De-Tailing and Sharpening of Response Peaks in Gas

De-Tailing and Sharpening of Response Peaks in Gas Chromatography J. W. ASHLEY, JR., and CHARLES N. REILLEY Department o f Chemistry, University of North Carolina, Chapel Hill, N. C.

b A general approach for designing active electronic networks to improve the shape of transient signals is presented. The method is demonstrated by two specific applications to gas chromatographic response signals: eliminating tailing of individual response peaks and partially sharpening individual peaks. Both of these applications can b e of significant value in chromatographic analyses. Also presented are some useful mathematical techniques for computing the moments of a transient response signal.

E

having finite response times can significantly contribute to the shape of the observed response in any physical experiment involving time-dependent functions. These undesirable distortions can originate anywhere between the original input and the final output. For example, some time-dependent distortions in gas liquid chromatography may result from rate of evaporation of a liquid sample, mixing in sample injection chamber, mixing due to irregular gas flow a t tubing connections and bends, processes occurring within the column, mixing in the detection system, rate of response of the detector element, and rate of response of electronic amplifying and recording systems. Of these distortions, only those occurring within the column are of interest to the chromatographer; all others are usually minimized in so far as practical. However, some extraneous distortions always remain and will affect the observed chromatogram to some extent. Also, some of the distortions occurring within the column are usually considered undesirable-i.e., an ideal column would only delay each component by its characteristic retention time, but would not alter the individual sample component shapes. Any real column does, however, broaden component peaks. When unwanted distortions are sufficient to obscure the desired information, the experimenter is faced with the problem of sorting out all the significant extraneous distortions and applying appropriate compensations to the observed data. In this article, a method for 626

XTRANEOUS ELEMENTS

ANALYTICAL CHEMISTRY

275 7 5

compensation before observation is illustrated by its application to a gas liquid chromatographic system. GENERAL METHOD

The basis of the compensating method is illustrated in Figure 1. Three assumptions are made : the amplitude of the actual response from the complete system is linearly related to the input amplitude; the system of interest is connected in series with the preceding and following extraneous elements as shown (extraneous elements may also be connected in series between two parts of the system of interest, but an element connected in parallel with any part of all of the system of interest introduces more rsstrictions) ; and the transfer function, E , of the lumped extraneous elements is (at least approximately) known or assumed. All extraneous elements, whether preceding or following the system of interest, can be treated as a single composite element. The Laplace transform, A , of the actual response is equal to the product of the transforms of the input, the

figure 1.

system of interest, and the extraneous elements.

.A

=

7%

(1)

The Laplace transform, R, of the desired response is equal to the product of the transforms of the input and the system of interest.

Thus, the desired response transform can be obtained from the actual response transform by multiplication by the reciprocal of the extraneous element transform.

(3) That is, the desired response will be obtained if a new comeonent having the transfer function ( E )-l is purposely inserted somewhere in the overall system. EFFECTS OF LAG DISTORTION

One type of distortion commonly encountered is the lag-that is, a distortion due to a component which, when excited by an ideal impulse, gives an

Method for eliminating unwanted distortions from response signal /(t) = input function A(t) = actual response function ![t)-= Aefired response function

E , , E2, S, E = transfer functions of indicated components A bar over a function indicates Laplace transform with respect to time, t.

r = ;[s) = y e - s l l ( t ) d t -m

where I is the Laplace transform variable

strates that the maxima and minima of the output of a lag element fall precisely on the input curve-regardless of its shape. This effect is easily derivecj as follows : the response transform, R, is related to the input transform, I,by

A

Figure 2. Effect of single lag element on typical chromatographic response signal

--_

Undistorted response signal Response signal after distortion by lag element having time constant, T , equal to (a) 1 sec., (b) 5 sec., (c) 10 sec. Me1 = methyl iodide, Et1 = ethyl iodide, MEK = = butonone

~

= I/(l

+ s7)

(6)

,4t the m3xima and minima of the response, SR = 0 (that is, dR/dt = 0) and thus R = I. Therefore, the input and response are equal a t maxima and minima of the latter. That the introduction of a lag alters the shape of a peak without changing its area simply means that all of the original sample is eventually accounted fore.g., the total amount of sample that comes out of a mixing chamber will eventually equal the total amount of sample that was put in. COMPENSATION FOR A LAG-DE-TAILING

exponentially decaying response, e--t'r, where the lifetime, 7 , is characteristic of the duration of the distortion effects. Some components which exhibit approximately this type of distortion are mixing chambers, detector elements (such as a thermistor), and the simple R-C integration circuit. The transfer function for such elements is 1/(1

