 4. Fourier transform smoothing of second harmonic ac polarographicdata A, original data: B. smoothed data: C, comparison of original and smoothed data: 0 = original data, - = smoothed data. System: same as Figure 3. Applied: same as Figure 3, except frequency = 228 Hz. Measured: quadrature current at 456 Hz (38)

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and which would involve no more run time than presently required to record a dc polarogram. Modern electrochemistry has been characterized by rather long time lags between demonstration of new, useful instrumental concepts to their routine application in assay procedures (e.g., approx. 20 years for differential pulse polarography), and F F T applications may be no exception. Nevertheless, it is expected that eventually FFT-based electroanalytical procedures will be commonplace in both kinetic-mechanistic and analytical applications of ERM.

systems might be handled by such procedures. While the promise and possibilities are significant, little attention has been accorded the use of standard-unknown correlation procedures in electrochemistry to date, except for the zero-order procedures of comparing unknown and standard voltammetric wave locations on the dc potential axis and their response magnitudes for quantitation purposes.

Conclusions The F F T can be applied to ERM in a variety of contexts, each representing an actual or incipient advance in the state-of-the-art. In addition to the relatively obvious and natural application to cell admittance measurements, the FFT can play an important role in electrochemical data processing because of its efficiency in assisting the mathematical procedures known as convolution, correlation, and their inverse operations, all of which can provide significant data enhancement. A simple extrapolation of the concepts advanced in this and the previous article leads to some rather interesting speculation regarding future electroanalytical procedures. One can envision scenarios for electroanalysis involving combinations of all the FFT-enabled data processing operations described in these two articles,

Acknowledgment Studies reported here which originated in our laboratory owe their success to the dedicated efforts of my coworkers A. M. Bond, K. R. Bullock, S. C. Creason, D. G. Glover, J. W. Hayes, and R. W. Schwall.

of Polarography”, Academic Press, New York, N.Y., 1966. (7) L. Gierst, L. Vanderberghen, and E. Nicolas, J . Electroanal. Chem., 12,462 (1966).

(8) R. N. Adams, “Electrochemistry at Solid Electrodes”. ChaD. 7 . Dekker. New York, N.Y., 1969. (9) C. N. Reilley and W. Stumm, in “Progress in Polarography”, p p 81-121, Interscience, New York, N.Y., 1962. (10) J. Kuta, in “Modern Aspects of Polarography”, pp 62-70, Plenum Press, New York, N.Y., 1966. (11) J.E.B. Randles and K. W. Somerton, Trans Faraday SOC., 48,937,951 (1952). (12) J. B. Flato, Anal Chem , 44 (11),75A (1972). (13) T. R. Copeland and R. K. Skogerboe, [ b i d , 46, 1257A (1974). (14) D. E. Smith, unpublished work,

Northwestern University, Evanston, Ill., 1974. (15) T. G. McCord and D. E. Smith, Anal Chem., 40,474 (1968). (16) R. A. Greinke and H. B. Mark. Jr.. i b d , 46,413R (1974). (17) H V. Malmstadt, C. J. Delaney. and

References

E A. Cordos CRC Crit Reb Anal Chem , 2,559 (1972). (18)H. B. Mark, Jr., Talanta, 19,717

(1) D. E. Smith, Anal. Chem., 48,221A

(1972). (19) R. K. Otnes and L. Enochson, “Digital

(1976).

(2) “Apollo Execution Times”, Raytheon

Data Systems, Norwood, Mass., 1975. (3) D. E. Smith, in “Electroanalytical Chemistry”, Vol 1, pp 1-155, A. J. Bard, Ed., Dekker, New York, N.Y., 1966. (4) M. Sluyters-Rehbach and J. H. Sluyters, ibid.,Vol4, pp 1-128, 1970. (5) D. E. Smith, Crit. Reu. Anal. Chem., 2, 247 (1971). (6) J . Heyrovsky and J . Kuta, “Principles

Time Series Analysis”, Wiley-Interscience, New York, N.Y., 1972. (20) L. R. Rabiner and B. Gold, “Theory and Application of Digital Signal Processing”, Prentice-Hall, Englewood Cliffs, N.J., 1975. (21) E. 0. Brigham, “The Fast Fourier Transform”, Prentice-Hall, Englewood Cliffs, N.J., 1974. (22) R. Bracewell, “The Fourier Transform and Its Applications”, McGrawHill, New York, N.Y., 1965. (23) R. Ramirez, Electronics, 48 (13), 98 (1975). (24) K. B. Oldham, Anal. Chem., 45,39 (1973). (25) M. Goto and K. B. Oldham, ibid.,p 2043. (26) M. Goto and K. B. Oldham, ibid.,46, 1522 (1974). (27) R. L. Birke, ibid.,45, 2292 (1973). (28) J. C. Imbeaux and J . M. Saveant, J . Electroanal. Chem., 44,169 (1973). (29) F. Ammar and J. M. Saveant, ibid., 47, 215 (1973). (30) L. Nadjo, J. M. Saveant, and D. Tressler, ibid., 52,403 (1974). (31) A. Sevcik, Coll. Czech. Chem. Commum, 44, 327 (1948).

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(32) C. N. Reilley, US.-Japan Seminar on Computer Assisted Chemical Research Design, Honolulu, Hawaii, July 2-6, 1973. (33) D. E. Smith, Symposium on Recent

1 Quantitation of TLC

Advances in Analytical Voltammetry, Division of Analytical Chemistry, 169th American Chemical Society Meeting, Philadelphia, Pa., April 6-11, 1975. (34) M. Goto and I. Ishii, J . Electroanal. Chem., 61,361 (1975). (35) A. den Harder and L. de Galan, Anal.

much more accurate with an SD-3000

Chem., 46,1464 (1974). (36) G . Horlick, ibid.,44,943 (1972). (37) G. Horlick, ibid.,43 (81, 61A (1971). (38) J. W.Hayes, D. E. Glover, D. E. Smith, and M. W. Overton, ibid., 45, 277 (1973). (39) C. A. Bush, ibid.,46,890 (1974). (40) D. E. Aspenes, ibid.,47,1181 (1975). (41) R. J. Schwall, I. Ruzic, and D. E. Smith, J . Electroanal. Chem., 60,117 (1975). (42) G. Horlick, Anal. Chem., 45,319 (1973).

This double-beam spectrodensitometeri fluorometer rapidly and accurately analyzes t h i n layer media in all classical modes. Its background discriminating capability has made the standard of R&D and quaiity contr

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Work supported by the National Science Foundation (Grants GP-28748X and MPS74-14597).

The enhancement of electroanalytical data by on-line fast Fourier

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