1879
Anal. Chem. 1991, 63,1879-1884
stationary phases that possess good film-forming ability and a wide applicable temperature range of 75-305 “C.The phases present excellent selectivity for alcohols and aromatic hydrocarbons, especially, and they exhibit unique selectivity for polar position isomers.
LITERATURE CITED KoHhff, I. M.; Chantoonl, M. K., Jr. Anal. Chem. 1979, 51, 1301- 1306. Rushong, LI. Crown Ether as Stationary UquM as GC. chin. J. Chro1980, 5, 304-305. Yongheo, Jln: Ruonong, Fu: ZaCfu, Huang. J . chrometop. 1989, 469, 153-169.
(4) Fine, D. D.; (kerhart, H. L., 11; Mottola, H. A. Talenta 1989, 32. 751-756. (5) Rouse, C. A.; Flnllson, A. C.; Tarbet. B. J.; Rxtox, J. C.; Djordjevlc, N. M.; M e r k M ~ ,K. E.: LOO. M. L. A M / . Chem. 1988, 60, 901-906. (6) CeCYhg, Wu: ChengAhg, Wang: Zhao-Rui, Zeng;Xue-Ran, Lu. Anal. Cbfn. 1990, 82, S68471. (7) CaCYlng. Wu; Hong-Yuan. U Yuan-Yln, Chen; XusRan. Lu. J . Chrometog*. 1990, 504, 279-286. (8) JoW, B. A.: Bradshaw, J. S.;et ai. J . Ckg. Chem. 1984, 49, 4947-4951 *
RECEIVED for review January 2,1991. Accepted May 17,1991. This work was supported by the National Science Foundation.
Mathematical Series for Signal Modeling Using Exponentially Modified Functions Alain Berthod Laboratoire des Sciences Analytiques, Universitg de Lyon 1, UA CNRS 435,69622 Villeurbanne Cedex, France
The rlmpie recursive genorating function of eq 8, with a timedependent function, €(I), a convoluted function, U ( t ) , corresponding to €(I), and the tlme interval between two measurcnnents, At, and a dhrenrrknleas parameter, A, linked to the time constant T by A = d A t 0.5, is demonstrated to produce U(t ), which k the discrete representation of the exponontlally modlfled € ( t ) lunctlon. It € ( I ) k a Gawslan function, then U ( t ) k the dkcrete represontation of the corr.rpondlng exponentially modlied Gaudan (EMG) func#on. The EMG function Is wkkly used In chromatography and flow injection analyrk (FIA) to model taiilng peaks. The proposed method can exponentially modify any Gawdan functions in seconds with a precldon comparable to the erf functlon and/or series expandon approximations commonly used. The convoMion series U ( t ) Is deflned. The ilnk with the exponentially modified function Is demonstrated for any E ( t ) function. The propaka of the ConVoMed U ( t )functbn are given. Three examples are developed: (I) the slngle square function whose exponentially modified form (EMS)is shown to be the equation of the one tank model in FIA; (11) the trlangle function: (111) the Gaussian functkn produdng the EMG function. The exponential modlflcation of very different functlons produces tailing peaks of simliar general shape.
+
INTRODUCTION The observation of a phenomenon necessarily induces a perturbation of the phenomenon. In quantum mechanics, the Heisenberg principle states that the momentum and the position of a particle cannot be known at the same time. Most analytical techniques do not need the use of quantum mechanics. However, the recorded signal of any experiment corresponds to the phenomenon under investigation with superimposed and indistinguishable noise or random errors (I).Peak-broadening processes or extracolumn effecte are an important problem in high-performance liquid chromatography (HPLC)(2). Dispersion and band broadening are also essential in flow injection analysis (FIA) (3). 0003-2700/91/0363-1879$02.50/0
It was shown that the exponentially modified Gaussian (EMG) function was an accurate model for real chromatographic peak shapes (2,4-12)and also FIA peak shapes (13, 14). The EMG function, h(t), is a regular Gaussian function,
at),
convoluted by an exponential decay term, H(t), of unit area
H(t)
=o
t CO
The expression of the EMG function, h(t), is H(t - t? dt’
h(t) = f G ( t ’ ) 0
or h(t) = i x t e x p [ 7
4
6
I’)“
-(
ex,[
If20
-(+’)I
(3)
dt’ (4)
where h(t) is the EMG peak height at time t, S the peak area, 7 the time constant of the exponential modifier, u the standard deviation of the Gaussian constituent, t, the peak retention time of the Gaussian constituent, and t’a dummy variable of integration. Equation 4 contains an integral form that is difficult to estimate numerically. Errors were noted in EMG function notation and evaluation (11,15). Series expansion is often used to numerically estimate EMG functions (11,16,17).Also, the error function, erf t, which has the form erf t =
-x G 2
t
exp[(-tq2] dt’
is often preaent as a resident subroutine in computers and was used to calculate h(t) by using the relationship (5,11, 15) 0 1991 Amerlcan Chemlcal Soclety
1880
ANALYTICAL CHEMISTRY, VOL.
