Black Box Linearization for Greater Linear Dynamic Range: The Effect

Anal. Chem. 2010, 82, 10143–10150

Black Box Linearization for Greater Linear Dynamic Range: The Effect of Power Transforms on the Representation of Data Purnendu K. Dasgupta,*,† Yongjing Chen,† Carlos A. Serrano,† Georges Guiochon,‡ Hanghui Liu,§ Jacob N. Fairchild,‡ and R. Andrew Shalliker| Department of Chemistry and Biochemistry, University of Texas at Arlington, Arlington, Texas 76019-0065, United States, Department of Chemistry, University of Tennessee, Knoxville, Tennessee, United States, Senomyx Incorporated, 4767 Nexus Centre Drive, San Diego, California 92121, United States, and Australian Centre for Research on Separation Science, School of Natural Sciences, University of Western Sydney, Parramatta, Locked Bag 1797, South Penrith Distribution Centre, South Penrith, New South Wales 1797, Australia Power transformations are commonly used in image processing techniques to manipulate image contrast. Many analytical results, including chromatograms, are essentially presented as images, often to convey qualitative information. Power transformations have remarkable effects on the appearance of the image, in chromatography, for example, increasing apparent resolution between peaks by the factor n and apparent column efficiency (plate counts) by a factor of n for an nth-power transform. The profile of a Gaussian peak is not qualitatively changed, but the peak becomes narrower, whereas for an exponentially tailing peak, asymmetry at the 10% peak height level changes markedly. Using several examples we show that power transforms also increase signal-to-noise ratio and make it easier to discern an event of detection. However, they may not improve the limit of detection. Power responses are intrinsic to some detection schemes, and in others they are imbedded in instrument firmware to increase apparent linear range that the casual user may not be aware of. The consequences are demonstrated and discussed. Despite caveats to the contrary, the first impression of a user looking at continuous data traces, e.g., in chromatography, is that the peak amplitudes directly reflect the respective concentrations. It can of course be grossly erroneous: detector response to two closely eluting analytes can be so different that neighboring peaks of equal magnitude may reflect concentrations that are order(s) of magnitude different. Universal detectors that exhibit reasonably uniform mass response are increasingly popular in the analysis of polymers and pharmaceuticals; the evaporative light scattering detector (ELSD1,2), the aerosol charge detector,3,4 and the * To whom correspondence should be addressed. E-mail: [email protected]. † University of Texas at Arlington. ‡ University of Tennessee. § Senomyx Incorporated. | Australian Centre for Research on Separation Science. (1) Mitchell, C. R.; Bao, Y.; Benz, N. J.; Zhang, S. J. Chromatogr., B 2009, 877, 4133–4139. (2) Takahashi, K.; Kinugasa, S.; Yoshihara, R.; Nakanishi, A.; Mosing, R. K.; Takahashi., R. J. Chromatogr., A 2009, 1216, 9008–9013. 10.1021/ac102242t  2010 American Chemical Society Published on Web 11/24/2010

condensation nucleation light scattering detector2,5,6 are particularly noteworthy. Although a particular analyte may exhibit a large linear dynamic range (LDR),3 often this ideal is not attained. Especially in a commercial instrument, the practicing analyst looks for a wide linear dynamic range; if the response behavior can be expressed by Y ) kCx

(1)

Y being the detector signal, C being the concentration, and k being a constant of proportionality, she desires x to be 1. Often, x is slightly 1. In either case, if the dependence is (3) (4) (5) (6) (7)

Dixon, R. W.; Peterson., D. S. Anal. Chem. 2002, 74, 2930–2937. Vehovec, T.; Obreza, A. J. Chromatogr., A 2010, 1217, 1549–1556. Allen, L. B.; Koropchak, J. A.; Szostek, B. Anal. Chem. 1995, 67, 659–666. Lu, Q.; Koropchak, J. A. Anal. Chem. 2004, 76, 5539–5546. D’Ottavio, Y.; Garber, R.; Tanner, R. L.; Newman, L. Atmos. Environ. 1981, 15, 197–203.

