Analytica Chimica Acta 854 (2015) 178–182
Contents lists available at ScienceDirect
Analytica Chimica Acta journal homepage: www.elsevier.com/locate/aca
The myth of data acquisition rate Attila Felinger a,b,∗ , Anikó Kilár a , Borbála Boros b a b
MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Data acquisition rate has no influence on band broadening and hence resolution. • Software packages often contain undocumented features. • Undisclosed digital filtering is often coupled with data acquisition.
a r t i c l e
i n f o
Article history: Received 6 October 2014 Received in revised form 9 November 2014 Accepted 11 November 2014 Available online 15 November 2014 Keywords: Sampling Data acquisition Trigonometric interpolation Digitalization
a b s t r a c t With the need for high-frequency data acquisition, the influence of the data acquisition rate on the quality of the digitized signal is often discussed and also misinterpreted. In this study we show that undersampling of the signal, i.e. low data acquisition rate will not cause band broadening. Users of modern instrumentation and authors are frequently misled by hidden features of the data handling software they use. Very often users are unaware of the noise filtering algorithms that run parallel with data acquisition and that lack of information misleads them. We also demonstrate that undersampled signals can be restored by a proper trigonometric interpolation. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The quality of digitized signals has been widely studied. Most often the effect of data acquisition rate is studied in terms of minimum data points per peak required for the accurate quantitation. Different studies based on the accuracy of the numerical integration of peak area state that at least 10 data per are required when a Gaussian peak is digitized [1–3]. Some authors claim that
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501500x24582; fax: +36 72 501518. E-mail address: [email protected] (A. Felinger). http://dx.doi.org/10.1016/j.aca.2014.11.014 0003-2670/© 2014 Elsevier B.V. All rights reserved.
6–7 points per peak, or one data per is sufficient for signal analysis [4], which latter density is, of course, in accordance with the Nyquist frequency, but such a low sampling rate is obviously improper when dealing with real signals where baseline noise is omnipresent [5,6]. The influence of data acquisition rate is often revisited as faster detectors become available. The need for high-frequency data acquisition is sometimes illustrated using fast detectors intentionally set to a low-frequency acquisition rate. In those instances the authors of scientific papers [7] or instrumentation brochures [8] claim that peak shapes become distorted and broad when the signal is sampled with a less-than-optimum sampling rate. The aim of this study is to shed light on this misconception that accompanies data acquisition.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
2. Materials and methods 2.1. Sampling theorem When merely the theory of digital signals is considered, the Nyquist or Shannon sampling theorem suggests that at least two samples for the highest frequency present in the continuous signal must be collected [5]: t ≤
1 2fmax
(1)
where t is the sampling time1 and fmax is the highest frequency to be preserved after digitalization. Obviously, in order to accurately determine the location of peak maxima or the peak area, a higher sampling frequency is required than it is established by Eq. (1). When the signal is recorded over a time period T, altogether np = T/t data points are acquired. The frequency resolution of the digitized signal will be f = 1/T. 2.2. Trigonometric interpolation The sampling rate and the total acquisition time of a digitized signal determine both the maximum observable frequency and the resolution of the Fourier spectrum. The maximum frequency fmax is inversely proportional to the sampling time t (see Eq. (1)). Low sampling rate will set a constraint the maximum observable frequency. The undersampled signal can be Fourier transformed, and the length of the Fourier transformed signal is increased by adding zeros after the last point, formally zero-intensity contribution of frequencies higher than fmax is added. When (k − 1)np zeros are added after fmax to extend the np -point Fourier spectrum, the maximum frequency in the spectrum becomes: fmax =
knp k = = kfmax 2T 2t
(2)
Therefore, when the zero-filled Fourier spectrum is inverse transformed, the data density will be k-fold compared to the original sampled signal. Thus, the above procedure is an interpolation of sampled signals. Interpolation by zero filling can be used to improve the resolution of insufficiently sampled signals [9–11]. It must be emphasized, however, that interpolation may help to isolate maxima or minima, but it is not a real resolution enhancement. When resolution is insufficient due to undersampling, although the max by adding imum observable frequency fmax is increased to fmax zeros, but all data within [fmax , . . ., fmax ] are zeros. Therefore, that frequency region contains no additional new information at all.