63, NO. 17, SEPTEMBER 1, 1991
-ii
Blond lot
1
2
4.
, 2-
w-
0
1 at
z nt
3 at
40t
0
Time Flgure 1. Illustration of the obtention of a convoluted E ( t ) function. U ( t )= U(t - At) 4- (E(t)- U(t - At))/A , A = 3. Only onethird ( V A , short vertical arrow) of the E ( ? )increase (long vertical arrow) is incremented to U(t). Stars and dashed line = E(t);closed squares and solid line = U ( t ) .
with
(7) The problem is that series, asymptotic, or continued fraction expansions are acceptable estimates of the EMG, the erf, or other functions, only over limited ranges of the argument. To make an approximation over a wide domain of the argument, the domain must be divided into regions, each possessing its own best approximation. Enough terms should be included to yield a reasonable result for series that may not converge rapidly (15,17). This work proposes a simple method to generate easily an acceptable approximation of exponentially modified functions. It is shown that the convolution can be done, for an actual set of values, by using a recursive series whose term n depends only on term n - 1 and on the nth original signal value.
CONVOLUTION SERIES Definition. E(t) is a time-dependent function representing the unconvoluted observed signal. E(t )can be any measurable quantity: a concentration, an absorbance, an electrical potential. E ( t ) is measured for a limited period of time [0, T] during which N points are stored. The time interval between two measurements is At = T / N . The convoluted set of points, U ( t ) ,derives from the N E @ )points. U ( t )is formed by using the recursive series
U ( t ) = U(t - At)
- At) + E ( t ) - U(t A
(8)
with A 1 1 and U(0) = E(0). A is a dimensionless factor. Figure 1 illustrates graphically how the convoluted points (closed squares) are obtained from the original set of E ( t ) values (stars). This series is based on the physical observation of a recorder pen. The recorder receives a signal, E@). On the paper, the trace drawn corresponds to a convoluted form, U ( t ) ,of E(t). At any time t, the pen follows E ( t ) ,starting from where it was at time t - At. It has to move from U(t - At) to E ( t ) . But it moves less, slowed down by a time constant. U ( t )follows E ( t ) with a time delay depending on A (Figure 1). The first convoluted point, U(O),corresponds to the finite time interval 0 I t I At, the second point, U(At)corresponds to the time interval At 5 t I 2At, and so on. To be exact, U(0)should be plotted on Figure 1 at time t = At/2, U(At)
10
30
20
SO
40
00
Time
Flguro 2. Square functlon (dashed line) and exponentially modified square functions (EMS, solM lines). Af = 1 time unit. EMS functions correspond to the one tank FIA model with T = V / F .