Analytical Chemistry, Vol. 82, No. 24, December 15, 2010

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nonlinear, many approaches to defining and calculating the LOD may bear further scrutiny.8 Regardless of the intrinsic operating principle, many detectors in practice do not exhibit linear behavior. Even the venerable optical absorbance detector response becomes sublinear at high absorbances due to stray lightsfor capillary-scale detectors the relative level of stray light tends to be higher and absorbance at which such departure from linearity perceptibly begins may not be all that high. Even in this case, a modest power transformation can increase the LDR. Power transformation of the signal thus has obvious effects on apparent S/N and also on the apparent resolution of adjacent chromatographic peaks. In a recent paper, we have discussed some of these aspects for both simulated and real separations, including multidimensional separations.9 In the present work, we examine the underlying theory in some detail: apply the principle to several sets of simulated and experimental data and experimentally examine the effect on S/N in an ELSD detection situation where an analyte is deliberately added postcolumn continuously to the column effluent. PRINCIPLES Chromatographic Peak Efficiency and Resolution. An ideal chromatographic peak has a Gaussian profile; many chromatographic peaks today come close to this ideal. A polynomialmodified Gaussian function can fit any real chromatographic peak.10 In fact the response profile to an analyte impulse in most chemical detection systems can be closely approximated by such a function. Presently, for simplicity we will assume a purely Gaussian response function and a baseline without significant drift. Previously real data where the baseline is far from flat has been considered;9 the basic considerations still hold. A Gaussian chromatographic peak can be described by the relationship Y ) Ymax exp(-((t - tR)/s)2)

(3)

Y being the response at any time t with a maximum value of Ymax at the retention time tR. The parameter s is related to the width or variance of the peak; the peak width increases with s. Normalizing (Ymax ) 1), eq 3 takes the simpler form Y ) exp(-((t - tR)/s)2)

(4)

Setting Y ) 0.5, the two possible values of t lead to the two solutions to the quadratic equation; the difference between them gives the peak half-width (Wh) in terms of s: t ) tR ( 0.693s

(5)

Wh ) 1.386s

(6)

The peak efficiency (plate count) N is given by (8) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A–724A. (9) Shalliker, R. A.; Stevenson, P. G.; Shock, D.; Mnatsakanyana, M.; Dasgupta, P. K.; Guiochon, G. J. Chromatogr., A 2010, 1217, 5693–5699. (10) Nikitas, P.; Pappa-Louisi, A.; Papageorgiou, A. J. Chromatogr., A 2001, 912, 13–29.

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N ) 5.54(tR /Wh)2

(7)

N is thus inversely proportional to s2: N ) 2.88(tR /s)2

(8)

The resolution Rs between any two peaks at tR′ and tR both with efficiency N can be given by Rs ) 0.5√N(tR' - tR)/(tR' + tR)

(9)

Since N is inversely proportional to s2, Rs is thus inversely proportional to s. If we transform Y to Yt by raising to any arbitrary power n, Yt ) Yn ) (exp(-((t - tR)/s)2))n ) exp(-((t tR)/(s/√n))2)

(10)

It is immediately observed that Yt is another Gaussian function, described by an equation identical to eq 4, except that s is replaced by s/n. In other words, applying an nth-power transform to chromatographic response data will result in an increase in apparent column efficiency and resolution by factors of n and n, respectively. Peak Asymmetry. Peak asymmetry (often measured at 10% of the peak height (As,0.1) as the width of the trailing half of the peak at this height, divided by that of the leading half) is unity for a symmetric purely Gaussian function. If we could simulate asymmetry by considering a peak where the s-term for the leading half of the peak (t < tR) is given as sl, whereas at t > tR the trailing half of the peak is represented by another, broader Gaussian distribution with the applicable s-term being st (st > sl), it is readily shown that As is equal to st/sl and is >1. If an nth-power transformation is carried out, it will reduce both st and sl by n; it follows that As will not change as a result of the transform. However, in reality a chromatographic peak with a high As value resembles an exponential decay on the tailing portion much more than another, more broadly distributed Gaussian. For an exponential function, the trailing peak width at any given height will decrease by a factor of n upon an nthpower transformation rather than n observed for a Gaussian peak. With the leading width decreasing by n and the trailing width decreasing by n, the peak asymmetry will improve by the factor n upon an nth-power transformation. RESULTS AND DISCUSSION Illustrative Examples. Figure 1 shows an example of actual experimental data where two peaks are poorly resolved and the second peak tails badly (in this case, as a result of overloading). All aspects discussed above regarding power transformation can be seen demonstrated in this example. We hasten to add that power transformation of the response of course does not improve the real separation or intrinsic chromatographic efficiency. Before one discards it as totally cosmetic, for many purposes (e.g., how many major components does it contain?) a representation in power-transformed format should be defensible solely on the grounds that it more readily conveys the desired information; after all, makers of real cosmetics, a multibillion dollar industry, need

