97 Analyst, February 1996, Vol. 121 (97-104) Deconvolution of Overlapping Chromatographic Peaks by Means of Fast Fourier and Hartley Transforms* Anastasios Economou, Peter R. Fielden? and Andrew J. Packham Department of Instrumentation and Analytical Science, UMIST, P.O. Box 88, Munchester, UK M60 IQD Chromatographic peaks usually exhibit an exponentially Gaussian modified peak profile that is characterized by a tailing behaviour at the end of the peaks. As a result of this modification of the ideal peak shape, the peaks lose their symmetry and, most importantly, become broader. An unfortunate consequence of this broadening effect is that the resolution between adjacent peaks decreases as compared to the ideal case in which purely Gaussian symmetrical peaks would be considered. In addition to various chemometrical techniques developed to increase the separation of overlapping peaks, methods based on deconvolution in the frequency domain can be exploited.The principle of this latter family of deconvolution methods rests on the division of the frequency spectrum of the signal to be deconvoluted by the frequency spectrum of a judiciously chosen deconvoluting signal. In this paper, the analytical characteristics and utility of deconvolution is assessed with emphasis on aspects of resolution enhancement, data distortion, linearity and signal-to-noise ratios. Keywords: Deconvolution; chromatogsaphy; tailing peak; Fourier- trunsform; chemometrics Introduction Fourier deconvolution of purely symmetrical Gaussian, Lorentzian2 and voltammetric3 peaks has been shown to result in narrower peak profiles, thereby improving the resolution between adjacent peaks in analytical chemistry.Traditionally, the deconvolution has been performed by deriving the discrete Fourier spectrum of the analytical signal and the discrete Fourier spectrum of an appropriate deconvoluting function. The deconvoluting function is usually a peak-shaped function with a peak shape similar (though not necessarily identical) to the expected peak shape of the analytical peaks. After some initial signal manipulation, the former spectrum is divided by the latter spectrum and the inverse Fourier transform is calculated to produce the final deconvoluted signal. The discrete Fourier transform, FV), of a digital time domain signal, x(k), is given by: N 1 F ( f ) = -c N- ‘s(k) N X = O forf = 0, I, 2 ,..., N - 1 ( I ) where N is the number of samples and k is the sample number.The inverse discrete Fourier transform reconstitutes the digital time domain signal from the frequency domain signal: * Presented at The SAC ’95 Meeting, Hull, UK, July 1 I-IS, 1995. + To whom correspondence should be addressed. isin - fork = 0, 1,2 ,..., N - 1 x(k) = C ” - ’ F ( f ) f = O ( c o s p - N 2n$.k N ) 2Jlfk (2) Earlier, a new deconvolution procedure for symmetrical peaks, based on the Hartley transform, rather than the conventional Fourier transform, has been proposed.4 The discrete Hartley transform H(f) of the time domain signal x(k) is given by.5 forf = 0, 1, 2 ,..., N - 1 (3) The inverse discrete Hartley transform is given by: N 2J-!$.k x(k) = x ” - l H ( I ) f = O (cos - N fork = 0, 1 , 2 ,..., N - 1 (4) For chromatographic peaks, the situation is more complex because the peak shape is more often asymmetrical.A sample introduced into a chromatographic system will be subject to various effects as a result of its interaction with this system. Consider a single active component and its concentration distribution with respect to time, t, measured from the point of injection. The input concentration of the system, i(t), can be represented by the rectangular sample plug introduced into the flowing carrier stream. The output concentration of the system, o(t), will be represented by the response recorded at the end of the experiment. The chromatographic system responds to the input pulse in a certain way given by the impulse response function, r(t).A chromatographic system can be described as a linear system. This is usually a valid approximation since, for analytical purposes, operation must be confined within the linear range of the chromatographic system. For such a linear system, the input, i(t), the output, o(t), and the impulse response, s(t), are related according to the general convolution theorem:6 o(t) = r(t) €3 i(t) ( 5 ) where €9 signifies convolution operations. Given that the input, i(t), is a very narrow, rectangular plug of sample, it can be approximated by a unit impulse function, 6, which, by definition, has unity value.7 Eqn. 5 then becomes: (6) Eqn. 6 implies that the output of the chromatographic system