should be plotted at time t = 3At/2, and any U t ’ ) point at a time t = t’+ At/2. Plotting U ( t ) at time t, as shown in Figure 1, produces a convoluted function translated by At/2 to the left. It would be easy to plot U ( t )at time t + At/2. However, most modern analytical techniques that produces digitized signals do it at high frequency. The actual signal is defined with a large number of points. The time interval, At, is small compared to the time scale of the whole signal and the A t / 2 left shift is small. In all examples illustrating this work, U ( t )was plotted at time t instead of time t + At/2. Time Constant Expression. To calculate the A value, eq 8 is applied to a single-step square signal (E(0) = 0; E ( t ) = 1, t > 0). With t = nAt, we obtain U(0) = 0, U(At) = 1/A, U(2At) = (1/A)(2 - l / A ) , and fl(n -j )
-
U(nAt) = U ( t ) =
?(-l)i+l’=a i=l
i!Ai
(9)
If At 0, then n = t / A t + m and eq 9 corresponds to the MacLaurin series of the exponential function (18),U ( t )= 1 - exp(-t/T), with the time constant of a single-step square signal, T , which is the time necessary to reach 6 3 % of the actual signal. Using the first point defined by n = 1 and t = At, it is -+
1 U(At) = A = 1 - exp(
$)
Rearranging, we form
A=
1 1 - exp(-At/T)
(11)
The dimensionless factor, A, is related to the time constant, T, and to the time interval, At, between the experimental points. Most integrators acquire data at a constant frequency or constant At. If this is not the case, to convolute the experimental set of points with a constant T value, A should be computed by using eq 11. T is defined by the relationship At (12) = ln [ A / ( A - 1)1 For A = 1, eq 8 produces a mathematical series, U(t),identical with the experimental set of points, E(t). This corresponds to a nil time constant for any At (eq 12). General Expression. The definition of the approximate generating function, U ( t ) (eq 8), can be rearranged as
( :)
U ( t ) = - + U(t - At) 1 - A
(13)
Developing for a set of points separated by a period of time, At, we obtained for the nth point
ANALYTICAL
CHEMISTRY,VOL. 03, NO. 17, SEPTEMBER 1, 1991 1881
Table I. Test of the Method’s Accuracy with a Square Function t
0
2 2.99 3 5 6 9 9.99 10 12 14 18 25
S(t)
0 0 0 10 10 10 10 10 0 0 0 0 0
At
Aa
2 0.99 0.01 2 1 3 0.99 0.01 2 2 4 7
2.541 4.561 400.5 2.541 4.521 1.895 4.561 400.5 2.541 2.541 1.582 1.210
expon modif function this methodb theoretical‘ 0 0 0 0.02496 3.94984 5.288 13 7.774 27 8.262 26 8.24163 4.99880 2.03193 1.11538 0.19382
0 0
0 0 3.93469 5.276 33 7.76870 8.25791 8.262 26 5.01131 3.039 51 1.119 38 0.19431
error
re1 error, %
0 0 0
0.0250 0.0151 0.0118 0.0055 0.0043 -0.0206 -0.0125 -0.0076 -0.0028 -0.0004
0.38 0.22 0.07 0.05 0.25 0.25 0.25 0.25 0.25
7=4 average error 0.0013 0.22 Calculated by using eq 11 with a time constant 7 = 4. Calculated by using eq 8 over the whole range. Calculated by using eqs 26-28 over the ranges t < 3,3 5 t 5 10,and t > 10,respectively. The data sampling interval was intentionally not constant. a
in the definition of U(t)(eq 17). I f E ( t )is not continuous (e.g., square function), U ( t )is continuous but the derivative, U’(t), is not continuous at times t corresponding to the discontinuities of E@).An example of each case is developed hereafter. Extrema of U ( t ) . Equation 8 contains the approximation of the derivative of U ( t )
By using eq 11, it becomes U(nAt) =
U ( t )- U(t - A t ) = Assume we are able to sample a phenomenon with an infinite number of points, each one separated by an infintely small time difference so that the discrete U ( t )function becomes a continuous one. Then At 0 and n m with nAt = t . Equation 11 shows that the limit of the A term is
- -
lim A = ? / A t
At -0
When At
-
E ( t ) - U(t - A t ) A
(18)
0, this difference is the derivative of U ( t )
(16)
Noting t‘, the product iAt,eq 15 yields
Equation 17 applies for any function E@),specially the Gaussian function, G(t). With E ( t ) = G ( t ) ,eq 17 is identical with eq 4, the EMG function. For a real set of points belonging to a Gaussian function, eq 8 can generate very easily the corresponding EMG set of points. In general, the convoluted set of points, U(t),is the exponentially modified (EM) form of E(t).