Figure 1. Separation of 2-nitrophenol (10 mg/L) and 3-nitrophenol (20 mg/L) detected at 325 nm. An amount of 50 µL of the aqueous mixture was injected on a 4 mm × 50 mm AG11 column (Dionex Corp.) and eluted with 20 mM Na2CO3 at 1 mL/min. The second peak maximum was 0.204 absorbance units.

only claim that appearance improves. A chromatogram is ultimately an image. A power transform increases image contrast and has long been used for that purpose11-13 (see Supporting Information Figures S1-S6 for an experiment with image contrast). If response peaks of very different magnitudes are present in the chromatogram, it is important to note that an n > 1 power transformation will only accentuate the original difference. No information is lost, however; in much the same way that very small peaks are looked at after magnification, the chromatograms can be depicted for the magnified insets. Figure S7 in the Supporting Information shows a very noisy chromatogram where the peaks vary considerably in height. Figure S8 in the Supporting Information shows the corresponding n ) 4 power transform, and the magnified view of the small peaks shows that there is decrease in relative noise for the small peaks as well. When two peaks of very different magnitude are overlapped, depending on the degree of the overlap, an n > 1 power transformation can make it more difficult to perceive two peaks or resolve them. However, a power transformation does not automatically connote n > 1; as Figure S9 in the Supporting Information indicates, a power transformation with 1 > n is beneficial in this case to better perceive the overlap and resolve them. Behavior of Real Detection Systems with Imbedded Blackbox Signal Processing. The caveat that the foregoing discussion underscores is that when calculating chromatographic figures of merit from the response of a detector, one should (11) Fisher, R.; Perkins, S.; Walker, A.; Wolfart, E. Image Processing Learning Resources. Exponential/‘Raise to Power’ Operator. http://homepages.inf.ed.ac.uk/rbf/HIPR2/pixexp.htm (accessed August 15, 2010). (12) O’Gorman, L.; Brotman, L. S. Proc. SPIEsInt. Soc. Opt. Eng. 1985, 575, 106–113. (13) Guo, L. J. Int. J. Remote Sens. 1991, 12, 2133–2151.

Figure 2. Response of a serially connected diode array detector (DAD) at 230 nm and an evaporative light scattering detector downstream of the DAD to a sample of verapamil hydrochloride (2 µg in 10 µL) injected into a carrier stream of 20% aqueous MeOH.

maintain a healthy degree of skepticism unless the response linearity of the detector and/or the nature of its signal processing is clearly known. Figure 2 shows overlaid responses from a UVabsorbance detector followed by that from an ELSD (detector brand names are not given for obvious reasons) that serially follows the absorbance detector (see Figure S10 in the Supporting Information for general layout; the auxiliary stream did not contain any analyte for this experiment). Obviously, the peak width can only increase in going from the absorbance detector to the ELSD due to unavoidable dispersion, whereas the data indicate that there is an ∼6% decrease in Wh (this difference is significant: RSD of Wh of 4 repeated injections on the same detector was ∼0.13%) and there is a more impressive decrease in As,0.1, from 2.34 to 1.75. Application in DNA Sequencing and Single-Molecule Detection. In applications such as sequencing where the primary information is essentially qualitative (presence or absence of a base) or in DNA fragment separations where semiquantitative information on the presence or absence of a peak at a particular retention time (this is controlled by size) is of interest, a powertransformed representation may be attractive. Figure 3 shows an example of a hydrodynamic separation of fluorescent dye labeled DNA fragments in a nanocapillary14 where we show both the original data and the fourth-power transform (both in normalized form). The power-transformed representation increases contrast. In fluorescence-based single-molecule detection experiments, photon counting is generally used and detection is ultimately shot noise limited.15 The detection of one or two fluorescein molecules in Figure 4 is much more apparent in the power-transformed representation than in the original. (14) Wang, X.; Veerappan, V.; Cheng, C.; Jiang, X.; Allen, R. D.; Dasgupta, P. K.; Liu, S. J. Am. Chem. Soc. 2010, 132, 40–41. (15) Burrows, S. M.; Reif, R. D.; Pappas, D. Anal. Chim. Acta 2007, 598, 135– 142.