will be a good estimate of the system impulse response, provided that the input concentration profile can be approxi- mated by a unit pulse.In a chromatographic system, the impulse response, r(t), is composed of two contributions. (a) A factor, g(t), representing the broadening of the concentration profile o(t) = 6 €3 r(t) = 1 @ r(t) = r(t)98 Analyst, February 1996, Vol. 121 due to several diffusion and mass-transfer mechanisms within the analytical column. This broadening contribution is usually approximated by a symmetrical peak profile. The peak shape is usually Gaussian.8 (7) where A is the peak height; t is the time measured from the moment of injection; tR is the retention time of the peak; and s is the standard deviation of the peak. (b) Various first-order extra-column effects (such as the detector dead volume; the injector dead volume; mixing effect of connectors; etc.).These effects will distort the symmetry of the peak by adding a tailing component at the end of the peak. The extra-column contribu- tions can be collectively represented by an exponential term, e(t).8 e(t) = exp (- z) where T is the time constant of the exponential contribution. The impulse response can then be expressed as the convolution of the two contributing effects. By combining eqns. 7 and 8, the following formula is derived: (9) r(t) = b(t) 63 e(t) Substituting eqn. 9 into eqn. 6: r 8 exp (- f ) Eqn. 10 suggests that the final recorded output will be a convolution of the Gaussian shape with an exponential term. The resulting peak will be termed an exponentially-modified Gaussian (EMG) peak.8 Thus: The shape of the EMG peak will resemble the familiar tailing peak shapes usually obtained in chromatography.The four most important parameters on an EMG peak are: (i) the retention time of the peak, tR (that determines the peak position); (ii) the standard deviation of the peak, s (that determines the peak width); (iii) the time constant of the exponential part, T (that determines the behaviour of the tailing part of the peak; and (iv) the peak height, A . The Fourier deconvolution of exponentially-modified peaks has been treated mathematically .9 For this purpose, simulated EMG peaks were deconvoluted by means of the discrete Fourier transform, using an EMG deconvoluting function. It has been shown that deconvolution had the effect of sharpening the peaks although the peak shapes and areas after deconvolution may be substantially modified with respect to the initial shapes.9 However, in most applications involving overlapping peaks, measurement of the peak heights will be more precise than the determination of peak areas.The use of peak areas is limited owing to the large errors associated with determining the peak areas resulting from minor errors in drawing the baseline.8 In this work, peak heights will be considered rather than peak areas. The deconvolution for tailing chromatographic peaks will be assessed with emphasis on the analytical aspects of the method such as the signal-to-noise ratios, linearity, resolution enhancement and potential data distortion.Deconvolution based on the fast Fourier transform (FFT) will be compared to deconvolution based on the more efficient fast Hartley trans- form (FHT). Finally, the method will be applied to some simple chromatographic data and the results will be discussed. Experimental Reagents All the chemicals in this work were of analytical-reagent grade. Water was purified in an Elgastat Option 3 water purifier equipped with a Maxima unit (Elga Ltd., High Wycombe, Buckinghams hire, UK). Apparatus The chromatographic system consisted of a Kontron Analytical (Zurich, Switzerland) (Model 410) LC peristaltic pump, a Rheodyne (Cotati, CA, USA) HPLC injection valve (Model 7125) equipped with a 10 p1 injection loop and a Philips Analytical (Cambridge, UK) LC-UV HPLC detector set to a detection wavelength of 250 nm.Separation of the phenol- chlorophenol mixture was performed using a Waters Millipore (Milford, MA, USA) Nova-Pak C18 column (150 X 3.9 mm) while the acetone-phenol-toluene mixture was separated in a Hichrom C8 column (Hichrom Ltd., Reading, Berkshire, UK). A Midas Chromatography Datastation (Comus Instruments Ltd., Hull, UK) was used to acquire the chromatograms. A mobile phase of methanol-water (8 + 2), de-gassed with helium, was used throughout at a flow rate of 1 ml min-1. Software Programming The software used for this work was LabVIEW for Windows (National Instruments, Austin, TX, USA). LabVIEW contains extended libraries for signal processing and was selected for convenience of use, speed, flexibility, and compatibility with