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Anal. Chem. 1991, 63,708-712
Digital Filtering Applied to Diode Array Fluorescence Spectroscopy B a r r y K. Lavine,* Ramesh Patel, and Abdullah Faruque
Department of Chemistry, Clarkson University, Potsdam, New York 13699 Jovan
M.Nedeljkovic
Boris Kidric Institute, Belgrade, Yugoslavia 11000 Seshardi Mohan
Department of Electrical and Computer Engineering, Clarkson University, Potsdam, New York 13699
The development of a soflware-based strategy for signal enhancement in fluorescence spectroscopy uslng spectral e& timation techdques and flntte impulse response (FIR) fllterlng is described. Autoregresslve moving average (ARMA) modeling, a spectral estimation technique, Is used to detect the frequency content of the dgnai in nolsy data. This Information is then used to design an FIR filter for the chemlcal system under investigation. Hence, a priori information about the signal is not required. The approach was successfully demonstrated, using a test data set of fluorescence spectra obtained wlth a gated linear diode array spectrometer for two fiuorophores, fluorescein and eosin Y. The combination of ARMA modeling and FIR filtering is a very powerful approach for capturing small signals from very noisy data sets and is applicable to a large number of spectroscopic techniques.
INTRODUCTION Due to its high sensitivity and specificity, fluorescence spectroscopy is a widely used analytical technique. It has been applied to many problems in chemical analyses, e.g., the identification of polychlorinated biphenyls (PCBs) and polynuclear aromatic hydrocarbons (PAHs) congeners in environmental effluents (l),and the detection of quinine, LSD, and other physiologically active agents in blood and urine samples (2). With the increasing emphasis of our society on solutions to environmental problems, the demands placed on fluorescence spectroscopy will be even greater in the near future. It will be necessary, to determine the presence of ultratrace organic contaminants in extremely complex matrices. Moreover, these analyses will also have to be global; Le., one will have to be able to detect and measure as many of the sample constituents as possible in the matrix. For the fluorescence spectroscopist, the task is clearly 2-fold: (1)to extend the limits of detection of the technique and (2) to improve the capability of fluorescence spectroscopy for multicomponent analysis. In order to extend detection limits in fluorescence spectroscopy, workers in the field have already taken advantage of the advances in laser technology and multiparametric detection devices ( 3 ) . This has resulted in a dramatic improvement in the signal-to-noiseratio, hence an improvement in the detection limit. However, signal enhancement in fluorescence spectroscopy ( 4 ) can also be achieved through the implementation of software-based strategies, i.e., approaches based on the separation of the significant data containing signals from the noise through the application of postmeasurement mathematical techniques. The use of high-speed A/D converters makes it convenient to acquire and 0003-2700/91/0383-0708$02.50/0
store signals in digitized form. Hence, the implementation of digital filtering for signal enhancement is both practical and convenient. In this study, the development of a software-based strategy for signal enhancement in fluorescence spectroscopy is presented. The strategy is based on the use of spectral estimation techniques (5) and low pass filters, i.e., finite impulse response (FIR) filters (6, 7). The spectral estimation techniques are used to detect the frequency content of the signal in noisy data. This frequency information is then used to design a low pass filter for the chemical system under investigation; hence, a priori information about the signal is not required. A successful application of this strategy is described, using fluorescence data obtained with a gated linear diode array spectrometer for two fluorophores: fluorescein and eosin Y. THEORY The most common type of digital filter is the FIR filter. The basic equation describing the filter is m
y ( n ) = %h(lz)x(n- k ) k=l
(1)
y ( n ) is the filtered data point, h(k) is the filter coefficient, and x(n - k ) is the raw data point. The order of the filter is determined by the number of filter coefficients used in the equation. These coefficients are obtained from a Fourier series expansion of the filter function. They define the impulse response function of the filter
h(k) = 1 / ( 2 7 r ) ~ w E e jdw wk -%
where -w, is the lower cut-off frequency and w, is the upper cut-off frequency of the FIR filter. For a low pass filter, the above equation reduces to sin 2rf& W )= ak (3) where f, is the cut-off frequency of the low pass filter. Typically, a hundred or so coefficients are used to define the impulse response function of the filter. This truncation, however, will lead to oscillations near the point of discontinuity. Therefore, workers in the field typically use a Hanning window (8) to minimize the effect. To determine the cut-off frequency of the FIR filter, a priori knowledge is needed about the bandwidth of the signal. This information, however, is often not available to the experimenter. Nevertheless, it is possible for the experimenter to extract this information directly from the raw data using spectral estimation techniques. Spectral estimation in this context can be viewed as a three-step procedure: (1)model selection, (2) model parameter estimation, and (3) power 0 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63,NO. 7, APRIL 1, 1991
spectral density (PSD) estimation using the model parameters. In this study, the PSD was determined by using an autoregressive moving average (ARMA) model to represent the experimental data. A PSD plot developed from the ARMA parameters shows the region(s) of the power concentration of the signal; hence, this plot can be used to determine the cut-off frequency of the filter. The ARMA process is based on the following linear difference equation P
~ [ n=]- C a [ k ] x [ n- k ] k=l
(5)
where
"I
r A q - 1) r&)
r,(q - p + r,(q - p + 2)
Wavelength (nm)
Absorbance spectra of (a)fluorescein and (b) eosin Y. The spectra were obtained by using a Perkin-Elmer 559A UV/vis absorbance spectrophotometer. Flgure 1.
Fourier transform on the a [ k ] and b[k] terms (see eq 7). In this study, a 1024-point discrete Fourier transform was implemented. The number 1024 corresponds to the number of discrete data points obtained from the gated linear diode array used to generate the experimental data. The relationship used to compute the PSD from the ARMA parameters is P
... ...