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Figure 3. Hydrodynamic separation of fluorescent dye (YOYO 1) tagged λ DNAsmono cut mix (N3019L) in a 5 µm i.d. capillary under 1 psi/cm pressure gradient with 50 cm total and 45 cm effective length. Sample injection: 4 s at 12 psi. Eluent was 10 mM Tris-HCl, 1 mM Na2EDTA, pH 8.0. Only a portion of this highly complex elution pattern is shown for clarity. Unpublished data, courtesy of X. Wang and S. Liu, University of Oklahoma, Norman, OK.

Figure 4. Original data is on left, the fourth-power transform is on right. The bottom data in each panel (in black) is for a blank (filtered pH 12 phosphate buffer). The top data (in red) in each panel is for a 14 pM fluorescein dissolved in the same buffer, acquired using a confocal microscope with 488 nm excitation and 525 nm emission equipped with a single-photon counting avalanche photodiode for single-molecule detection. The scaling for the left and right ordinate in each panel is identical, except for direction. The results are acquired in 1 ms bins. Unpublished data, courtesy of S. M. Burrows and D. Pappas, Texas Tech University, Lubbock, TX.

Non-Gaussian Functions. Some mathematical distributions functions, e.g., Gaussian, do not change their profile upon power transformation, as previously noted. Some others, e.g., Lorentzian, change in a very minor fashion. In applications where the ordinate signal as a function of the X-variable is expected to fit a Gaussian or Lorentzian shape, power transformation of noisy raw data may be beneficial. It is not our purpose here to compare the relative merits of different filtering techniques with power transformation to improve the signal-to-noise ratioswe clearly have not done that. But mathematically it is much easier to understand and explain what power transformation does relative to, e.g., the operation of 10146


Figure 5. X-ray photoelectron spectroscopy (XPS data) for a sample containing C, Fe, O, and traces of F. The data shown related to the 1s electron of F and the original signal is very noisy, but the transformed signal readily fits a Lorentzian line shape. Unpublished data, courtesy of C. R. Savage, The University of Texas at Arlington.

the Savitzky-Golay filter,16 and any spreadsheet software is sufficient to perform the power transformation operation. In Figure 5 we show raw X-ray photoelectron spectroscopy (XPS) data where the original signal is very noisy but the transformed signal readily fits the expected Lorentzian line shape. Quantitation. For quantitation, it is essential that the mean baseline is set to zero before the transform. Quantitation based on peak height is obviously straightforwardsthe nth-power transformed peak height is readily reverse-transformed by taking the nth root of the observed peak amplitude. The data for a sixanion chromatogram spanning a large concentration range is shown in Supporting Information Figure S11a. The corresponding peak heights and the amplitudes of the baseline-corrected peaks after raising to powers 2, 4, and 8 for three of the anions are shown in Supporting Information Table S1 along with the data for square, fourth, and eighth root of the transformed peak amplitudes. It will be observed that, except for the lowest concentration where it is difficult to accurately correct for baseline because of noise, there is essentially perfect agreement. For peak area based quantitation, considerations are also relatively simple for a purely Gaussian peak as given in eq 3; the peak area Apeak is given by Apeak ) Ymax(s√π)

(11)

Upon an nth-power transform of eq 3, as in eq 10, but without normalizing the pre-exponential to unity, we have Yn ) Ymaxn exp(-((t - tR)/(s/√n))2)

(12)

This produces a transformed peak area Apeak,t Apeak,t ) Ymaxn(s√π/n) (16) Savitzky, A.; Golay, M. J. E. Anal. Chem. 1964, 36, 1627–1639.

(13)

Figure 6. Peak areas of chloride and formate for the data in Supporting Information Figure S11a. The solid line is the line drawn through the original data. The symbols are the areas calculated after second-, fourth-, and eighth-degree power transforms following baseline correction, processed according to eq 15. Except at very low concentrations where baseline correction is difficult, the processed values provide virtually the same results.

