Application of correlation analysis for signal-to-noise enhancement in

Feb 1, 1973 - Correlation-based detection of spectral information using microliter volumes, in-torch vaporization (ITV) sample introduction and induct...
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hour so that it should be suitable for analytical control during the industrial extraction of TiOzfrom its ores. The carbon filament method of atomization offers certain advantages over the use of a flame cell in that the solutions after standard additions can be limited to only 5 ml each. A nebulizer/flame atomization system would require larger volumes and this would necessitate the fusion of a much larger sample of T i o n . Also, the large excess of sodium present in the solutions as a result of the fusion requires no preliminary

extraction, since it is easily separated by selective volatilization from the filament.

RECEIVED for review July 31, 1972. Accepted October 2, 1972. Thanks are due to Laporte Industries Ltd. for assistance and financial support for this program, and also to the Science Research Council for the provision of a CAPS award for K.W.J.

Application of Correlation Analysis for Signal-to-Noise Enhancement in Flame Spectrometry Use of Correlation in Determination of Rhodium by Atomic Fluorescence G. M. Hieftje Department o j Chemistry, Indiana University, Bloomington, Ind. 47401

R. I. Bystroff and Robert Lim Lawrence Livermore Laboratory, Lioermore, Calif, 945.50

Signal processing by cross-correlation is directly compared to lock-in amplification for atomic fluorescence photosignals. The determination of rhodium with a non-optimized atomic fluorescence spectrometric system serves to illustrate the inherent advantages of cross-correlation. A detection limit of 0.16 ppm is achieved for rhodium, representing a five-fold improvement in the signal-to-noise ratio. Freedom from impulse noise and direct digital compatibility are additional advantages.

MOST FLAME SPECTROMETRIC SYSTEMS in use today employ some method of instrumental signal-to-noise enhancement. Commercial atomic absorption spectrophotometers, for example, are almost universally equipped with choppers and phase sensitive detectors (lock-in amplifiers), which serve to increase stability and precision and decrease detection limits. Unfortunately, because of the success which has been obtained with these systems, little attention has been given to the limitations of lock-in amplification or to the utilization of alternative methods of signal improvement. Of the various signal improvement techniques which could be applied to flame spectrometry (1-3), tuned (frequency selective) amplification (4), lock-in amplification (9,signal averaging (6),dc integration (7), boxcar integration (8), and photon counting (9-13) have been investigated with varying (1) D. J. Fisher, Chem. Instrum., 2, l(1969). (2) G. M. Hieftje, ANAL.CHEM.,44 (6), 81A (1972). (3) Ibid., 44 (7), 69A (1972). (4) G. F. Box and A. Walsh, Spectrochim. Acta, 16,255 (1960). (5) B. J. Russell, J. P. Shelton, and A. Walsh, ibid., 8, 317 (1957). (6) D . J. Fisher, R. W. Stelzner, and H. C. Jones, Chem. Instrum., 2,51(1969). (7) W. W. Harrison and F. E. Berry, Anal. Chim. Acta, 47, 415 (1969). ( 8 ) L. M. Fraser and J. D. Winefordner, ANAL.CHEM.,43, 1693 ( 1971). (9) D. Alger, R. M. Dagnall, B. L. Sharp, and T. S. West, Anal. Chim. Acta, 57, 1 (1971). (10) D. 0. Cooke, R. M. Dagnall, B. L. Sharp, and T. S . West, Spectrosc. Lett., 4, 91 (1972).

degrees of success. Notable in its absence from this list is the powerful tool of correlation analysis (14) which includes the techniques of autocorrelation and cross-correlation. Although autocorrelation and cross-correlation have been employed to considerable advantage in some fields (15, 16), their use has unfortunately been limited by a lack of understanding of the somewhat complex principles involved in computation and measurement of the correlation function. Recently, however, the development of simple correlation computers (17) and the introduction of moderately priced commercial correlators (18) has made the use of correlation for signal-to-noise enhancement much more attractive. Also, the increasing use of small digital computers in the analytical laboratory now renders a software approach to correlation convenient and inexpensive. As an illustration of the power of correlation techniques for signal-to-noise enhancement, a study has been performed in which auto- and cross-correlation were applied to the determination of rhodium by atomic fluorescence flame spectrometry. In this study, a commercial correlator was found to be simple to use and excellent in its ability to extract weak atomic fluorescence from the noise inherent in a completely non-optimized flame spectrometer. Using a turbulent air/Hz flame from a total consumption burner, it has been found possible with this system to improve the detection limit for rhodium using the 343.5-nm resonance line to 0.16 ppm. A further advantage of the use of a correlation technique