179
Diode Array detector (Agilent 1100) at 261 nm, and an autosampler (Agilent 1100). The Agilent ChemStation (B.01.03(204) version) software was applied on the HPLC system. The Shimadzu UFLC XR system consisted of a liquid chromatograph (Prominence Liquid Chromatograph LC-20 ADXR, Shimadzu), a DGU-20 A3 micro vacuum degasser, an SIL-20 ACXR auto sampler, an SPD-M20A diode array detector, a CTO-20 AC column oven, and a CBM-20 controller. LabSolutions (Shimadzu) software was used to control the UFLC system and for data processing. The same Ascentis Express C18 column (50 × 2.1 mm, 2.7 m, Supelco, Bellefonte USA) was used with both instruments. For the isocratic separations, the mobile phase was methanol:water = 80:20 (v/v %). Sample concentration was 0.13 (v/v%) of toluene and ethylbenzene in the solvent of the mobile phase. Operating conditions were as follows: flow rate 0.5 mL min−1 , column temperature was 25 ◦ C and injection volume of the standards solvent was 1 L (three replicate injections). The data acquisition frequency of the diode array detector was changed from 0.3 to 80 Hz on the Agilent instrument and the optical (cell) slit widths were 1 and 4 nm. The maximum data acquisition frequency in the Shimadzu instrument was 40 Hz. 3. Results and discussion 3.1. Misleading experimental data In order to reproduce the effect of sampling frequency on peak width, first we carried out the separation of a toluene–ethylbenzene mixture on an Ascentis Express C18 (50 × 2.1 mm, 2.7 m) column using an Agilent 1100 instrument. The mobile phase was MeOH:H2 O = 80:20 (v/v%). The chromatograms recorded at 80 Hz and 1.25 Hz sampling rates are plotted in Fig. 1. When the sampling rate is 80 Hz, the number of theoretical plates is N = 1078 for toluene and it is N = 1344 for ethylbenzene. The resolution factor for this separation is Rs = 1.60. Note that we did not attempt to optimize the separation efficiency and the extra-column band broadening caused by the instrumentation is significant. Albeit this fast separation is a good demonstration
80 Hz 1.25 Hz
3
2.5
2.3. Chemicals and reagents Toluene (99.5%) was purchased from Sigma–Aldrich (St. Louis, MO, USA). Ethylbenzene (>99%) was from Fluka (Buchs, Switzerland). Methanol (Chromasolv for gradient elution) and water (Chromasolv Plus for HPLC) were obtained from Sigma–Aldrich (St. Louis, MO, USA).
mAU
2
1.5
1
2.4. Instrumentation and conditions The analyses were performed on an Agilent 1100/1200 Series HPLC and a Shimadzu Prominence UFLC XR systems. The Agilent HPLC system consisted of a micro vacuum degasser (Agilent 1100, Agilent, Palo Alto, CA, USA), binary pumps (Agilent 1100), thermostated column compartment (Agilent 1200),
0.5
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min)
1
The data acquisition or sampling rate (frequency) is defined as 1/t.
Fig. 1. Chromatograms recorded on the Agilent 1100 instrument at 80 Hz and at 1.25 Hz sampling rate.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
example to illustrate the effect of data acquisition rate on signal quality. When the sampling rate is reduced to 1.25 Hz – and otherwise the experimental conditions are unchanged – it is indeed shocking to see that peaks become broader and resolution drops. At a sampling rate of 1.25 Hz, the number of theoretical plates is N = 205 for toluene and it is N = 308 for ethylbenzene. The resolution factor for this separation is now Rs = 0.75. This experimental result clearly contradicts the sampling theorem. The 4–5-fold drop in separation efficiency cannot be attributed to the difference in sampling frequency, even though this observation is in accordance with the claims discussed above [7,8]. The change of peak width with data acquisition frequency is an artifact and this is discussed below. The nature of the noise in an electronic instrumentation may have various power spectra. Most often a pink or 1/f noise is observed in spectrophotometric detectors [5], although brown noise has also been identified [12]. A visual inspection of the baseline in Fig. 1 shows that the noise level also changes when sampling rate is altered. This should not happen, the frequency of data acquisition should not affect the noise level. The chromatograms plotted in Fig. 1 thus show two important features: (i) there is an apparent increase of band broadening and (ii) decrease of noise level as we lower the sampling frequency. This paradox can be resolved when we look at the two phenomena simultaneously and trace what type of undocumented feature is hidden in the data station software provided by the instrument vendor. We repeated the above separation at nine different sampling rates between 0.3 Hz and 80 Hz. The variance of the baseline noise was calculated for every chromatogram. We observed a monotonous decrease of noise level as the sampling frequency was decreased. This result suggests that when fewer data points are requested by setting a lower sampling frequency, the raw digitized data are filtered and thus baseline noise is smoothed by a digital filter. When a window that contains n points is involved in boxcar averaging or digital filtering with a rectangular window, an n-fold decrease of the original variance is observed due to smoothing: n2 =
2 n
2
-4
10-6
-7
1
10
100
number of points in boxcar averaging Fig. 2. Influence of the number of points of digital filtering on the variance of baseline noise.