The original peak area Apeak can be readily computed from the observed peak area Apeak,t. Note that dividing eq 11 by eq 13 gives Apeak ) (Apeak,t √n)Ymax1-n

(14)

In practice the reverse transformation will be easier in terms of the transformed peak amplitude Ymaxn: Apeak ) (Apeak,t √n)(Ymaxn)(1-n)/n

A general expression for a skewed or tailing Gaussian peak has not been worked out at this time. However, such a peak can be approximated by the sum of a finite number of Gaussian peaks.17 A reverse transformation to the original peak area should not have insurmountable problems in the numerical domain. Signal to Noise. Power transforms increase contrast and hence increase the signal-to-noise ratio in the transformed data. Consider a baseline with a mean value of b, a standard deviation of σ, and a putative analyte signal that is pσ above this at maximum, such that the mean total signal value is b + pσ. The base case “S/N” is thus p, and the relative standard deviation of the baseline is σ/b. If an nth-power transform is applied, the mean baseline value, as a first approximation (see the discussion in the Supporting Information), will be bn. We will consider the noise term (baseline standard deviation) as an uncertainty. Since raising to a power is equivalent to sequential multiplication, the uncertainty will propagate as in a standard multiplication operation. In a multiplication operation, the relative uncertainty is preserved and in the final product is equal to the square root of the sum of squares of all of the individual relative uncertainty terms. The relative uncertainty of the final product upon raising the baseline to the nth power is therefore (σ/ b)n, and multiplying by the absolute value of the powertransformed baseline, the absolute value of the noise is bn-1σn. Similarly, as a first approximation, neglecting the effect of noise atop the signal peak of total value b + pσ, the value of this upon nth-power transform will be (b + pσ)n. The net signal (amplitude above baseline) will be given by (b + pσ)n - bn. The apparent “S/N” on the nth-power transform, (S/N)n, is then given by S N

( )

[( ∑ ( ) n

) n

k)0

)]

n

∑

n n-k k k 1 b p σ - bn /[σbn-1 √n] ) k √n k)1

()

n 1-k k k-1 b pσ (16) k

(15)

Figure 6 shows the area responses of chloride and formate as originally obtained and after second-, fourth-, and eighth-degree power transform with the data processed according to eq 15. It will be observed that, except at very low concentrations where baseline correction is difficult, the processed values provide exactly the same results. Similar results are observed for nitrate; all these data appear in the Supporting Information as ac102242t_si_ 002.pdf. Note that these peaks are not at all perfectly Gaussian; the asymmetries at 5% height range from 0.64 for formate to 0.72 for chloride to 1.14 for nitrate for the 100 µM sample. The plate counts N also vary directly with n; Supporting Information Figure S11b shows that N/n is essentially invariant. Figure S12 in the Supporting Information represents another chromatogram, that of an essential oil mixture. Consider, for example, the peak at tR 1.789 min (assuming a Gaussian profile); the measured Ymax is 154, the efficiency N is 27 800 plates, and the peak area is 16.5 (V · s). Taking the same peak from the n ) 2 transform, the measured Ymax is 23 700, the efficiency is 57 000 plates, with an area of 1780 (V2 · s). On the basis of eq 13, the area of the transformed peak should be 1800 (V2 · s) and N is should be twice as large, 55 600 plates. These expectations are again very close to what is experimentally observed.

The gain in S/N as a result of the power transform, Gn, is given by Gn )

1 S p N

( )

) n

1 ) √n

∑ (nk )b n

1-k

(pσ)k-1

(17)

k)1

Let us imagine that the mean baseline b is set to zero, and then an offset yσ is added to it to ensure that the numerically most of the baseline data lie above zero. If baseline noise follows a normal distribution, a choice of y ) 2.6 will ensure that 99+% of the baseline data will be above zero. Putting b ) yσ in eq 16 results in Gn )

1 ) √n

∑ (nk )(p/y) n

k-1

(18)

k)1

Numerical values for Gn for n ) 1-10 and p/y ) 0.01-10 are given in Table 1. The lower limit for Gn is n. The case where the baseline is offset by 2.6σ and the signal is only 1σ higher results in a S/N gain of 1.7-22 for n ) 2-10. (17) Stone, M. H. Trans. Am. Math. Soc. 1937, 41, 375–481.


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Table 1. Numerical Values of the Gain in S/N (Gn in Eq 18) as a Function of the Power and the Parameter p/y power