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(11) R. M. Dagnall, B. L. Sharp, and T. S . West, Nature Phys. Sci., 234,69 (1971). (12) Ibid., 235, 65 (1972). (13) B. J. Dawson, Method. Phys. Anal., Num. Special, Sept. 1971, p 32. (14) Y. W. Lee, T. P. Cheatham, Jr., and J. B. Wiesner, Proc. IRE, 38, 1165 (1950). (15) S . Ichamaru, J. Phys. Soc. Jap., 19, 1207 (1964). (16) W. A. Rosenblith, “Processing Neuroelectronic Data,” MIT Press, Cambridge, Mass., 1962. (17) M. Fukao, Rea. Sci. Instrum., 42,783 (1971). (18) P. J. Garforth, Electron. Instrum. Digest, 6(8), 7 (1970).

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was to provide freedom from interference of impulse noise as well as from other noise sources. Impulse noise is particularly troublesome in an instrument employing phase-sensitive detection. Other advantages obtained with the correlator include automatic background correction, digital compatibility, computational flexibility, direct measurement of signal and noise power, and minimization of drift problems. These advantages and others possible using correlation analysis will be discussed and application of the technique to other areas of chemical measurement will be considered. CORRELATION ANALYSIS Because the techniques of auto- and cross-correlation are unfamiliar to most spectroscopists, it will be instructive to briefly consider the bases of these techniques. Although correlation analysis has been broadly applied to theoretical problems in physics and chemistry [cf, for example (Is)], its use as a signal-enhancement technique is more recent (14) and is only now becoming popular (3, 20). The use of correlation for signal enhancement is an outgrowth from information theory and is based on the fact that signals, being periodic or coherent, are predictable while random noise is generally not. By examining the time correlation within a signal or between a signal and a reference, it is therefore possible to differentiate the signal information from any noise contribution which is present. To understand this better, consider first the autocorrelation of a sine wave. In this discussion, it will be helpful to refer to Figure 4 of reference 3. Basically, autocorrelation produces a time average of the product of a signal multiplied by a delayed version of itself, plotted as a function of the delay. This plot is called an autocorrelogram. For example, one point of the autocorrelogram of a sine wave will be obtained at zero delay. To obtain this point, the sine wave is multiplied by itself in phase and then averaged to produce a value equal to the mean square of the sine wave. Because this point was obtained at zero delay, it will be plotted at the zero point on the delay (7) axis of the correlogram. A second point can then be obtained by moving a replica of the sine wave a selected distance along a time axis (7) and again multiplying the two waves. Because the waves are no longer in phase, the time average of this product will no longer be the mean square of the sine wave but will be some other, smaller value. The value is also plotted at the appropriate position on the delay (7) axis of the correlogram. By calculating a large number of points at various values of 7, the entire autocorrelogram can be obtained in this way. It can be appreciated from this simple example that as long as the delay increments chosen for computation are significantly less than the period of the sine wave being examined, the autocorrelogram will be periodic. This is caused by the periodicity of the sine wave itself. The same values for the autocorrelation will be obtained at all values of 7 equal to multiples of the sine wave period. Thus, the autocorrelogram will have the same value at 7 = 0 as for T = to, 2t0, 3to, etc., where to is the period of the sine wave. In fact, for a sinewave signal, it will be found that the autocorrelogram is also sinusoidal, with a period (in terms of 7) equal to io, the (19) “Selected Papers on Noise and Stochastic Processes” N. Wax, Ed., Dover Publications, Inc., New York, N. Y., 1954. (20) I. H. F.H. Lange, “Correlation Techniques,” Van Nostrand, Princeton, N. J., 1967. 254