PROPERTIES OF EXPONENTIALLY MODIFIED FUNCTIONS Equation 8 should be used with a limited set of N E ( t ) points. It produces an N-point sampled representation of the U t )function. It is worthwhile to study the U ( t )properties assuming we are working with an infinite number of E ( t ) points, i.e., assuming U(t)is not a discrete representation of the EM form of E ( t ) but U ( t )is a continuous function, the EM form of E(t). With this assumption, the continuity, the extrema, the derivative, and the integrated form of U(t)can be studied. Continuity. E ( t ) should be a defined function; if not, it cannot be convoluted. If E ( t ) is continuous and continuously derivable (e.g., Gaussian function), U(t)is also continuous and continuously derivable. If E ( t ) is continuous but not continuously derivable (e.g., triangle function), U ( t )is still continuous and continuously derivable due to the integral form
which is nil when E ( t ) = U(t),for any A value (A I 1). This means U ( t )always crosses the original function, E(t),with a nil slope. In other words, the extrema of U ( t )fall on the contour of E(t). The maximum peak height of a EMG peak in chromatography occurs at a retention time, t,, for which the maximum peak height, EMG(tJ, corresponds to the height of the Gaussian constituent, G(t,), at the same time, t, (4). This property of EMG functions was used in chromatography to extract the T and u values of a peak and to model peak tailings (7,19). We notice that if E(t)is continuous, then U t ) is continuous (eq 8), and U’(t) is continuous since it depends on two continuous functions,E(t)and U(t)(eq 19), even when E’(t) is not continuous. U’(t) depends only on E(t). Also a high A value (high time constant) produces low U’(t) values, which means a “flat” U ( t )signal as illustrated below. Area, The area under U ( t )is not necessarily equal to the area under E(t). The equality
is true if and only if it exists a time T with E(0) = E ( T ) = 0 and E ( t ) = 0 for t < 0 and t > T. Practically, this is the case of G ( t ) for which a peak end time, T, can be defined: G(T) < e, with e being a given number small enough to consider G(t > 7‘) = 0. The triangle and the single square functions also fulfill the conditions. Practically, the upper limit of integration of U ( t )can be T + 5u instead of + m . Relation between A and T. Equation 11expresses A as a function of T and At. For A t / r < 1, which is actually always true, eq 11 can be developed to the second order
A N A L Y T I C 4 CHEMISTRY, VOL. 63, NO. 17, SEPTEMBER 1, 1991
1882 10
SIgnaI
at 'A I
7. 2
,
.
2 \ n 0
10
32 30
20
60
40
80
Time
Flgw 3. Exponentleiymodsfkd functkns of the Figue 2 EMS tvlctkn wlth T = 4. TMs CaTesponds to the several tanks In Serieg FIA model.
l
o
b
The EMS function is continuous. The maximum EMS value occura a t time t = 10 for which the derivative,EMS'@), is not nil and has a differing value approaching 10 from the left than from the right. This is due to the discontinuityof S(t)at time t = 10 (eq 19). Note that EMS(10) is not equal to S(l0). Table I lista the values of S(t)(eqs 23-25), the corresponding EMS(t) values obtained through eqs 26-28, and the EMS@) values obtained with the proposed method (eq 8). The data sampling interval was intentionally not constant. The error was lower than 2.5 X lo-, with a relative error lower than 0.38% (average relative error 0.22%). Equation 8 is able to generate with an acceptable accuracy the exponentially modified form of a square function (Figure 2). The EMS function is very important in FIA studies. It corresponds to the concentration evolution of an injected sample plug flowing into a mixing chamber. The model is called the "one tank model" (3). As described by Tyson (20), before the sample reaches the mixing chamber, Cl(t) = 0. When the sample flows into the mixing chamber, the concentration, C,(t), increases
[
C,(t) = c, 1 -
0
10
20
50
W
60
40
Time
Figure 4. Triangle function (dashed Ilne) and exponentially modified triangle functions (EMT). T Is Indicated on the figure for each convoluted EMT function (solid lines). At = 1 time unit.