Band broadening of undersampled signals is caused by the convolutional nature of digital smoothing. It is entirely unrelated with the sampling rate itself. 3.2. The true effect of data acquisition frequency The same experiments were repeated on a Shimadzu Prominence instrument. That instrument is equipped with a detector with a 40 Hz maximum data acquisition rate. Fig. 3 shows the chromatograms recorded at 40 Hz and 1.5 Hz sampling rate, respectively. We can observe that the serious undersampling – only 7 or 8 points per peak – does not introduce any band broadening or efficiency loss. When the sampling rate is 40 Hz, the efficiency is N = 1837 for toluene and it is N = 2322 for ethylbenzene. The resolution factor for this separation is Rs = 1.71. When the sampling rate is reduced to 1.5 Hz. At a sampling rate of 1.5 Hz, peaks are not at 1000
(3)
n S/N
-5
10
10
40 Hz 1.5 Hz 800
(4)
Thus, when instead of 80 Hz, data are digitized at 1.25 Hz, an 8fold improvement of S N−1 would be observed if the peak heights were unaffected. Unfortunately, when data are digitized at 1.25 Hz, the filtering window is so wide that its large time constant will distort the peak. In Fig. 2, the symbols show the change of baseline noise level with the width of the smoothing window. The solid line indicates the theoretical change of variance calculated with Eq. (3). The excellent agreement between the experimental and calculated variances confirms that when lower sampling rate is set, the data are subjected to digital filtering. The time constant of the digital filter depends on the data acquisition frequency. In some analytical instruments when the data acquisition frequency is set to a value lower than the maximum sampling frequency, digitization still happens at the highest possible rate. Depending on the sampling rate, a digital filtering is applied. This is an undocumented feature of the software, which can deceive the users and they will misinterpret the effect.
600
signal
√
σ /n
10
√ When peak height is unchanged after digital filtering, an n-fold improvement of the signal-to-noise ratio will be observed: S/Nn =
2
σn
variance of baseline noise
180
400
200
0
-200 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min) Fig. 3. Chromatograms recorded on the Shimadzu Prominence instrument at 40 Hz and at 1.5 Hz sampling rate.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
181
the 40-Hz signal is merely 7.7% larger than that of the 1.5-Hz signal.
900 800
3.3. Reconstruction of undersampled signals 700 600
signal
500 400 300 200 100 0 -100 0.3
0.35
0.4
0.45
0.5
0.55
0.6
time (min) Fig. 4. Chromatogram recorded on the Shimadzu Prominence instrument. Original sampling rate was 4 Hz (black symbols) and a ten-fold numerical interpolation to 40 Hz (solid line) was calculated.
Trigonometric interpolation can be used via Fourier transform, to improve the properties of undersampled signals. As it is illustrated in Fig. 4, when the sampling rate is 4 Hz, too few data are collected and the peak apices will be cut off. The peak apices (location and amplitude as well) can, however, be restored by calculating the Fourier transform of the digitized signal, applying the zerofilling method and calculating the inverse Fourier transform. As Fig. 4 demonstrates, not only the slightly irregular peak shape, but also the slowly varying baseline in the 0.3–0.35 min range is reconstructed. With this calculation we restored the signal to data point density of 40 Hz. Of course the reconstructed, interpolated signal does not contain information components beyond 2 Hz (see Eq. 1), nevertheless, the location of the peak apex and the peak amplitude can accurately be determined from the reconstructed signal. When we repeat the same measurement with 40 Hz sampling rate (see the solid line in Fig. 5), the signal recorded at 40 Hz, and the one that was recorded at 4 Hz but was interpolated to 40 Hz (see the red symbols in Fig. 5) cannot be distinguished. This is an additional confirmation that undersampling does not affect the peak shape. 4. Conclusions
all broader; the number of theoretical plates is N = 1931 for toluene and it is N = 2780 for ethylbenzene. The resolution factor for this separation is only slightly affected and definitely did not decrease: Rs = 1.78. Fig. 3 illustrates the true effect of undersampling. When the data acquisition frequency is too low, only a few points per peak are recorded. We will miss the peak apices, numerical integration of the signal will be highly inaccurate, but peaks remain as sharp as they were at the highest sampling rate. The noise level of the signals sampled at 40 Hz or 1.5 Hz are practically the same: the standard deviation of the baseline of 900 4 Hz interpolated to 40 Hz 4 Hz 40 Hz
800
Acknowledgments
700
300
This research was realized in the frames of TÁMOP 4.2.4.A/211-1-2012-0001 “National Excellence Program – Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and co-financed by the European Social Fund. The research infrastructure was supported by the Hungarian Scientific Research Fund (OTKA K 106044).