period of the sine-wave signal, and of peak amplitude equal

to the mean square of the sine wave. This same behavior will be exhibited by all periodic signals since they must be made up of sine waves themselves. The only difference which will be observed between a sine wave and other periodic signals is that with many signals, the correlogram waveform will be different from the original waveform, even though the frequency content of the original and correlogram waveforms will be identical. This difference is caused by the property of autocorrelation which phase relates all frequency components of a signal at the point 7 = 0. Because a sine wave contains only a single frequency, its waveform obviously cannot change by phase alteration although for a multiple frequency waveform such as a square wave, autocorrelation produces a significant change in waveshape, in this case to a triangular wave (3). For a nonperiodic or random waveform (i.e., random noise), autocorrelation produces a notably different result from that described above for periodic signals. A random, stationary wave has a time average of zero and only by squaring the wave can any nonzero time average be obtained. Multiplying a random wave by another random wave produces a waveform which is also random and again, has a time average of zero. Therefore, autocorrelation of random noise generates a value at 7 = 0 of the mean square of the noise and a value of zero for all other values of 7. This is because at 7 = 0, the noise waveform is effectively squared and averaged to generate the mean square. For all other values of 7, the noise is multiplied by a delayed version of itself. Because the noise is incoherent, the delayed version represents a completely unrelated random process so that the product of the two random functions will have a time average of zero. From the behavior of periodic signals and random waveforms (noise) during autocorrelation, it can be understood how autocorrelation can provide signal-to-noise enhancement. Random noise such as generally encountered in flame spectrometry will contribute to an autocorrelogram only at values of 7 very close to zero while the coherent signal will continue to contribute at much greater delays. Therefore, the autocorrelogram of a “noisy” sine-wave signal will have an appearance similar to the sine wave except for the region near zero delay, where the noise contribution will be evident. By examining the correlogram only at larger values of 7, therefore, it is possible to determine the original sine-wave signal frequency and amplitude ecen in the presence of enormous amounts of noise. For noisy nonsinusoidal (e.g., square wave) signals, of course, the find1 correlogram waveform will be different as discussed above. However, as with the sine-wave signal, the signal information can be extracted from the noise merely by monitoring the correlogram only at nonzero delay values. Autocorrelation provides several advantages over the more conventional methods of signal-to-noise enhancement (e.g., lock-in amplification) (3). Among these are direct background correction, obviation of the need for a reference or synchronizing wave, retention of all signal frequencies, digital compatibility, elimination of drift, and minimization of the effects of impulse and interference types of noise. The importance of these characteristics in the application of correlation to atomic fluorescence flame spectrometry will become evident in the text of this paper. Cross-correlation analysis is similar in concept and structure to autocorrelation except that in cross-correlation the correlation between two waveforms is examined rather than