...
...
I
(4)
Rx,A = -rxx
+ 1)
1
P
+ k=O C b [ k ] u [ n- k]
where x [ n - k ] and x[n] are the original data, n is the number of data points used to represent the spectra, a[k] are the autoregressive coefficients, b[k ] are the moving average coefficients,and u[n- k] is the white noise. The autoregressive component of the ARMA process is Ca[k]x[n - k], and it represents the signal component of the data. The moving average component is Cb[k]u[n - k], and it represents the noise. T o compute a [ k ] and b[k], it is necessary to perform an autocorrelation function on the raw spectral data. The a[k] terms are then computed directly from the autocorrelation matrix by using the modified Yule-Walker relationship
rdq
80.0
709
...
+ kC= l b ( k ) exp(-j2?rfk)12
11
PSDmMA(f) =
r,(q + P - 1) m ( q + P - 2) m ( q )
U'
P
11
+ kC= la ( k ) exp(-j2?rfk)12
(7)
where
.
I
...
I
=
I1...
I
I
-.*
rXx(q + P)
The variance of the white noise in the data is u2, r,[O] is the first term of the autocorrelation function, and r,[k] is the kth term of the function. The white noise is assumed to be Gaussian with a mean of zero and standard deviation of u. The ARMA model is capable of detecting the power concentration of signals that have either a broad or narrow frequency range. Therefore, it is well suited for the application described herein.
I
J
p is the order of the autoregressive part of the ARMA process, q is the order of the moving average part of the ARMA process, and rJq) is the qth term of the autocorrelation matrix of the original data. (The user must select the values for p and 4.) Because R, is a non-Hermitian Toeplitz matrix, the modified YuleWalker relationship is solved for the a[k] terms by using an extension of the Levinson recursion algorithm (9). The b[k] terms, i.e., the moving average coefficients, are determined by the Durbins method (IO). First, an autoregressive model of order L is developed from the raw data where p
E
-=
b
L
--
Figure 10. Intensky of the filtered signal at diode position 250 plotted against exposure time for fluorescein (slope = 1.322) and eosin Y (slope = 11.23)
two fluorophores. The plots of the other diodes showed similar characteristics. The correlation between signal intensity and exposure time is very high; in fact, the r2 value for this relationship is near unity for all of the non-zero-filled diodes, regardless of the fluorophore. These results indicate that signal intensity is linearly related to exposure time for the filtered data, even if the excitation wavelength used to excite the fluorophore does not correspond to the absorbance maximum of the compound, e.g., eosin Y. The results also show that the sensitivity of the signal, i.e., the rate of change of the signal with respect to the exposure time, is characteristic of the fluorophore (see Figure 10). This property is by no means restricted to the DARSS and can be found with photomultiplier tubes (PMTs) as well. In the latter case, the gain of the PMTs can be adjusted by the applied high voltage or by using a switch-selectable dynode system (1I, 12). For applications such as on-line high-performance liquid chromatography (HPLC) detection, the DARSS may be advantageous insofar as complete spectra can be recorded in very short times.
A R M modeling and FIR filtering offer a general solution to the problem of low signal to noise in fluorescence spectroscopy. However, it is important to note that this approach can also be used to improve the capability of fluorescence spectroscopy to perform multicomponent analysis. Digital filtering in combination with spectral subtraction or selfmodeling curve resolution (13) can be a very useful method for deconvoluting overlapping fluorescent bands. (Techniques like self-modeling curve resolution work best when the data have high signal to noise.) In fact, the use of signal-processing techniques will make the development of an on-line detection scheme for HPLC which employs a DARSS to resolve overlapping chromatographic bands possible. Such a scheme will be important in the analysis of complex mixtures. (Giddings and Davis (14) have shown that the likelihood of overlapping chromatographic components is much higher than previously realized.) A study is presently underway in our laboratory to evaluate the viability of this scheme.
LITERATURE CITED Lee, M. L.: Novotny, M.; Bartle, K. D. Analyfical Chemistry of Polycyclic Aromatic Compounds; Academic Press: New York, 1981; 462 PP. Guilbault, G. G. Practical Fluorescence: Theory, Methods, and Technique;Marcel Dekker: New York, 1973. Talmi, Y., Ed. Multichannel Image Detectors ; ACS Symposium Series 102; American Chemical Society: Washington, DC, 1979; 351 pp. Jagannathan, S.; Patel, R. C. Anal. Chem. 1886, 58, 421-427. Key, S. M. Modern Spectral Estimafion ; Signal Processing Series; Prentice Hall: Englewocd Cliffs, NJ, 1988; 542 pp. Cappellini, V.; Constantinides, A. G.; Emillanl, P. Dlgital Filters and Their Applications; Academic Press: New York, 1978; p 210. Balkowski, S. E. Anal. Chem. 1888, 60 (5),355A-361A. Harris. F. J. On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transforms. R o c . I€€€ 1978, 6 , 51-83. Trench, W. F. J. Soc. Ind. Appl. Math. 1964, 72,515-522. Durbins, J. Biometrika 1858, 46, 306-316. Patel, R. C. J. Chem Inst. 1876, 7 , 83-90. Chu, P.; Patel, R. C.; Matijevic, E. Appl. Spechosc. 1987, 4 7 , 402. Layon, W. H.; Sylvester, E. A. Technometrics 1971, 13, 617. Davis, J. M.; Giddings, J. C. Anal. Chem. 1883, 55, 418.
RECEIVED for review August 20, 1990. Accepted January 7, 1991.