the Windows environment and platform independence capabil- ities. The data acquired by the Midas system were initially converted into an ASCII spreadsheet format and could then be directly imported into LabVIEW.The structure of the deconvo- lution program was similar for the deconvolution of simulated peaks or acquired data and is shown in Fig. 1. Simulated peaks were generated by convoluting symmetrical peaks with an exponential term. All the parameters of the peaks and exponential term were user-adjustable. The signal array was rotated-translated (as will be discussed below) and padded with zeros to at least double its length so that the total number of points was a power of two. The deconvoluting peak was generated with the same number of points as the signal to be deconvoluted and wrapped around.Padding and wrapping around was necessary in order to convert circular deconvolution to linear, as discussed elsewhere.10 The total number of points should be a power of two to take advantage of the fast implementations of the Fourier and Hartley transforms. The signal spectrum (Fourier or Hartley) was calculated and divided by the spectrum of the deconvoluting function. The deconvo- luted signal was inverse-transformed, inverse-rotated-trans- lated and filtered before display. Results and Discussion Effect of the Deconvoluting Time Constant In order to investigate the effect of the deconvoluting peak time constant, a simulated system consisting of two identical EMG peaks with standard deviation s, and time constant t, has been deconvoluted with an EMG peak with standard deviation sd = s,.The value of the deconvoluting peak time constant, zd, was varied and its effect on the deconvoluted signal was observed. Although the heights of the two peaks were identical in the original data, the peak height of the second peak appearedAnalyst, February 1996, Vol. 121 Generate Gaussian PeaW 99 Generate exponential term(s) higher owing to the contribution of the tailing part of the first peak. Three regimes can be distinguished: (i) when the time constant of the deconvoluting peak is lower than the time constant of the analytical peaks (i.e., td < t,), the deconvoluted peaks retain their EMG shape after deconvolution [Fig. 2(a)]. This observation can be verified by noticing that, after deconvolution, the second peak is still higher than the first peak, indicating that the deconvoluted peaks do possess a tailing end.The same observation can be visually made for the exposed tailing end of the second peak. As td increases, the time constant of the deconvoluted peak decreases as indicated by the fact that the peak height of the second peak after deconvolution decreases [Fig. 2(b)]. As a result, higher values of zd benefit resolution by causing a decrease in the tailing of the deconvo- luted peaks; (ii) when the time constant of the deconvoluting peak became equal to the time constant of the analytical peaks (i.e., td za), the tailing part of the deconvoluted peaks is completely eliminated and the deconvoluted peaks are turned to Gaussian [Fig.2(c)]. This property is verified by the observa- tion that both peaks in the deconvoluted signal have the same height. The resolution within this regime is already significant; and (iii) when the time constant of the deconvoluting peak is greater than the time constant of the analytical peaks (i.e., td > t,), the deconvoluted peaks remain Gaussian but a negative sidelobe appears at the end of the tailing end of the peaks (this sidelobe is only visually apparent at the end of the second peak [Fig. 2(6)]. This effect has been observed previously.9 The resolution keeps increasing with increasing zd. Ideally, the peaks after deconvolution should be Gaussian so that the better resolution can be achieved and contributions from tailing ends are eliminated.So, one should operate in the region in which td = t,. However, analytical signals usually consist of multiple peaks with different standard deviations. In this case the selection of the best time constant of the deconvoluting peak becomes more complicated. If too low a time constant is Convolute Gaussian peak(s) with exponential term(s) selected, narrow peaks will be turned into Gaussian after deconvolution but wider peaks will not be significantly affected. Conversely, if too high a time constant is chosen, wider peaks will be turned into Gaussian, as well as narrow peaks, but negative sidelobes will appear at the tailing ends of the narrower peaks. Therefore, the selected value of Td will be a compromise between elimination of the tailing of the deconvoluted peaks and minimization of the