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2 , FEBRUARY 1973

coherence within a waveform. More descriptively, a crosscorrelogram contains information about the frequencies which are common to two waveforms, one of which is usually the signal and the other a reference wave. For example, if the signal and reference waves are both sinusoidal and of the same frequency, the situation is similar to that discussed earlier concerning autocorrelation. Here, however, the reference wave is clean (noiseless) so that no delay point can be found where the noise accompanying the signal will be coherent with anything in the reference. Even if some noise exists on the reference wave, it will be incoherent with the noise on the signal. Therefore, the time average of the product of the noises or of the signal noise with the reference will be zero, regardless of the delay chosen. From this discussion, the cross-correlogram of a noisy sine-wave signal with a sinusoidal reference wave of the same frequency will be a nearly noise-free sine wave of the same frequency and of amplitude equal to the product of the signal and reference-wave amplitudes. For nonsinusoidal signals or reference waves, the situation will be more complex, of course, although the cross-correlogram will again be nearly noise-free and will contain all frequency components common to both signal and reference waves. Because only sine-wave signal and reference waves have been employed in this study, these other, more complex cases need not be considered here but can be understood from some other relatively simple discussions of correlation theory (3,12,21). The advantages of cross-correlation as a signal-enhancement technique are similar to those of autocorrelation, although cross-correlation, as mentioned before, requires a reference wave which is synchronized to the signal. Crosscorrelation has the added advantage, however, of being able to provide greater signal levels and thereby greater signal-tonoise ratios than autocorrelation. The reason is that in cross-correlation, the reference wave can be of considerably greater amplitude than the signal. This will produce a correlogram whose signal information is enhanced by an amount proportional to the reference-waveform amplitude, while the noise is held to a low level. This advantage is especially important in the measurement of small signals which carry a great deal of noise--i.e., in a situation where the signal-tonoise ratio is less than 1. This case, which arises in most analytical techniques near their detection limits, often results in overload or saturation of the signal-conditioning or amplifying system because of the large amount of noise present. In order to obtain a measurable signal, a rather considerable amount of amplification must often be employed in these cases. The noise, being even larger than the signal, can then overload the amplifier. The noise-free character of the reference wave used in cross-correlation eliminates this problem in at least one correlation channel, so that greater output signal levels can result. Application of autocorrelation and cross-correlation to atomic fluorescence flame spectrometry is straightforward. The necessary instrumentation is simple and flexible, with the only added component being a correlation computer. Although auto- and cross-correlation can both be performed on a digital computer, a hardware correlator was employed in this investigation. In this way, the correlator can be directly compared to other signal-measurement systems, in particular, to the commonly-used lock-in amplifier. The components and equipment used in these studies are discussed in the next section. -

(21) B. LuBow, Elecfronics, Oct. 31, 1966, p 75.

Source current

Correlotor

amp

0.25 m Ebert spectrometer

-

generator 310 Hz

regulator

-

Lock-in amp1ifier

-

Ref. out.

-b

-

-

I

Recorder

I

Figure 1. Schematic of the experimental arrangement EXPERIMENTAL

Because only a comparison of techniques was sought, the experimental system used in this study was chosen and designed for convenience and simplicity rather than for optimum performance. High quality components were used throughout but no attempt was made at complete optimization or thorough characterization of the components, their operating parameters, or the system as a whole. To illustrate the effectiveness of the correlation techniques in flame spectrometry, two readout systems were simultaneously connected to the assembled atomic fluorescence unit. The two readout systems, employing lock-in amplification and the selected correlation technique, were both used to constantly monitor the fluorescence signal, so that any difference observed between the two could be confidently ascribed to actual signal processing characteristics rather than to an unknown instrumental artifact. The actual instrumental components used in this study are illustrated in block diagram form in Figure 1. Atom Reservoir. The atom reservoir used throughout this work was an air/hydrogen turbulent flame produced by a total consumption atomizer-burner (No. 4020, Beckman Instruments). Although this flame is certainly not optimal in its atom formation characteristics and its freedom from interelement interferences, it is simple and safe to maintain and employ and has been found to provide excellent fluorescence yields in previous work (22-24). Pre-purified fuel and oxidant gases were supplied from compressed gas cylinders via double stage regulators; gas flows were controlled by needle valves and monitored by dual-float rotameters. In accordance with the burner manufacturer's suggestions, air was employed as the aspirating gas and was delivered at 23 psig. Hydrogen was supplied at 12 psig and 13.5 liters/min. Optical System. The primary light source employed was an ASL (Atomic Spectral Lamps) rhodium hollow cathode discharge lamp. The choice of rhodium as a test element in this work was arbitrary and was based primarily on an inhouse need. The lamp was powered electronically by a modulatable power supply which was designed for an earlier investigation (25). The supply was sinusoidally modulated (22) M. P. Bratzel and J. D. Winefordner, A m l . Lett., 1, 43 (1967). (23) D. W. Ellis and D. R. Demers, ANAL.CHEM., 38, 1943 (1966). (24) J. D. Winefordner and R. C . Elser, ibid., 43(4), 24A (1971). (25) B. E. Holder, R. Lim, A. Maddux, and G. M. Hieftje, ibid., 44, 1716 (1972).