p/y value

n

0.01 0.03 0.1

0.3

0.4

1

3

10

1 2 3 4 5 6 7 8 9 10

1 1.42 1.75 2.03 2.28 2.51 2.73 2.93 3.12 3.31

1 1.63 2.30 3.09 4.04 5.21 6.65 8.43 10.67 13.48

1 1.70 2.52 3.55 4.90 6.66 9.02 12.16 16.38 22.08

1 2.12 4.04 7.50 13.86 25.72 48.00 90.16 170.3 323.5

1 3.54 12.12 42.50 152.5 557.3 2064 7723 2.91 × 104 1.11 × 105

1 8.49 76.79 732.00 7202 7.23 × 104 7.37 × 105 7.58 × 106 7.86 × 107 8.20 × 108

1 1.44 1.78 2.09 2.37 2.64 2.90 3.14 3.39 3.63

1 1.48 1.91 2.32 2.73 3.15 3.59 4.04 4.53 5.04

Increase in S/N with Power Transform Means Improved Contrast, but Does It Improve LOD? The IUPAC concept of detection limit based on replicate measurement of the blank (b) with a standard deviation of σ invokes that at a minimally measured sample value of b + 3σ, the sample is statistically different from the blank with 99% certainty, assuming that the blank measurements are normally distributed. This approach is valid if the major source of uncertainty is in the blank.8 If we are concerned with distinguishing a peak from random baseline noise, this condition applies to a baseline with negligible drift. Often the 99% probability is translated to what we refer to as the S/N ) 3 criterion. Although this criterion is generally accepted in analytical chemistry, it should not be regarded as “generally accepted”; physicists routinely report detection events with less stringent S/N criterion. Most filtering techniques to improve S/N work primarily to reduce σ. The filtering process is irreversible; the original signal cannot be recovered from the filtered data. Power transforms are readily reversible and have a beneficial effect on S/N for a different reason. By definition, the signal is always greater than the blank. In cases of indirect detection that result in negative going analyte peaks, the signal can always be inverted, and an nth-power transform where n > 1 will always increase the signal more than it increases the baseline and the noise. In other words, it will increase the contrast between the signal and the background. However, noise is not only present on the baseline; it is also present atop the signal. In shot noise limited detection situations,18 the absolute (but not the relative) noise may actually increase on the top of the signal. In the Supporting Information we analyze this in some detail: the mean of the power-transformed signal is greater than the mean signal raised to the nth power because of the presence of noise atop the signal. Obviously, the greater the amount of the noise in the measured value, and greater the value of n, the bigger will be the influence of this noise in the power transform. If absolute noise was the same on the baseline and atop a peak, after the power transform, the noise atop the peak will increase relative to the power-transformed baseline noise. Although the S/N for the putative peak measured as the mean of the signal at the peak top ratioed to the baseline noise undoubtedly increases upon power transformation, the probability that spurious noise can be mistaken as a peak also increases. One potential way out of this dilemma is to exploit the difference between the frequency domain of the signal and the noise, which we intend (18) Horowitz, P.; Hill, W. The Art of Electronics, 2nd ed.; Cambridge University Press: New York, 1989; pp 431-432.

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Figure 7. Left panel shows data magnified from ref 18, Figure 2. The right two panels show y2 and y4 transforms.

to explore in the future. Whether the LOD actually improves depends on the original S/N and the degree of power transform applied. Figure 7 shows a situation where a recombinant green fluorescent protein is eluting in a 1.5 µm bore capillary and fluorescence detection is being performed with a confocal laser excitation arrangement; it has been shown that the peak, ca. 50 s in half-width, fits a Gaussian profile well.19 The estimated S/N ratio in the original is ∼4.2 and increases to ∼20 and ∼>200 in the y2 and y4 transforms. The p/y value, as defined in eqs 16 and 18 and referring to Table 1, is estimated to be between 3 and 10, and the observed gains are quite similar to those estimated in Table 1. In this case the shape of the original Gaussian signal remains unaltered even though the relative noise atop the peaks that are shown in a normalized scale does predictably increase. Figure 8 shows the results of power transformation for two different values of n, 2 and 4, for two different pretransform S/N values: 0.5 (Figure 8a) and 2.0 (Figure 8b). In each case, the left panel shows simulated random noise with a Gaussian distribution (mean 0.51, standard deviation 0.29) generated by the NORMINV(RAND (),mean, sd) function in Microsoft Excel.20 The bold (red) trace shows the simulated Gaussian signal that will be added to this random noise. The total (normalized to a maximum of 1) is shown in gray the next panel to the right, along with a 10 point moving average depicted in bold (blue). The two rightmost panels show y2 and y4 signal transforms, with respective moving averages in bold (blue). The case for identical noise but four times greater signal (S/N ) 2) is shown in Figure 8b. Note that, by definition, neither case meets the S/N ) 3 criteria for a positive detection. In keeping with the more detailed discussion in the Supporting Information, for the S/N ) 2 case there would appear to be a clear improvement in S/N when the starting S/N is 2 (Figure 8b), whereas for a starting S/N of 0.5, spurious peaks begin to appear, especially with the fourth-degree transform. Intermediate cases of S/N ) 1 and S/N ) 1.5 are shown in the (19) Wang, X.; Cheng, C.; Wang, S.-L.; Zhao, M. P.; Dasgupta, P. K.; Liu, S. Anal. Chem. 2009, 81, 7428–7435. (20) Kyd, C. An Introduction to Excel’s Normal Distribution Functions. http:// www.exceluser.com/explore/statsnormal.htm (accessed October 20, 2010).