sidelobes at the end of the peaks.For n multiple analytical peaks with time constants T , , ~ , T , , ~ , . . ., z ~ , ~ , and provided that the ratio of the maximum over the minimum time constants among the peaks is less than 2 [i.e., max (t,,l, z,2 . . ., z,,,)/min (za,1, ‘ta,2, . . ., z,,J S 21 the optimum time constant of the deconvoluting peak, zd, should lie between the minimum and maximum time constants of the analytical peaks, i.e. min @,,I, ~ , 2 , .. ., ta,J < zd max (%,I, ~ , 2 ..., L,J (12) Generate noise Generate baseline Effect of the Deconvoluting Standard Deviation To assess the effect of the deconvoluting peak standard deviation, sd, a simulated system consisting of two identical overlapping EMG peaks with standard deviation s, and time constant t, have been deconvoluted with an EMG peak with time constant td = t,.The standard deviation of the deconvoluting EMG peak, sd, was varied and its effect on the deconvoluted system was observed. Two regimes could be observed in terms of the effect of s, on the deconvoluted signal: (i) when the deconvoluting peak standard deviation is much lower than the standard deviation of the analytical peaks (i.e., sd << so), the deconvolution causes only a marginal improvement in the resolution between the peaks [Fig. 3(a)]. However, as sd increases with respect to s,, the resolution between the two peaks increases after deconvolution [Fig. 3(b) and (c)]; and (ii) as soon as the standard deviation of the deconvoluting peak data Add noise and baseline to multipeak response Import data from spreadsheet file I Derive the FFT of the I - Rotate and translate the signal Generate deconvoluting Generate exponential symmetrical peak analytical signal Convolute symmetrical peak with exponential term to the same number of points as the analvtical sianal I signal J.Inverse rotate-translate the deconvoluted signal Wrap around the deconvoluting signal . -W Display results Derive the FFT of the deconvoluting signal ,-d Divide the FFT’s I - - I I Derive the IFFT L 1 1 Filter the deconvoluted 1 \ / f I100 Analyst, February 1996, Vol. 121 0.040 0.030 0.020 0.010 0.000 -0.010 3 -0.020 c .- C -0.030 .+ -0.040 $ 0.040 0.030 0.020 0.010 0.000 -0.01 0 -0.020 -0.030 -0.040 100 150 200 250 300 350 400 0.040 0.030 0.020 0.010 0.000 -0.01 0 3 -0.020 - .- S 3 -0.030 $ .= -0.040 e 100 150 200 250 300 350 400 (b) - - - - C - - - 1 1 1 1 1 v a, 0.040 g 0.030 v) v) LT 0.020 0.01 0 0.000 -0.010 -0.020 -0.030 -0.040 100 150 200 250 300 350 400 0.040 0.030 0.020 0.010 0.000 -0.010 -0.020 -0.030 -0.040 100 150 200 250 300 350 400 Time (arbitrary units) Fig.2 The effect of td of the deconvoluting EMG peak with sd = 8 points on the deconvolution of two EMG peaks located at points 230 and 260 with standard deviations s , , ~ = s,,~ = 10 points and time constants = T , . ~ = 10 points. Time constant of the deconvoluting peaks: (a) td = 2 points; (h) td = 5 points; (c) td = 10 points; and (d) td = 15 points. Traces: A, individual peaks; B, signal before deconvolution; and C , signal after deconvolution multiplied by - 1.0.040 0.030 0.020 0.01 0 0.000 -0.010 -0.020 -0.030 0.040 0.030 0.020 0.010 0.000 -0.01 0 -0.020 -0.030 -0.040 100 150 200 250 300 350 400 100 150 200 250 300 350 400 Time (arbitrary units) Fig. 3 The effect of sd of the deconvoluting EMG peak with td = 10 points on the deconvolution of two EMG peaks located at points 230 and 260 with standard deviations = s , , ~ = 10 points and time constants. T , , ~ = T , , ~ = 10 points. Standard deviation of the deconvoluting peak: (a) sd = 2 points; (h) sCf = 5 points; ( c ) sd = 8 points; and (6) sd = 10 points. Traces: A, individual peaks; B, signal before deconvolution; and C , signal after deconvolution multiplied by - 1.Analyst, February 1996, Vol. 121 101 becomes equal (or greater than) the standard deviation of the analytical peaks (i.e., sd 3 s,), the deconvoluted peaks are oversharpened and sidelobes start to appear at the foot of the deconvoluted signal [Fig.3(d)]. The magnitude of the sidelobes increases as the value of sd increases above the value of s,. This latter situation corresponds to the physically meaningless operation of trying to produce deconvoluted peaks of negative width.3 In order to achieve the maximum resolution without oscillations, the deconvoluting peak standard deviation should be slightly lower than the standard deviation of the analytical peaks. In the case of n multiple peaks with different standard deviation s ~ , ~ , s,2 . . ., s , , ~ the standard deviation of the deconvoluting peak should be lower than the standard deviation of the narrowest peak, i.e.