T o compensate for the distortions due to a single lag, the appropriate antidistortion element is one having the s7. transfer function ( E ) - 1 = 1

+

sr). r

The effect of a single lag element in a GLC network is shown in Figure 2, where lags of various lifetimes, r , were purposely introduced to give different degrees of distortion to the chromatogram. As the lifetime of the lag increases, each peak is shifted to later times and becomes more broadened and tailed. The contributions of the lag element to delay and variance of each response peak are easily computed from Equations 41 and 6h. The contribution to delay is -1im (1 s+o

+ s7) -dds (1 +

+

+

+

(a) DE-TAILER

+

v1 =

Hence, de-tailing is effected by adding the actual signal and a portion of its first derivative as illustrated in Figure 3a. The appropriate weighting factor, r , is incorporated through the time constant, T = RC, of the differentiator by setting T = r . When the experimenter is primarily interested in an integrated response, the integrating and de-tailing can be combined into one operatio!. In this_ case, the desired signal is R / s T = A ( l sT)/sT and the transfer function of the sT)/sT = integrating de-tailer is (1 (l/sT) 1. Thus, de-tailing and integration are simultaneously effected by adding a portion of the observed signal to its integral as illustrated in Figure 3b. Combination of the de-tailing and integrating components has two instrumental advantages : incorporation of the integrator requires no new components and the de-tailing circuit ordinarily requires the inclusion of extra damping components to offset the amplification of high-frequency noise in the differentiating part of the circuit. Thus, in practice, the integrating de-tailer re-

-------

SIMPLIFIED CIRCUIT :

-I

is+

ff =-H(l+sTI

s7)-l

-

(4)

(b) INTEGRATING DE-TAILER

and the contribution to variance is

-

T2

(5)

Thus, the introduction of a lag element having lifetime r delays each peak by 7, increases the variance of each peak by 9, and has no effect on the total area under each individual peak (if they could be observed separately). Sternberg ( 5 ) has recognized that when a gaussian-shaped peak is applied to a lag element, the maximum of the response peak will fall exactly on the original gaussian. Figure 2 demon-

SIMPLIFIED CIRCUIT :

A

3 BRZ--(l+sT)

nT

Figure 3. Schematic circuit diagrams illustrating how derivative augmenting effects de-tailing

-D-

.. .

1

Slgn-Inverting operational amplifier Slgnal a t indicated part of circuit (ignoring sign-inversions) .Desired signal (included for reference) T h e constant-product of resistance and capacitance

VOL. 37, NO. 6, MAY 1965

a

627

I? = I. Therefore, input and response are equal a t maxima and minima of the former.

and the contribution to variance is p' = Y2

-

Y12

PEAK SHARPENING

=

1 0

20

10

30

40

TIME, s e t

Figure 4. Effect of single de-tailing element on typical chromatographic response signal

---

Actual response signal (includes 5-sec. lag distortion) Response signal after (anti-) distortion b y de-tailing element having time constant, 1, equal to (a) 2 sec., ( b ) 5 sec., (c) 10 sec.

quires fewer parts than the de-tailer and does not introduce extra lag distortions. Although the integrating de-tailer produces a larger signal-to-noise ratio, this effect results from the attenuation of high-frequency noise coincident with the integration process and does not constitute a fair comparison of informationto-noise ratios. Before the de-tailer can be used effectively, it is necessary to determine the value of 7 (the amount of tailing) in the observed signal. While it may be possible to estimate an appropriate value from a knowledge of the apparatus and operating conditions (S), a simple, empirical approach is readily applicable for improving the resolution of component peaks. Figure 4 illustrates the effect of adding too much ( T = 10 sec.), too little ( T = 2 see.), and,just the right amount ( T = 5 sec.) of derivative to comliensate for the tailing in the actual response curve. Over-compensationi.e., too much derivative or too large a time constant in the de-tailer-yields negative values in the response curves; the signal begins to look too much like the derivative. Thus, the appropriate value for the time constant of the detailer can be found by first using too large a value and then decreasing the time constant until the negative portion of the signal just disappears. An upper limit for the appropriate time constant can be quickly determined by measuring the width-at-half-height of the sharpest peak-i.e., the air peak. The appropriate time constant must be less than this width divided by In 2. The contributions of the de-tailing element to the delay and variance of each peak are easily computed from Equations 4.i and 6 h . The contribution to delay is v1