or
(A - 0.5)At
T
(22)
EXAMPLES To demonstrate the method's accuracy and the relative independence of ita results on data sampling interval, two sets of points of simple functions, the square function and the triangle function, were convoluted with eq 8. The convoluted points were compared to the exact values obtained with the analytical integrated EM form of the two functions. Square Function. Figure 2 shows the square function, S(t), with a dashed line. S(t) = 0 t 20 (33) The five solid lines correspond to the exponentially modified triangle (EMT) function obtained by using eq 8. Analytical equations of the EMT function can be mathematically derived in three steps. For t E IO, 101,
- t')]dt'
EMTl(t) = l l t t ' e x p [ - ( t T
O
EMT,(t) = t - T
0 It I10 (34)
+ T exp(-t/T)
(35)
F o r t E [lo, 201,
-(t - 10)
) + -(t7) 'S'(-t'+20) exp( - t? (36)
EMT2(t) = EMT1(lO) exp(
T
10
ANALYTICAL CHEMISTRY, VOL. 03, NO. 17, SEPTEMBER 1, 1991 Signal
Table 11. Test of the Method’s Accuracy with a Triangle Function t
T(t)
0
0 1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
expon modif function this methodo theoreticalb
0 0.22120 0.61467 1.14230 1.77442 2.48792 3.26479 4.09101 4.95568 5.85028 6.76812 7.261 86 7.425 14 7.331 10 7.03666 6.58615 6.01409
0 0.24916 0.641 05 1.16745 1.79861 2.511 36 3.28765 4.11342 4.97773 5.87206 6.78975 7.227 35 7.39365 7.30197 7.00937 6.56030 5.98936
average error
I.
error abs
re1 %
-0,02795 -0.0263 7 -0.0251 4 -0.0241 8 -0.0234 4 -0.02285 -0.0224 0 -0.0220 5 -0.0217 7 -0.0215 6 0.03451 0.03148 0.029 12 0.02785 0.02585 0.02473
11.22 4.115 2.154 1.344 0.933 0.695 0.544 0.443 0.370 0.317 0.477 0.425 0.398 0.389 0.394 0.413
-0.00557
1.539
aCalculated by using eq 8 with T = 4, At = 1, and A = 4.5210. *Calculatedby using eq 35 up to t = 10,then using eq 37.
0.6
0.4
0.2
0
EMT,(t) =
10
by
)
0 6 : +D5+(SB6-05)/D;1 A
B
Y’
2
A a1gn.1
1.541
t 1 . C
1
a
3
RE
-
4
0
0
5 6 1 8 9 10 11
1 2 3 4 5
;
6
7 8 9
I
I
5 6 7 8 9
13 14
io
io
15 16
11 12 13
9
I4
6
I5
5
8 i
?
E
0
8
H
0 16
32
4.511 8.510 16.505 Convoluted triangle sIgn.1 0.000 0.000 0.000 0.000 0.393 o.aai 0.118 0.061
32.503
4
0.000 0.031
;:zF R P E:: :::::
1.661 3.585 4.535 5.505 6.481 7.476 8.469 8.618 8.411 7.156 7.116 6.1~9 5.389
1.774
1.488
3.265 4.091 4.956 5.850 6.768 1.162 7.425 1.331 1.037 6.586 6.014
1.045 1.510 2.037
a.6ao 3.153 3.9111 4.641
5.154 5.488 5.666 3.705 5.612 5.431
E: 0 . m
0.510 0.839 1.151 1.506
0.443 0.614 0.010
1.099 1.319
1.031 1.177 1.515 1.175 1.966 1.111 2.140 2.315 1.317
1.194 3.170 3.463 3.675 3.118 3.809
3.896
and continuously derivable because the triangle function is continuous (eqs 8 and 19). EMT’l(0) = 0, and EMT’&) is nil for
Table I1 compares the analytical EMT values obtained by using eqs 35,37, and 39 with T = 4 to the ones obtained with a value eq 8. The absolute error was as low as f2.5 X comparable to the EMS error. The error was relatively constant over the whole domain. This produced a high 11% relative error with an argument value of 0.24 for t = 1(Table 11). The same absolute error produced only 0.3% relative error with a 6.7 argument value (t = 10, Table 11). The average relative error was 1.5%. The derivatives are obtained by deriving directly the EMT(t) expressions or, more easily, using eq 19. dEMTl/dt = EMTtl = 1- exp
)(:
t
c [O, 101 (40)
-(t
C
(38)
EMT,(t) =
2 ex.(
60
60
Figure 3-5 are similar.