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Rhodium concentration, ppm

Figure 4. Rhodium working curves for atomic fluorescence 0 Obtained using cross-correlation 0

Obtained using a lock-in amplifier

D e l a y T i m e (7)

Figure 2. Cross correlogram of a sine-wave reference signal and atomic fluorescence signal obtained using 100 ppm Rh solution Effective time constant, 13 seconds Cl ,z(r)= cross-correlationfunction =

0

1

2

vi(t)vz(f - 7)df

3

4

Time (minutes)

Figure 3. Lock-in amplifier output obtained from the atomic fluorescence of a 100-ppm Rh solution Lock-in amplifier time constant, 10 seconds

by the output of a suitable oscillator (Model 3300A, HewlettPackard Co.), which also provided a synchronized 10-V, p-p reference wave when it was needed. To avoid lamp damage, and to simulate expected “normal” operating conditions, the sinusoidal current to the lamp was regulated at 20 mA rms. A simple optical system was constructed which consisted of a 50-mm focal length plano-convex quartz lens used to image the hollow cathode lamp radiation onto the flame at a position 3.7 cm above the burner tip. The fluorescent radiation from the flame was by placing the burner as as possible to the entrance slit of the monochromator. This configuration, although primitive, gave excellent fluorescent signal levels. 256

Readout. The 0.25-meter monochromator which was selected (Model 82-410, Jarrell-Ash Co.) was provided wit! fixed 2000-micgron slits to give a spectral bandpass of 33 A at the 3435 A rhodium resonance line, which was used throughout the study. An R106 photomultiplier detector (Hamamatsu TV Co.) was operated at 500 V, supplied by a high voltage power supply (Model 412, John Fluke, Inc.). The resulting photocurrent was fed to a photometric preamplifier (Model 184, Princeton Applied Research Corp.) and a correlation computer (Model 3721A, Hewlett-Packard Co.) or a lock-in amplifier (Model 126, Princeton Applied Research Corp.). These items were chosen to represent their respective classes of modern sophisticated instrumentation. In particular, the Hewlett-Packard correlator was chosen for its ability to perform both auto- and cross-correlation and to display the resulting correlograms in real time on an integral cathode ray tube display. Alternatively, the computed correlogram can be conveniently plotted automatically on a strip chart recorder using an appropriate output terminal on the correlator. The correlograms can be computed using either a total number of sampled signals or, alternatively, with an exponential weighting system which enables continuous updating of the correlogram. Reagents. Rhodium standard solutions were prepared by dilution from a stock which was obtained by dissolution of the reagent grade chloride salt. Measurement of S/N Ratios and Determination of Detection Limits. The measurement of signal-to-noise ratios (S/N) is essential for the determination of the detection limit (26, 27) and for a meaningful comparison of any two analytical techniques. In this study, the signal-to-noise ratio at the outputs of two distinctly different signal processing devices was evaluated. From an autocorrelogram, it is a simple matter to calculate S/N for the original signal being correlated (3) ; to do so for the final correlogram is less straightforward. In this work, the S/N of the output of the lock-in amplifier was obtained by dividing the dc signal level by the peak-to-peak excursions in the signal (28). A similar procedure was used for the correlation techniques, where the amplitude of the correlated signal was divided by the peak-to-peak deviations from the recorded correlogram. Detection limits were defined as the concentration of rhodium which generated an S/N = 2. (26) P. A. St. John, W. J. McCarthy, and J. D. Winefordner, ANAL. CHEM.,39,1495 (1967). (27) R.Gabriels, ibid., 42, 1439 (1970). (28) V. D. Landon, Proc. IRE, 50 (February, 1941).

ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973

3.21 mrec*

0

1

2

3

4

Time, m i n Delay Time

(7)

Figure 5. Illustration of the effect of impulse noise on a Cross Correlogram. Signal due to 100 ppm rhodium solution Effective time constant, 13 seconds

Figure 6. Chart records illustrating the effect of impulse noise on a lock-in amplifier output. Arrows indicate introduction of impulses A. B.