(13) Sd < min (S,,lr sa,2, . . * ? Sa,n) Effect of the Deconvoluting Peak Type Two deconvoluting functions have been used; EMG shape (eqn. 11) and the exponentially modified Lorentzian (EML) function. The latter type of peak can be generated by convoluting a Lorentzian function, l(t), with an exponential term, e(t). The Lorentzian function, l(t), is given by the formula] *: A l(t) = (1+-) The exponential contribution is given by eqn. 8. Hence, the EML peak can be expressed as: Deconvolution, with both the deconvolution peak types (i.e. EMG and EML), exhibits a marked improvement in resolution. The resolution enhancement is very similar for both deconvolut- ing functions (Fig. 4). As expected, the Gaussian deconvolution peak produced the best baseline [Fig.4(a)] while the Lorentzian peak produced a negative going oscillation at the front and tailing ends of the deconvoluted peaks [Fig. 4(b)]. The effects of the time constant, zd, and standard deviation, s,, of the deconvoluting EML function are similar to the effects of these parameters on the EMG deconvoluting function, as discussed in previous sections. Linearity The measurement of the height of overlapping peaks usually involves errors associated with the incomplete separation of the peaks. Indeed, as the simple case of two overlapping peaks demonstrates, the height of the second peak tends to be enhanced by the contribution of the tailing part of the first peak. This effect gives rise to a systematic positive error in the determination of the second peak height.The relative values of this error will mainly depend on the time constants, the retention times and the relative peak heights of the two peaks: the higher the time constant and peak height of the first peak and the closer the retention times, the more pronounced the enhancing effect on the second peak will be. Thus, the effect will be more pronounced in the case of small and narrow peaks occurring close to the tailing end of much wider and taller peaks. However, the first peak will only be marginally affected. So, even if the underlying mechanism producing the peaks is linear (i.e., the height of individual peaks increases linearly with concentration), the determination of peak heights may involve large errors.Deconvolution will be, in principle, able to correct or minimize the systematic positive error in determining the height of the second peak, at least to some extent. The errors in the determination of the peak height of the second peak, before and after deconvolution, are compared in Fig. 5. As expected, a wider first peak, Fig. 5(b), gives rise to higher errors than narrow first peaks, Fig. 5(a). However, the errors after deconvolution are much lower than the errors before deconvolu- tion. The choice of zd and sd of the deconvoluting peak will be crucial in this step since both these parameters will affect the resolution, as discussed previously. Under suitable circum- stances, the measurement of the deconvoluted peak heights will not involve any errors and the heights will be accurately determined.The example in Fig. 6 demonstrates this latter point; by observing the recommendations in eqns. 12 and 13 for the selection of sd and td, the error in the determination of the second peak was eliminated completely after deconvolution at the expense of a small sidelobe after the second peak (this sidelobe is due to the fact that td is higher than z,,Z). Before deconvolution, an error of 40% was calculated for the determination of the peak height of compound 2. It is interesting to note that, as in the case of symmetrical peaks, the absolute peak ratios after deconvolution are not preserved but the linearity of all the peaks is maintained.' Comparison Between the Fourier and Hartley Transforms It has been shown previously that the use of the conceptually simpler Hartley transform can be used for the deconvolution of symmetrical Gaussian, Lorentzian and voltammetric peaks.4 The FHT is twice as fast as the FFT and uses half the memory 0.040 0.030 0.020 0.010 0.000 -0.01 0 -0.020 5 -0.030 .g -0.040 2 100 150 200 250 300 350 400 - 0.040 0.030 - .- $? % = 0.020 v) 0.010 0.000 -0.010 -0.020 -0.030 -0.040 I I 1 I I I 1 100 150 200 250 300 350 400 Time (arbitrary units) Fig.4 Comparison between (a) EMG; and (b) EML deconvoluting peak with standard sd = 8 points and td = 10 points for the deconvolution of two EMG peaks located at points 230 and 260 with standard deviations sU,l = sUs2 = 10 points and time constants T ~ , ~ = T , , ~ = 10 points. Traces: A, individual peaks; B, signal before deconvolution; and C, signal after deconvolution multiplied by - 1.102 Analyst, Februaiy 1996, Vol.121 than a conventional FFT method. However, an important requirement for the application of the Hartley transform is that the deconvoluting function is even (i.e., symmetrical). In the particular case of symmetrical peaks, it can be shown that deconvolution using the Hartley transform is mathematically equivalent to using the Fourier tran~form.