mm/min, this corresponds to 0.2 ml/mm X 3.13 mm = 0.626 ml = 626 pl. Aspiration of a 0.01 mg/l. (Cu) copper-NTA SOlution a t a rate of 2.00 ml/min for 0.313 min corresponds to aspiration of a total of 626 pl of copper-containing solution. This should yield the "square-wave" signal in Figure 2. The amount of copper which would elute during this period would equal 626 p1 X 0.01 ng/pl = 6.26 ng. Assuming a triangular peak, the area of a chromatographic peak of copper-NTA which would reach the detection limit value at maximum peak height is half of the area of the "square-wave" peak. This is shown in Figure 3. Therefore, the minimum detectable amount of copper-NTA corresponds to 3.13 ng of copper. A similar calculation for copper-EDTA on the column used in this study predicts a detection limit of 6.72 ng of copper for this peak.
RESULTS AND CONCLUSIONS The validity of the above calculations was tested by comparison to actual detection limits from known amounts of the copper chelates of NTA and EDTA. Following the arguments advanced above, the minimum detectable amount of copper-NTA would generate a triangular peak of 30 mm2 area and the area of the copper-EDTA peak would be 68 mm2. Experimentally it was found that 100 p1 of a copper-NTA solution gave a peak with an area of 280 mm2. The area of a corresponding copper-EDTA peak was 288 mm2. By proportion, the copper content of the copper-EDTA peak at the detection limit is, 100 ng 30 mm2 x -- 10.7 ng Cu for copper-NTA 280 mm2 and the copper content of the copper-EDTA peak at the detection limit is the following: 100 ng 68 mm2 X = 23.6 ng Cu for copper-EDTA 288 mm2 Comparison to the calculated values of 3.13 ng of copper for ~
copper-NTA and 6.72 ng for copper-EDTA shows that the experimentally determined detection limits are 3.4 and 3.5 times as high as the calculated values for the NTA and EDTA chelates, respectively. The consistency of these two ratios lends credibility to the approach. The differences between calculated and experimental values could be due to less than ideal behavior near the detection limit for copper. In addition, the assumption that the detection limit for a metal at the height of a chromatographic peak is as low as that for continuous aspiration is likely to be too optimistic. In summary, flame spectrometric methods of detection for HSLC have unique applications for the determination of certain types of species. Unfavorable detection limits can be a major handicap for this application. A method has been presented for the calculation of these limits from known chromatographic and spectrophotometric parameters.
LITERATURE CITED (1) S. E. Manahan and D. R. Jones IV, Anal. Lett., 6,745 (1973). (2)M. Unebayashi and K. Kitagishi, 5th International Conference on Atomic Spectroscopy, Monash University, Melbourne, Australia, Aug. 25-29, 1975. (3) D. R. Jones iV and S.E. Manahan, Anal. Lett., 8, 569 (1975). (4)D. R. Jones IV, H. C. Tung, and S. E. Manahan, Anal. Chem., 48, 7 (1976). (5) D. R . Jones IV and S. E. Manahan, Anal. Chem., 48, 502 (1976). (6)D. J. Freed, Anal. Chem., 47, 186 (1975). (7)A. Y. Cantilb and D. A. Segar, Proceedings of the InternationalConference On Heavy Metals in Environment, Toronto, Canada, 1976. ( 8 ) A. J. P. Martin and R. L. M. Synge, Biochem. J., 35, 1358 (1941). (9)H. C. Thomas, J. Am. Chem. Soc., 66,1664 (1944). (IO)C. E. Boyd, L. S.Meyers, and A. W. Adamson, J. Am. Chem. Soc., 6% 2849 (1947). (11)I. M. Kolthoff, E. B. Sandell, E. J. Meehan, and Stanley Bruckenstein, "Quantitative Chemical Analysis", 4th ed., The Macmillan Company, New York, N.Y., 1969. (12) Perkin-Elmer Corp.. "Anal. Methods for AAS," "Reference Manual" for Perkin-Elmer Model 403 AAS, March 1971.
RECEIVEDfor review July 7,1976. Accepted August 13,1976. This research was supported by National Science Foundation Grant No. MPS75-03330 and USDI OWRT Matching Grant B-095-MO.
Autocorrelation Analysis of Noisy Periodic Signals Utilizing a Serial Analog Memory K. R. Betty and Gary Horlick" Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada, T6G 2E1
The nature of autocorrelation analysis is briefly reviewed and then a new type of Integrated circuit Is described that is uniquely suited to the development of autocorrelationinstrumentation. This integrated circuit can be generally classified as a discrete time analog signal processlng device and is called a serlal analog memory. The serlal analog memory can temporarily store 128 consecutivesignal samples directly as analog levels. An autocorrelator has been constructed using the serial analog memory as a variable analog delay line. The operation of the autocorrelator is illustrated using several periodic signals. A noisy sine wave signal ( S I N < 0.2) can easily be recovered using this autocorrelator.