I
-(t - 20)
40
Figure 5. Gaussian functbn (dashed line) and exponentially modified Gaussian (EMG) functions (solid lines). t , = 10 time units, S = 10, u = 3 tkne units, At = 1 time unit, and T is shown on the figwe for each convoluted EMG function (solid lines). The shapes of the peaks shown
17 18 19
For t > 20,
so
20
Time
ia
EMT,(t) = EMT2(20) ex,(
188s
lo)) - 1 - exp(
):
t C [lo, 201 (41)
dEMTs/dt = EMT’S =
t > 20 (42) We notice that EMTrl(lO) = EMT’,(lO) and EMTZ(2O) = EMTdZO). As demonstrated,the EMT function is continuous
-
tM is the time for the EMT convoluted peak maximum. For T < 8, tM 10 + 0.697. Also for any T value EMT,(t,) = - t ~ 20 = T(tM) (44)
+
The maximum of the EMT function falls on the contour of the triangle function. It ia straightforward to demonstrate that the area of the EMT function described is 10 on the time interval [O,+m] for any value of T. The properties of the EMT function can be compared with those of the EMG function. Gaussian Function. Figure 5 shows G(t)with u = 3 time units, t, = 10 time units, and S = 10 (dashed line). The five solid lines correspond to the EMG peaks obtained by convoluting the Gaussian function by using eq 8 and the time constants of 2,4,8,16, and 32 time units. The interest and properties of the EMG functions are very important for band-broadening characterization in chromatography. The moment method (21) is the only valid method to characterize skewed peaks. The peak skew and peak excess are obtained through the third and fourth central momenta, M 3and M,, respectively. The exactitude of thew higher moments critically depends on the peak start and peak end (21, 22). The mathematical properties of the EMG function allow us to calculate easily these moments M and the related chromatographic figures of merit (4, 6-10): retention time = M1 = t , T
+ peak variance = M 2= u2 +
T~
1884
AMI.
chem.1991, 63, 1884-1889
peak
peak excess = M 4 / M $ - 3
M4 = 3a4 + 6a2r2+ 9~~
The asymmetry ratio, bla, obtained from the peak width at 10% of the peak height, WO.lh, with a + b = WO.lh, was empirically related to the EMG 7 / u ratio by (19)
--4l
parameter change versus time. Observing Figures 2-5, it appears that the general shape of an exponentially modified perturbation (positive variation followed by a baseline return) is a tailing peak. This is the case when the underlying functions are as different as the square function (Figures 2 and 3), the triangle function (Figure 4), or the Gaussian function (Figure 5). It is possible that exponentially modified Gaussians are an ideal peak shape model. If it is not the case, the use of the EMG model to describe any tailing peak may be questionable. FIA peaks may be more accurately described and modeled by using exponentially modified square functions than EMG functions.
LITERATURE CITED PRACTICAL IMPLEMENTATION
Programming the mathematical recursive series described by eq 8 is straightforward in any computer language. It needs few program lines. If the acquisition frequency is not constant, one more line shall be added to calculate the adequate parameter, A, by using eq ll with the desired constant value, 7 . Equation 8 is so simple that it is often not necessary to write a computer program to obtain the exponentially modified function. Modern electronic spreadsheets can be conveniently used. Figure 6 is a screen hardcopy of the spreadsheet used to draw Figure 4 with the software Lotus 123 (Lotus Development Corp., Cambridge, MA). Columns A and B contain the time, t, and signal, E @ ) ,respectively. Columns C-G contain the eq 8 convoluted signal, U t ) . The time constants, 7 , and corresponding A terms are contained in rows 1and 2, respectively. Cell D6 is highlighted to show (Figure 6, top left) how eq 8 is easily introduced in such a software. Changing a time constant in row 1 instantly produces the corresponding EMT values and graph. Tables I and I1 were obtained by using the same software.