~ Although pure Gaussian, Lorentzian and voltammetric deconvoluting peaks do fulfill this requirement, their exponentially modified versions do not. Obviously, the higher the ratio of sd over ~d of the exponential term, the more the EMG deconvoluting peak approaches a symmetrical shape. Therefore, the success of applying the Hartley transform will depend on the requirements placed upon the value of the ratio S,/Q of the deconvoluting peak. An empirical recommendation is that the FHT can be used if the condition s,/T~ 3 4 is fulfilled.Background Correction The implementation of the discrete Fourier and Hartley algorithms requires that the numerical values of the first and last points of the recorded signal are equal to zero. If this requirement is not met, serious errors in the calculation of the frequency spectra will result. 12 Two methods that can be used to accomplish this task are rotation-translation and windowing of the signal. The first method involves subtracting from the signal array, x(k), the straight line, h(k), that passes through the first W, I 0 1 0 2 0 ~ 4 0 m ~ 8 0 o Separation (tH,,-tH,Z), points "0, 1 5 1 * c 100 i .1 A\ ', I 0 I I I , 1.- A \I 10 20 30 40 W W 7 0 Sci)arntion (tU.I-tR.2), points Fig. 5 Comparison of errors in determining the peak height of an EMG peak 2 at the tailing end of another EMG peak 2A, before and B, after deconvolution as a function of the separation between the two peaks. Time constants of both peaks tu,i = = 10 points. (a) Peaks 1 and 2, s,, I = s,,~ = 10 points, and (b) peak 1 = 20 points and peak 2 s,,~ = 10 points. Height ratios: peak 2/peak 1 = 1/3. The deconvoluting peak was an EMG peak with sd = 8 points and td = 10 points. 0.mO- 0.m- 2 I \ 1 .P: 0.8 0.7 5 0.6 0.5 2 0.4 $ 0.3 - - D - c g 0.2 g 0.1 0.0 -2 -1 0 1 2 3 4 Concentration of compound 2 (arbitrary units) Fig. 6 (a) Simulated standard additions for an EMG peak 2 located at point 260 at the tailing end of a wider EMG peak 1 located at point 220 before (A) and after (B) deconvolution with a EMG peak with sd = 8 points and td = 8 points. Conditions of the overlapping peaks: height ratio: peak 2/peak 1 = 1/3; s,,~ = 12 points and su,2 = 10 points; tu,l = 10 points and peak 2 z,.~ = 5 points. (b) Standard additions plots A, before and B, after deconvolution.1 ;A I ' , I ' I ', 0.020 0.010 rn 0.000 -0.01 0 -0.020 5 -0.030 h c .- -0.040 .- c 100 150 200 250 300 350 400 2 0.040 I c ." - - - v 8 0.030 0 U E 0.020 U 0.010 0.000 -0.010 -0.020 -0.030 -0.040 100 150 200 250 300 350 400 Time (arbitrary units) Fig. 7 Comparison between an (a) Hamming and (b) rectangular FIR filter for lowpass filtering of deconvoluted data.Initial S/N ratio = 20. Low-pass cut-off frequency at the 46th harmonic. Overlapped EMG peaks located at points 260 and 230 with s,,~ = s , , ~ = 10 points and T~,] = T , , ~ = 10 points. The deconvoluting peaks was also EMG with sd = 8 points and td = 10 points. Traces: before (A) and after (B) deconvolution.Analyst, February 1996, Vol. 121 103 1.4 and last points. A new background-subtracted array, x’(k), is thus formed12: x[(n - 1)l- -do) ~ ’ ( k ) = ~ ( k ) - b(k) = ~ ( k ) -x(O) - N - 1 f o r k = 0 , 1 , 2 ,..., N - 1 (16) where x(0) is the first point of the data array and x[(n - l)] is the last point of the data array. The new array, x’(k), is used for the deconvolution operation; after deconvolution, the straight line, b(k), is added back to the deconvoluted signal (an operation referred to as inverse rotation-translation).The second method involves multiplication of the data array with a window function that approaches smoothly the value of zero at the two ends of the array. A comparison of the two methods has shown that the rotation-translation operation is superior to the window method, especially in the presence of very sloping baseline^.^ Thus, background correction in this work was accomplished by means of the rotation-translation method. - ‘ phenol (4 B Effect of Noise It has been widely recognized that deconvolution by means of Fourier transform is sensitive to noise in the original data.99’3 This sensitivity has been attributed to the division operation of small numbers in the frequency domain that give rise to artifact noise components in the spectrum before the inverse trans- f ~ r m .