Figure 9. Response to 200 ng (10 µL) of verapamil hydrochloride by an ELSD to three conditions: bottom trace, background contains no analyte; middle trace, background contains 0.5 mg/L analyte; top trace, background contains 2 mg/L analyte.

Supporting Information in Figure S15, parts a and b s it will be readily apparent that this is a gradual process and S/N ) 1 may be a turning point. Power Transform to Increase Linear Dynamic Range. An actual detector calibration plot over 2 orders of magnitude range of analyte concentration C is shown in Figure S16 in the Supporting Information. The original response is perfectly linear (eq 1, x ) 1.00) up to 10 mg/dL, and then the response slowly falls below the original slope, x being 0.98, 0.94, and 0.91 at C ) 20, 50, and 100 mg/dL. This type of response behavior, with the response slope decreasing at higher concentrations, is common for many type of detectors for a variety of reasons. This figure also shows what happens when a modest degree of power transformation (n > 1) is applied. Because upon such transformation the higher values increase to a greater degree than the lower values, the overall response becomes more linear as indicated by most measures, including the r2 value of the best linear fit of the data (although other ways to test linearity21 would have

been much preferred). Whether or not one approves of power transforms, such manipulation is often imbedded in detector firmware because the linearization technique described in eqs 1 and 2 can be effective to increases the LDR. One aspect that is not often realized, however, is that at the low end where the response was originally linear now becomes supralinear (x > 1 in eq 1) as demonstrated in the inset of Supporting Information Figure S16. In discussing Figure 2, we have alluded to how this can result in apparently better figures for chromatographic efficiency or band asymmetry. This low end supralinear response results in paradoxical improvement of both the signal and S/N with the addition of some analyte to the detector background, much like the SFPD. Figure 9 shows a flow injection experiment in which of verapamil hydrochloride (see the Supporting Information for the chemical name), 200 ng in 10 µL, was injected to a 20% MeOH carrier stream flowing at 0.95 mL/min. Downstream of the injector and prior to the detector, an auxiliary 20% MeOH stream flowing at 0.05 mL/min was tee’d in. This stream contained 0, 10, or 40 mg/L verapamil hydrochloride. The results shown clearly indicate an improvement in S/N, and likely LOD, upon analyte addition to the background. Detailed reasons are the same as those for the SFPD, discussed in ref 7 and much other early literature on the SFPD. In summary, power transforms are presently not commonly used as a representational tool for depicting or examining data generated during chromatography or in other experiments. It is interesting to note that, long before microprocessors, Brown22 patented a squaring circuit for hardware implementation, claiming to improve S/N in analog circuits. In some cases a nonunity power dependence of the response on concentration is intrinsic to a detection principle. In others, especially in certain types of universal detectors used in liquid chromatography, power transforms may be imbedded in instrumentation firmware. This paper

(21) Cassidy, R. M.; Chen, L. C. LC · GC Mag. 1992, 10, 692–696.

(22) Brown, M. K. U.S. Patent 4,046,961, September 6, 1977.

Figure 8. (a) Left panel shows simulated random noise (gray) and in the bold (red) trace a Gaussian signal one-half the average noise amplitude. The ordinate is normalized to unity in the other plots. The total is shown in the next panel (gray) along with a 10 point moving average depicted in bold (blue). Similarly the next two panels show y2 and y4 signal transforms, with respective moving averages in bold (blue). Part b is identical to part a, except that the initial signal-tonoise ratio is 2.


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is an effort to delineate both the advantages and limitations of this technique, heretofore rarely discussed in data presentation and processing in the analytical sciences.

SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGMENT This work was supported by National Science Foundation Grant CHE-0821969. We thank anonymous reviewers for constructive and meaningful criticism.

Received for review August 26, 2010. Accepted October 31, 2010.

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AC102242T

Black Box Linearization for Greater Linear Dynamic Range: The Effect

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