(1) SaVbky, A.; W Y , M. J. E. Anal. Cbm. 1964. 36, 1627-1839. (2) Kkkland. J. J.; Yau. W. W.; Stoklosa, H. T.; Dilks, C. H. J. J . ChromeSCl. 1977, 15, 303-316. ( 3 ) Ruzlcka, J.; Hansen, E. H. Fbw In@tbn Anal)rslp: 2nd ed.:Chemlcal Analysis Serbs; Wlky: New York, 1988; Vol. 62. (4) Sternberg, J. C. I n Advances h chnwnetogaphy; Giddings, J. C., Keller, R. A., Eds.; Dekker: New York, 1960; Vol. 2, pp 205-270. (5) Gladmy, H. M.; Dowden. 8. F.; Swalen, J. D. Anal. Cl”.1969, 4 1 , 883-888. (6) Grushka, E. Anal. Chem. 1972, 44, 1733-1738. (7) Yau, W. W. Anal. Chem. 1077, 40, 395-398. (8) Pauls, R. E.: ROQWS,L. B. Anal. Chem. 1977. 40, 825-626. (9) Barber, W. E.; Can. P. W. AMI. Chem. 1981, 53, 1939-1942. (10) Fdey, J. P.; Dorwy, J. G. Anal. (XlCHn. 1963, 55, 730-737. (11) FObY, J. P.: DO~SOY. J. 0. J . chnwnew. Scl. 1964, 22, 40-48. (12) Fdey, J. P. AMI. Chem. 1967, 5 0 , 1984-1987. (13) Hemandez-Torres, M. A.; Khaledi, M. G.; Dorsey, J. 0. Anal. Chlm. ACIB 1967, 2Q1, 67-76. (14) B r W k S , S. H.; DOr~ey,J. 0. Anal. chkn. Acta 1990, 220, 35-46. (15) HanWl. D.; Can. P. W. AMI. Cbm. 1961, 57, 2394-2395. (16) Delley, R. chrometm@?k 1964. 18. 374-382. (17) M k y , R. Anal. chsm.1965. 57. 388. (18) Selby, M. S. StandardMeUmmatlCel Tabks; CRC Press: Boca Raton, FL, 1974. (19) Berthed, A. J . Liq. Chrometog. 1969, 12. 1187-1201. (20) Tyson, J. F. Anal. chkn.Acta 1966, 170, 131-148. (21) Grushka. E.; Myers, M. N.; Schetter, P. D.; Glddlngs, J. C. Anal. Chem. 1969, 41, 889-892. (22) Bldllngmeyer, B. A.; Warren, F. V. Anal. Chem. 1964, 56, 1583A1598A.
m.
CONCLUSION
Equation 8 describes a simple generating function that can easily produce the EM form of any function representing a
RECEIV~,for review January 3,1991. Accepted May 30,1991.
CORRESPONDENCE Capillary Isotachophoresis with Concentration Gradient Detection Sir: Recently, capillary zone electrophoresis (CZE) has attracted significant attention from the analytical community (1,2). This method performs rapid, high-resolution separation of minute sample quantities and has important applications in biochemical and medical separations. Another interesting capillary electrophoretictechnique is capillary isotachophoresis (CITP) (3-8).CITP has certain advantages over CZE since it has less need for a sharp injection of high-concentration samples. Samples can be introduced with simple injection devices (3-5). Conventional detectors for CITP such as UV or conductivity offer sufficient sensitivity to monitor the high concentration of analytes in the separated zones. However, no high-spatial-resolution, sensitive detector has been proposed that would also allow the universal analysis of low0003-2700/91/0363-1884$02.50/0
concentration sample components. The major disadvantage of CITP is the necessity of using a discontinuous buffer system to separate anions or cations. A leading electrolyte (LE) and tailing electrolyte (TE) bracket the sample analytes and are chosen primarily on the basis of their mobility. The leading electrolyte has the highest mobility and migrates first in the separation capillary and is followed by the sample zones and the tailing electrolyte. During the course of the separation, each sample component becomes separated into a pure zone, stacked sequentially between the leading and tailing electrolytes, according to its mobility and degree of ionization at a given pH. After the steady state is reached, the zones migrate toward the detector. For example, if a sample originaUy contains two components S1and S2,then, 0 1991 American Chemical Society