200
References
600 500 signal
Data acquisition rate has no influence at all on band broadening and hence resolution. What some researchers observe is not the effect of sampling rate but it is the consequence of undocumented software features. Loss of efficiency or resolution may indeed be observed when a modern detector is set to a slow sampling rate. The loss of efficiency, however, occurs when instrument manufacturers think that a good signal-to-noise ratio is more important than faithful representation of the original signal and they sacrifice peak shape by coupling a digital filtering to data acquisition. Another important result is that a modern fast detector cannot always be used to emulate the performance of slow detectors, since when a fast detector is run at slow data acquisition rate, the inner clock still digitalizes the analog signal at a fast rate, but the digital manipulation or signal processing step alters the shape of the signal.
400
100 0 -100 0.3
0.35
0.4
0.45 time (min)
0.5
0.55
0.6
Fig. 5. Chromatogram recorded on the Shimadzu Prominence instrument. Original sampling rate was 4 Hz (black symbols) or 40 Hz (solid line). Ten-fold interpolation of the 4 Hz data to 40 Hz (red symbols) coincides with the signal that was indeed sampled at 40 Hz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
[1] S.N. Chesler, S.P. Cram, Effect of peak sensing and random noise on the precision and accuracy of statistical moment analyses from digital chromatographic data, Anal. Chem. 43 (14) (1971) 1922–1933. [2] M. Goedert, G. Guiochon, Sources of error in chromatographic analysis: effect of sampling parameters on the accuracy of numerical measurements, Chromatographia 6 (2) (1973) 76–83. [3] G. Guiochon, M.J. Sepaniak, Exchange of comments on data acquisition for chromatographic peaks, Anal. Chem. 63 (1) (1991) 73. [4] P.J.H. Scheeren, P. Barna, H.C. Smit, A software package for the evaluation of peak parameters in an analytical signal based on a non-linear regression method, Anal. Chim. Acta 167 (1985) 65–80. [5] A. Felinger, Data Analysis and Signal Processing in Chromatography, Elsevier, Amsterdam, 1998. [6] N. Dyson, Chromatographic Integration Methods, Royal Society of Chemistry, Cambridge, 1990.
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[7] K.J. Fountain, U.D. Neue, E.S. Grumbach, D.M. Diehl, Effects of extra-column band spreading, liquid chromatography system operating pressure, and column temperature on the performance of sub-2-m porous particles, J. Chromatogr. A 1216 (32) (2009) 5979–5988. [8] Publication Number 5989-5033EN, Technical Note, Agilent Technologies, 2006. [9] G. Horlick, W.K. Yuen, Fourier domain interpolation of sampled spectral signals, Anal. Chem. 48 (11) (1976) 1643–1644.
[10] R.J. O’Halloran, D.E. Smith, Fast Fourier transform based interpolation of sampled electrochemical data, Anal. Chem. 50 (9) (1978) 1391–1394. [11] M.B. Comisarow, J.D. Melka, Error estimates for finite zero-filling in Fourier transform spectrometry, Anal. Chem. 51 (13) (1979) 2190–2203. [12] A. Felinger, M. Káré, Wavelet analysis of the baseline noise in HPLC, Chemometr. Intell. Lab. Syst. 72 (2) (2004) 225–232.
Contents lists available at ScienceDirect
Analytica Chimica Acta journal homepage: www.elsevier.com/locate/aca
The myth of data acquisition rate Attila Felinger a,b,∗ , Anikó Kilár a , Borbála Boros b a b
MTA–PTE Molecular Interactions in Separation Science Research Group, Ifjúság útja 6, H-7624 Pécs, Hungary Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary
h i g h l i g h t s
g r a p h i c a l
a b s t r a c t
• Data acquisition rate has no influence on band broadening and hence resolution. • Software packages often contain undocumented features. • Undisclosed digital filtering is often coupled with data acquisition.