10-second time constant 1-second time constant

RESULTS AND DISCUSSION

Using sinusoidally modulated hollow cathode radiation and a sine wave reference where needed, lock-in amplification was simultaneously used with either auto- or cross-correlation to determine the detection limit attainable with each technique. Typical outputs are shown in Figures 2 and 3 for a cross-correlogram and lock-in amplifier output, respectively, each of which was obtained using a solution containing 100 ppm rhodium. The autocorrelogram which was obtained under these conditions is similar to Figure 2, except for its somewhat smaller amplitude and for the presence of a “spike” at zero delay indicating the random noise contribution. For purposes of comparing signal-to-noise ratios, the preamplifier gain was increased to a point where l/f noise (i.e., drift) in the signal became evident. Signal records of typically 2.2 minutes were obtained for all measurements to include this drift and enable a meaningful estimate of S/N. It was immediately found that, because a reference waveform was conveniently available, autocorrelation possessed no advantages over cross-correlation, with the latter technique invariably exhibiting slightly better readout signal-to-noise ratios. For this reason, all further comparisions were made between the cross-correlation technique and lock-in amplification. In all signal determinations, it was necessary to subtract a blank reading caused by scattering. Scattering of the primary source radiation has been recognized as a serious problem in atomic fluorescence flame spectrometry since its inception (29, 30) and is especially troublesome near the de(29) J. D. Winefordner and T. J. Vickers, ANAL.CHEM., 36, 161 (1964). (30) J. D. Winefordner and R. A. Staab, ibid., p 165.

tection limit, where the fluorescence signal is only slightly larger than the scattered radiation. This problem is aggravated by the use of total consumption burners, which generate rather large droplets to increase the total amount of scattering (24, 31). Source modulation, as used in this study, is of no help in alleviating this problem, because scattering and fluorescence are both modulated identically and cannot be differentiated on that basis. To correct for scattering, blank readings were obtained during aspiration of water and were subtracted from signal readings. The signal readings, corrected for scatter, were obtained for a number of rhodium solution concentrations using both signal processing techniques. The resulting analytical curve is displayed in Figure 4 for the correlator and lock-in amplifier processors. The curve in Figure 4 indicates the response characteristics of the two readout systems to be similar on a relative basis and that easily measurable fluorescence signals can be obtained for a broad range of rhodium solution concentrations. These signal levels, however, cannot be indefinitely amplified for the detection of increasingly dilute solutions. For very low concentrations and for the evaluation of a detection limit, the noise output of each signal processing system must be considered. In the determinations indicated in Figure 4, the noise level at the output of the lock-in amplifier was several times that displayed by the correlator for a given concentration and signal level. From these noise levels, the detection limit for rhodium was 0.16 ppm using the correlator readout although with the lock-in amplifier only 0.91 ppm could be detected. These detection limits indicate the superiority of crosscorrelation over lock-in amplification as a signal processing (31) D. R. Demers and D. W. Ellis, ibid., 40, 860 (1968).

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system for conventional atomic fluorescence spectrometry. Additionally, in many applications, especially those involving significant amounts of impulse noise, a further advantage will be realized. Impulse noise is generated by such sources as spark generators, lightning flashes, and the start-up of large equipment (e.g, motors) and is characterized by a short, powerful pulse. This pulse is easily picked up by electronic instrumentation where it appears as noise. In this study, impulse noise was simulated by striking a brief spark from a Tesla coil to the grounded photomultiplier housing. Sufficient transmission occurs through the housing to cause a brief noise pulse on the measured photocurrent while the grounding serves to protect the preamplifier from an excessive current surge. The effects of the impulse on the readout from the correlator and the lock-in amplifier are shown in Figures 5 and 6, respectively. The traces in Figures 5 and 6 represent the readout signal obtained from a 100-ppm rhodium solution, in the presence of an impulse noise source. In Figure 6, it is seen that large excursions are produced in the output of the lock-in amplifier whenever an impulse (marked by arrow) occurs. When several impulses occur, it becomes rather difficult to establish a “true” level for the output. This situation, which often occurs when impulse noise is present, is seen to greatly increase the readout noise and can lead to considerable error in the recorded signal value. The correlator output, displayed in Figure 5 , is considerably less susceptible to impulse noise. The impulses, which still cause deviation in the output signal, can now be simply discarded as being obviously deviant from the otherwise smooth sinusoidally varying waveform. Using simple testing criteria, even large amounts of impulse noise need cause no error in this type of readout, thereby effectively eliminating its influence. Even with the advantages presented above, it is questionable that many flame spectrometric applications will justify the additional expenditure necessary to purchase a correlator over a lock-in amplifier. However, with the increasing introduction and use of small digital computers into the analytical