=

-lim (1 s+o

+ S T ) - ' ds~d - ( 1+ s T )

- T2

(8)

Thus, the introduction of a de-tailing element having time constant T advances each peak by T , decreases the variance of each peak by T 2 , and has no effect on the total area under each peak. Ailso, application of the superposition principles (Appendix) demonstrates that the contributions of the detailing element to delay and variance cancel the contributions of the lag element if T = 7. An additional feature demonstrated in Figure 4 is that the output of a detailing element passes exactly through each maxima and minima of the input curve. This effect is readily derivtd as follows: the response transform, R , is related to the input transform, I , by

R

=

ANALYTICAL CHEMISTRY

+ sT)

(9)

At the maxima and minima of the input, SI = 0 (that is, d I / d t = 0) and thus

and the desired response transform is =

I,-StR

(a) D'-SHARFENER

SIMPLIFIED CIRCUIT:

R :-fi(I-szTz)

(b) INTEGRATING D2-SHARPENER

SIMPLIFIED CIRCUIT

:

I

~

R = -A(i-#Tz) ST

Figure 5. Schematic circuit diagrams illustrating how second derivative augmenting effects peak sharpening __ Signal at indicated part of circuit (ignoring sign-inversions)

... .

628

I(1

*is demonstrated by hllen, Gladney, and Glarum ( I ) , the appearance of a symmetrical peak may be arbitrarily sharpened by adding to t'he peak positive or negative portions of its even derivatives. These authors considered that the ultimately desired signal should consist of infinitely thin lines (Dirac delta functions) rather than peaks of finite width. In terms of gas chromatography, selective retention of each component by the column is desirable, but the accompanying broadening effects (longitudinal diffusion, etc.) are undesirable distortions. If the actual chromatographic peaks are assumed to be (approximately) gaussian, t,hen the actual response transform, for a single component, is (2,4)

Input rlgnol or its integral (included for reference)

(11)

of high-frequency noise in the differentiating sections of the circuits. Figure 6 shows the effects of both 0 2 and D4-sharpening using the coefficients of Equation 13. For the Dz-sharpener, the contributions to delay and variance are d ds

(1 - s2T2)-l- (1

uI = -1im 8-0

=o

W

Ln

- s2T2) (14)

0

Lo

112

IT W

= =

UP

lim (1

- s2T2)-I d 2 (1 - sZT2) ds2

8+O

For the D4-sharpener, the contributions are u1 = 1

1

1

2

,

0

1

I

I

-1im (1

,

8-+0

4

2

- s2T12+

TIME. sec.

Figure 7. Effects of de-tailing and D2sharpening of typical chromatographic response signal

TIME/c

Figure 6. Effects of D2 and D4-sharpening elements (a) on gaussian-shaped peak and (b)on its integral

....(a) Gaussian,G = ( o . \ / ~ ) - 1 ( e - t 2 / 2 ~ 2 ) a n d (b)integral ---(a) D2-sharpened gaussian, G - ( u 2 / 2 ) (d2G/dt2)and ( b ) integral -(a) D4-sharpened gaussian, G - (u2/2) (dZG/dt2) (u4/8)(d4G/dt4)and ( b )integral

+

Hence, the appropriate antidistortion element (or sharpener) should have the transfer function

Because a n analog circuit having exactly this transfer function is not practical, an approximation is used. Expansion of the exponential gives

(E)-l = Saul

I--+--.

2

s4u4 8

,

,

(13)

Thus, partial sharpening effects may be achieved by subtracting a portion of second derivative from the initial signal as shown in Figure 5a; better sharpening may be obtained by also including a portion of the fourth derivative, etc. The relative amounts of each derivative to be included are determined by the value of U , a measure of the peak width. When the experimenter is primarily interestpd in the integrated response, the sharpening and integrating components may be combined as shown in Figure 5b. In both the second-derivativt. sharpener (D2-sharpener) and the integrating D2-sharpener, it is usually necessary to include estra damping components to offset the amplification

=o 112

= v2 =

(16) 8+0

=

-2T1'

+

lim(1 - s2TI2

(17)