Correlation techniques have long been used to measure and process signals in the chemical, biological, physical, and engineering fields. Lee (1)discussed the application of correlation analysis to the detection of periodic communication sig-
nals rather early and later reviewed the topic in some detail ( 2 ) .Correlation techniques, again applied primarily to the communication field, have also been discussed by Lange ( 3 ) and were applied at a relatively early stage to the analysis of electroencephalographic data ( 4 , 5 ) .Correlation techniques have been applied to the measurement and processing of spectrochemical data (6, 7) and the application of correlation methods to chemical data measurement has recently been reviewed (8). Although correlation techniques have been employed in these and other fields with considerable advantage and success, their use has unfortunately been somewhat limited primarily by a lack of effective methods and instrumentation for the rapid, automatic evaluation of correlation functions. Present developments in certain large scale integrated circuits such as diode arrays, charge coupled devices, and bucket brigade devices are now beginning to provide very inexpensive instrumentation capable of sophisticated real time correlation operations (9). These integrated circuits can be generally classified as discrete time analog signal processing
ANALYTICAL CHEMISTRY, VOL. 48, NO. 13, NOVEMBER 1976
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Flgure 1. ( a )Autocorrelation of a sine wave. ( b ) Autocorrelation of a
square wave devices. One such device is a serial analog memory. When operated as a variable analog delay line, it allows real time autocorrelation processing of noisy periodic signals. In order to facilitate discussion of this application a brief outline of autocorrelation analysis is presented in the next section. This discussion is based in part on material presented in References
6-8,lO. WHAT IS AUTOCORRELATION? Correlation analysis provides information about the coherence within a signal or between two signals. The correlation function of two signals is obtained by evaluating the time averaged or integrated product of the two signals as a function of their relative displacement. Mathematically, correlation can be expressed as:
where c a b (7)is the correlation function between the two signals a ( t ) and b ( t ) , and r is their relative displacement. The signals can be a function of essentially any variable, e.g. wavelength, retardation, frequency, accelerating voltage, time, etc. Thus, if a and b are considered to be functions of time, the correlation function c a b will be related to and plotted vs. the relative time delay between the two signals. Two different correlation operations can be identified. If a ( t ) and b ( t ) are identical (Le., if a = b ) , an autocorrelation function is obtained by application of Equation 1. Thus, autocorrelation indicates whether coherence exists within a signal. In contrast, a cross-correlation function, produced if a ( t ) and b ( t ) are different, shows the similarities between two signals. To illustrate the process of correlation, let us consider the autocorrelation of a sine wave as shown in Figure la. In the first frame, the sine wave is multiplied by itself in phase, i.e., 7 = 0'. The result as illustrated is a sine2 wave. To complete the evaluation of the autocorrelation function at this relative displacement, the average value of the sine2 wave must be determined (see Equation 1).The dashed line indicates this value. This first point of the autocorrelation is plotted in the lower part of Figure l a and represents the mean square value of the original sine wave. 1900
In the successive frames of Figure l a , the evaluation of the autocorrelation function a t 7 values of 90°, 180°, and 270° is illustrated. From the plot of the autocorrelation function C,, in Figure l a , it should be apparent that the autocorrelation operation applied to any sinusoidal wave (irrespective of its phase) converts it to a cosine wave. Thus phase information contained in the original sine wave is lost. However, the amplitude and frequency (or period) of the autocorrelation function are unambiguously related to those of the original sine wave. The periods of the two are identical while the amplitude of the autocorrelation function is the mean square of the original sine wave. The autocorrelation function of any periodic waveform will display similar characteristics since any such waveform, as can be shown by Fourier analysis, is just a combination of a number of sine waves, each with its own amplitude and phase characteristics. But, as mentioned above, although frequency and amplitude information about the original waveform are carried in the autocorrelation function, the phase information in the original signal is lost which results in a loss of knowledge about the exact shape of the original signal. This fact is emphasized by the autocorrelation function of a square wave generated as shown in Figure l b . The triangular wave autocorrelation function contains the same frequencies (Le., sine wave components) as the original square wave; however, in the triangular wave, all the sine wave components are in phase a t r = Oo and have different relative amplitudes related to the mean square of the amplitudes of the square waves sine wave components. Thus they produce a different summation wave shape, i.e., the triangular wave, from which the original signal cannot be regenerated without some a priori information about the phase relationship of the sine wave components in the original signal. Autocorrelation of nonperiodic or random (noise) waveforms produces markedly different results from those obtained for periodic waves. To understand this, we need only recognize again that any wave, whether periodic or not, is composed of sine wave components. Each of these components, when autocorrelated, will produce a cosine wave beginning at r = 0. When many such components are present in a parent waveform, the resulting autocorrelation will be the sum of the autocorrelation functions of the components. Four such autocorrelation images are shown in Figure 2a. A true random waveform (white noise) contains all frequencies. The cosine autocorrelation images of all these frequencies reinforce a t r = 0 to produce a value equal to the mean square of the original random signal and a t any point beyond r = 0 they destructively interfere to produce a time averaged value of zero (see Figure 2b). In order to obtain this ideal autocorrelation function of random noise, the bandwidth of the noise must be infinite. Of course, no process in nature is truly random. Real random waveforms are "band-limited" and do not produce a single spike upon autocorrelation but instead a shape such as portrayed in Figure 2c. The functional form is a decaying exponential (for low pass filtered random noise) whose width is inversely proportional to the bandwidth of the noise waveform. When the autocorrelation functions of bandlimited random noise (Figure 2c) and of periodic signals (Figure 1)are compared, it is clear how autocorrelation can be a powerful technique for the extraction of periodic signals from noise. Random noise contributes to the autocorrelation function only at very small values of r. A periodic signal, in contrast, will continue to contribute even at very large values of 7. The autocorrelation function of a noisy sinusoid is shown schematically in Figure 2d. The amplitude and frequency of the periodic signal buried in noise can be determined simply by examining the autocorrelation function a t a location well removed from
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