laboratory, it is likely that most flame spectrometers will eventually be interfaced for purposes of sample handling and data collection and processing. With the availability of such systems, the purchase of a hardware correlator will no longer be necessary. Auto- and cross-correlation, unlike lock-in amplification, can be conveniently performed numerically on a digital computer. If such a digital system is already available, adoption of a correlation technique for signal processing could even result in a savings over the cost necessary for a lock-in readout system. For the calculation of a cross-correlogram, the necessary reference waveform could even be stored digitally in the computer memory, to obviate the need for repeated sampling of the reference waveform. This approach, while requiring a very stable source modulation frequency, is typicai of the advantages which can be derived from a software approach to correlation. Whether a hardware or software system is employed, correlation has been shown in this study to be a realistic and often advantageous alternative to conventional readout systems. In particular, cross-correlation can provide increased signalto-noise ratios and lower detection limits than a commercial lock-in amplifier. The additional advantage of freedom from the effects of impulse noise also makes cross-correlation the signal processing technique of choice in many applications. Freedom from drift, direct digital compatibility, and the ability to handle nonsinusoidal signals, all combine to encourage further use of this technique in other studies in flame spectrometry and other methods of chemical analysis.

RECEIVED for review August 28, 1972. Accepted October 11, 1972. One of the authors (GMH) wishes to acknowledge the partial support of this work through National Institutes of Health Grant GM 17904-01. Work performed under the auspices of the Atomic Energy Commission. Reference to a company or product name does not imply approval or recommendation of the product to the exclusion of others that may be suitable.

Electrochemical Characteristics of the Gold Micromesh Electrode W. J. Blaedel and S. L. Boyer Department of Chemistry, University of Wisconsin, Madison, Wis. 53706

The design and construction of a flow-through gold micromesh electrode are described. Current-voltage curves are reported for various flow rates. Measured limiting currents are shown to be directly proportional to the number of screens ( N ) in the electrode, to the concentration of electroactive material (c), and to the cube root of the volume flow rate (V,) of solution through the electrode. Various mesh sizes are examined. Application is made to the measurement of su bmicromolar concentrations.

electrodes, such as higher sensitivity, simplicity, and versatility (3). The platinum rotating electrode ( I , 4) and the platinum tubular electrode (PtTE) (3, 5 ) have been utilized for quantitation purposes. Fundamental studies of these two electrodes have shown that the limiting current for a reversible reaction controlled only by mass transfer may be represented by the following equations: iL = 1.88

x

105 nDZl3 v-l’B r2

wlI2

C (rotating electrode) (1)

FORCED-CONVECTION ELECTRODES have been reviewed by Adams ( I ) and Nicholson (2). Convective transport gives these electrodes several analytical advantages over stationary

iL = 5.306 x l o 5 nD213 X213 C (PtTE) (2) Each hydrodynamic relationship contains a dependence upon electrode velocity (a) or solution flow rate (V,), electrode

( I ) R. N. Adams, “Electrochemistry at Solid Electrodes,” Marcel Dekker, New York, N.Y., 1969. (2) R. S. Nicholson, ANAL.CHEM., 44 ( 5 ) , 478R (1972).

(3) W. J. Rlaedeland S. L. Boyer, ANAL.CHEM., 43,1538 (1971). (4) V. G. Levich, “Physiochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N.J., 1962. ( 5 ) W. J. Blaedel and L. Klatt, ANAL.CHEM., 38,879 (1966).

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