Thus, the introduction of either a D2or a D4-sharpener has no effect on the position of each peak, decreases the variance of each peak by 2T12,and has no effect on the total area under each individual peak. Inclusion of only the first two terms of Equation 13 is sufficient to reduce the variance of the response peak to zero-i.e., superposition of the variance, a2, of the initial gaussian peak with the variance contribution, -2TI2, of a D2"sharpener shows that the variance of the response peak is zero when T12 = u2/2 as indicated in Equation 13. Inclusion of the first three terms reduces to zero the values of both the variance and the fourth moment about the peak center. Similarly, as each successive term is included, the value of the corresponding moment about the peak center is reduced to zero. Because each analog differentiation decreases the signal-to-noise ratio, instrumentation involving more than about two differentiations is difficult, so sharpening higher than D 2 will not be considered further. Because of the approximation involved in dropping off higher-order terms of Equation 13, the relative amount of second derivative to be used in D2-sharpening is not uniquely determined. With D2-sharpened Lorentzian peaks ( I ) , the dips on the sides of the sharpened peaks can be avoided by not using excessive amounts of second

(a)

Actual response signal (includes a 1-sec. lag distortion) ( b ) De-tailed ( T = 1 sec.) response signal (c) De-tailed ( T = 1 sec.) and D2-sharpened ( T 2 = 0.2 set.*) response signal. MMK = Acetone

derivative. But with D2-sharpened gaussian peaks, some dip will always occur regardless of how small an amount of second derivative is used. Alllen,Gladney, and Glarum (1) have demonstrated that D2-sharpening can significantly enhance the resolution of overlapping Lorentzian peaks having approximately the same width. However, D2-sharpening of overlapping gaussian peaks is somewhat less effective, partly because of the negative dip and partly because of the increased extension of the peak edges (Figure 6). Figure 6 implies that D2-sharpening of overlapping gaussian peaks may sometimes be advantageous; however, negative dips can be a source of serious errors in quantitative measurements. These errors tend .to be mutually cancelling when the overlapping peaks are not too different in size and symmetrical shape. Figure 7 demonstrates the enhancement in apparent resolution achieved by both de-tailing and D2-sharpening. The relative amount of second derivative was arbitrarily chosen to give an apparently complete resolution of the MJIK and Et1 peaks. The sharper peaks (smaller U ) contain too much second derivative. DISCUSSION

The de-tailer is most useful when all the peaks in the original chromatogram lag (or tail) by the same extent-as caused, for example, by a mixing volume or a detector time constant. Thus, detailing should yield significant improveVOL. 37, N O . 6, MAY 1965

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629

ments to chromatograms obtained with capillary or other low-volume columns. Kieselbach ( S ) , to compensate for a detector time constant, employed a circuit essentially equivalent to the one shown in Figure 3a, but he reported no attempt to compensate electronically for apparatus mixing volumes. When two or more lags are significant, it may be desirable to eliminate their effects by using two or more de-tailing circuits in series. Although the preceding discussion has been confined to linear systems, the de-tailer circuit may also be empirically applied to reduce the effects of tailing resulting from nonlinear isotherms or slow kinetic steps, as in solid surface adsorption. The de-tailing operation does not affect the total area under each peak but merely redistributes that area in a characteristic way. Thus, the detailer may be used advantageously in quantitative analyses even though the tailing is not strictly a lag and is not necessarily the same for each peak. Of course, it will be to the chromatographer's advantage to design a more appropriate circuit when the shape of the distortion differs too much from that of a simple lag. A further complication to the applicability of the Dz-sharpener is inherent in isothermal chromatography. Because the optimum amount of second derivative to be used depends on the peak width and peak width generally increases with retention time, a Dzsharpener adjusted to be most effective in a given portion of a chromatogram will incorporate too much second derivative into earlier peaks and too little into later peaks. Thus, it is necessary either to sharpen each set of unresolved peaks in separate experiments or to design the sharpener with continuously varying time constants so as to incorporate the appropriate amount of second derivative into each peak. This complication can be avoided in temperature-programmed gas chromatography because the individual peak widths are usually approximately the same.

transform. Laplace transformation also leads to a mathematically convenient method for computing the moments of a function, which are directly related to delay and variance. For these reasons, the following mathematical definitions and derivations are presented. The Laplace transform, P ( s ) , of a function of time, F ( t ) , is defined by

,

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ANALYTICAL CHEMISTRY

s(s).

R = IS (7N Differentiation and rearrangement gives 1

dR

R

as

-

dI

1

1

dS

I Z+lS'Z

J - m

The nth derivative of the transformed function is

J-, (-t)ne-S'F(t)dt +m