~ . ’ ~ Under all circumstances, the signal-to-noise (S/N) ratio after deconvolution will be lower than the initial S/N ratio. In general, attempts to increase the resolution (by using higher 6.0000 5.7500 5.5000 sd values) will result in even poorer S/N ratios.4.13 The artifact deconvolution noise will be distributed in the whole frequency range but only components of relatively low frequency are of importance for the accurate representation of the analytical peaks. This property allows the application of low-pass filtering of the deconvoluted signal. Different types of filters, both infinite impulse response (FIRF) and finite impulse response (FIRF), have been assessed.The usual limitations of selectivity versus spectral leakage associated with digital filters must be considered.14 Filters that have steeper cut-off slopes (such as a rectangular FIRF) will be more frequency-selective at the expense of leakage effects that produce oscillations in the deconvoluted signal. Less selective filters (such as Butterworth IIRF’s or Hamming and Kaiser FIRF’s) are preferable as the S/N ratios after deconvolution are lower. An example is illustrated in Fig. 7, in which a rectangular and a Hamming filter are compared for filtering deconvoluted EMG peaks with a S/N ratio of 20 in the original data. The low-pass filter cut-off frequency will need to be carefully controlled and can be selected by deriving and comparing the power spectra of the raw data and the deconvoluted signal; the useful frequency compo- nents that must be retained will be confined to the lower range of the frequency spectrum.It should be noted that filtering will introduce a group delay in the deconvoluted signal which will manifest itself as a shift of the peak locations. This shift can be easily compensated for by a reverse shift of the deconvoluted array. phenol / \ I \ - - “ 1 \ - ‘? ‘\ A a a 6.2500 6.5000 I . . (4 - chlorophenol (4 I 6.2500 I- 5.2500 - 5.0000 - 4.7500 - 4.5000 - - - - 4.2500 - 4.0000 - 1 1 1 1 -1 n 50 60 80 100 120 140 160 180 200 220 250 6.0000 - 5.7500 - 5.5000 - -- - - _ ---_- ‘- 5.2500 - 5.0000 - 4.7500 - 4.5000 _ - - - - p ~ ’ 42500 - 50 60 80 100 120 140 160 180 200 220 250 Time/102 min Fig.8 Application of the deconvolution procedure to overlapping phenol- chlorophenol chromatographic peaks. Concentrations: phenol 2.8 X g ml-1, chlorophenol 2.5 X 10-3 g ml-1. Deconvoluting peaks: (a) EMG peak with sd = 3 points and td = 4 points, and (b) EML peak with sd = 3 points and td = 4 points. Traces: before (A) and after (B) deconvolution. The deconvoluted data were filtered with a Hamming low-pass FIRF with cut-off at the 50th harmonic. 1.2 1 .o 0.8 0.6 0.4 0.2 a 2 -0.0 50 100 150 200 250 300 350 400 450 $ 0.7 9 0.6 0.5 0.4 0.3 0.2 0.1 -0.0 50 120 140 160 180 200 220 240 260 280 300 Time/102 min Fig. 9 Application of the deconvolution procedure to a chromatogram for the separation of 4.1 x 10-3 g ml-* acetone, 2 X g ml-l phenol and 1 .O x 10-3 g ml-1 toluene.The deconvoluting peak was an EMG peak with sd = 2 points and td = 0.5 points. (a) The whole chromatogram, and (b) magnified region between 1 and 3 min. Traces: before (A) and after (B) deconvoluted. The deconvoluted data were filtered with a Hamming low- pass FIRF with cut-off at the 50th harmonic.104 Analyst, February 1996, Vol. 121 Applications The proposed procedure has been applied to simple chromat- ographic separations. Fig. 8 illustrates the effect of deconvolu- tion on the separation of phenol and chlorophenol using an EMG and EML deconvoluting function. Owing to the sig- nificant overlap between the two peaks and their tailing nature, the height of the second peak appears higher than it should be (as compared with the same concentration of the pure compound). Deconvolution corrected for this effect as a result of the narrower response-time profile afforded by the operation.The sidelobes appearing after deconvolution are a combined result of the filtering step and the amplification of pump pulsations. A second application was the separation of a sample involving the components: acetone; phenol; and toluene (Fig. 9). After deconvolution, the resolution between acetone and phenol was improved, no distortions of the data occurred, the initial S / N ratio was only marginally decreased and the peak ratios for all the peaks remained constant. Moreover, the low sd/ ‘td ratio of the deconvoluting peak (in this case the s&d ratio was set to the value of 4) allowed the application of the FHT. References 1 2 Engblom, S. O., J . Electroanal. 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