the Krebs-cycle acids. Although purification has been achieved, with good recovery of some of the acids, many hours are required for the preliminary preparation of the sample. An improved method involving less time that will extract the organic acids from tissue in form sufficiently purified for G L C with good recovery for all of the acids still remains to be found. Improvement of these preparatory steps is being continued.
ACKNOWLEDGMENT
The author wishes to express her gratitude to A. Cazorla and C. Krumdieck for their support and interest. RECEIVED for review July 27, 1970. Accepted March 26, 1971. This work was aided in part by grant 66-G-123 from the U.S. Army Element, Defense Research Office, Latin America, Rio de Janeiro, Brasil.
Resolution Enhancement of Chromatograph Peaks Dale W . Kirmse and Arthur W . Westerberg Department of Chemical Engineering, University of Florida, Gainesville, Fla 32601 This paper presents work on a new method for area resolution of overlapped gas chromatograph peaks. The method is resolution enhancement utilizing the Fast Fourier Transform. Overlapped peaks are mathematically narrowed or sharpened, while retaining their individual areas unchanged, and thus become visually resolved from each other. The appearance of noise in the data and the use of spectral windows to counter its effect are illustrated.
If the transform of peak shape function H ( f ) is known, we can in principle convert the observed chromatogram represented by y ( t ) into a set of ideal pulses. We form
A CHROMATOGRAPH COLUMN can be considered to be a system which converts an input signal x(t) into an output chromatogram y(t). The input is typically a single, almost impulselike, peak whereas the output is a series of L widened peaks appearing at different delay times. A model of this system could be written
(7)
which has the inverse Fourier Transform
where
and S(r) dt = 1
We define a Fourier transform by (1)
XU) = F[x(t)l
=
exp(-2?rift)dt
(2)
x ( f )exp(27rif)df
(3)
x(t) S-mm
and its inverse by x(r)
=
F-'[XCf)]
=
J-ma
The time of each pulse is 7 1 ,the delay or elution time for component I. It is evident at this point that converting Y ( f ) t o Y d f > does not require a knowledge of the elution times for the peaks. The factor al is, under appropriate scaling for H m ,the area under peak 1. We can demonstrate this by the following. The contribution to Equation 6 by peak 1 is
where Y I ( ~ )
i is 4 - 1 , and
Using these definitions, the Fourier Transform of Equation 1 is Hle-2wifrr
=
J-m
yl(t)e-2"ifidt
(4)
If we assume for the purposes of illustration that all components will elute with the same peak shape except for a scale parameter, we can write
Yl(0) =
J m -m
yr(t)dt = Area of Peak
I
(6)
With the single assumption that H(0) is unity, we see that at is simply the area under the unsharpened peak 1 and also the area under the sharpened peak y,,(f) in Equation 7. The above peak sharpening process is known as resolution enhancement. Allen, Gladney, and Glarum (2) used both digital and analog filter techniques based on differentiation
G. M. Jenkins and D. G. Watts, "Spectral Analysis and Its Applications," Holden-Day, San Francisco, Calif., 1969,p 44.
(2) L. C. Allen, H. M. Gladney, and S. H. Glarurn, J. Chem. Phys., 40, 3135 (1964).
H d f ) = atH(f) and we get for Equation 4
(5)
L
Y ( f )= H ( f ) (1)
Ydf) which for f = 0 yields
L 1=1
H(f)ale-2Tifr1
Using Equation 2 we can write
f is frequency in cycles/unit time
y(f)=
=
1=1
ANALYTICAL CHEMISTRY, VOL. 43, NO. 8, JULY 1971
0
1035
Each peak in y s ( t ) has its variance reduced by u 2 and thus will appear as a sharper peak. We note by setting f = 0 in both Equations 8 and 10 that the areas under both the unsharpened and sharpened peak I are equal to a [ . This result occurs again because H,q(O) = 1 is used. This example indicates restrictions on u 2 for H&). In order that Equation 10 has an inverse u 2 _