a r t i c l e
i n f o
Article history: Received 6 October 2014 Received in revised form 9 November 2014 Accepted 11 November 2014 Available online 15 November 2014 Keywords: Sampling Data acquisition Trigonometric interpolation Digitalization
a b s t r a c t With the need for high-frequency data acquisition, the influence of the data acquisition rate on the quality of the digitized signal is often discussed and also misinterpreted. In this study we show that undersampling of the signal, i.e. low data acquisition rate will not cause band broadening. Users of modern instrumentation and authors are frequently misled by hidden features of the data handling software they use. Very often users are unaware of the noise filtering algorithms that run parallel with data acquisition and that lack of information misleads them. We also demonstrate that undersampled signals can be restored by a proper trigonometric interpolation. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The quality of digitized signals has been widely studied. Most often the effect of data acquisition rate is studied in terms of minimum data points per peak required for the accurate quantitation. Different studies based on the accuracy of the numerical integration of peak area state that at least 10 data per are required when a Gaussian peak is digitized [1–3]. Some authors claim that
∗ Corresponding author at: Department of Analytical and Environmental Chemistry and Szentágothai Research Center, University of Pécs, Ifjúság útja 6, H-7624 Pécs, Hungary. Tel.: +36 72 501500x24582; fax: +36 72 501518. E-mail address: [email protected] (A. Felinger). http://dx.doi.org/10.1016/j.aca.2014.11.014 0003-2670/© 2014 Elsevier B.V. All rights reserved.
6–7 points per peak, or one data per is sufficient for signal analysis [4], which latter density is, of course, in accordance with the Nyquist frequency, but such a low sampling rate is obviously improper when dealing with real signals where baseline noise is omnipresent [5,6]. The influence of data acquisition rate is often revisited as faster detectors become available. The need for high-frequency data acquisition is sometimes illustrated using fast detectors intentionally set to a low-frequency acquisition rate. In those instances the authors of scientific papers [7] or instrumentation brochures [8] claim that peak shapes become distorted and broad when the signal is sampled with a less-than-optimum sampling rate. The aim of this study is to shed light on this misconception that accompanies data acquisition.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
2. Materials and methods 2.1. Sampling theorem When merely the theory of digital signals is considered, the Nyquist or Shannon sampling theorem suggests that at least two samples for the highest frequency present in the continuous signal must be collected [5]: t ≤
1 2fmax
(1)
where t is the sampling time1 and fmax is the highest frequency to be preserved after digitalization. Obviously, in order to accurately determine the location of peak maxima or the peak area, a higher sampling frequency is required than it is established by Eq. (1). When the signal is recorded over a time period T, altogether np = T/t data points are acquired. The frequency resolution of the digitized signal will be f = 1/T. 2.2. Trigonometric interpolation The sampling rate and the total acquisition time of a digitized signal determine both the maximum observable frequency and the resolution of the Fourier spectrum. The maximum frequency fmax is inversely proportional to the sampling time t (see Eq. (1)). Low sampling rate will set a constraint the maximum observable frequency. The undersampled signal can be Fourier transformed, and the length of the Fourier transformed signal is increased by adding zeros after the last point, formally zero-intensity contribution of frequencies higher than fmax is added. When (k − 1)np zeros are added after fmax to extend the np -point Fourier spectrum, the maximum frequency in the spectrum becomes: fmax =
knp k = = kfmax 2T 2t
(2)
Therefore, when the zero-filled Fourier spectrum is inverse transformed, the data density will be k-fold compared to the original sampled signal. Thus, the above procedure is an interpolation of sampled signals. Interpolation by zero filling can be used to improve the resolution of insufficiently sampled signals [9–11]. It must be emphasized, however, that interpolation may help to isolate maxima or minima, but it is not a real resolution enhancement. When resolution is insufficient due to undersampling, although the max by adding imum observable frequency fmax is increased to fmax zeros, but all data within [fmax , . . ., fmax ] are zeros. Therefore, that frequency region contains no additional new information at all.
179
Diode Array detector (Agilent 1100) at 261 nm, and an autosampler (Agilent 1100). The Agilent ChemStation (B.01.03(204) version) software was applied on the HPLC system. The Shimadzu UFLC XR system consisted of a liquid chromatograph (Prominence Liquid Chromatograph LC-20 ADXR, Shimadzu), a DGU-20 A3 micro vacuum degasser, an SIL-20 ACXR auto sampler, an SPD-M20A diode array detector, a CTO-20 AC column oven, and a CBM-20 controller. LabSolutions (Shimadzu) software was used to control the UFLC system and for data processing. The same Ascentis Express C18 column (50 × 2.1 mm, 2.7 m, Supelco, Bellefonte USA) was used with both instruments. For the isocratic separations, the mobile phase was methanol:water = 80:20 (v/v %). Sample concentration was 0.13 (v/v%) of toluene and ethylbenzene in the solvent of the mobile phase. Operating conditions were as follows: flow rate 0.5 mL min−1 , column temperature was 25 ◦ C and injection volume of the standards solvent was 1 L (three replicate injections). The data acquisition frequency of the diode array detector was changed from 0.3 to 80 Hz on the Agilent instrument and the optical (cell) slit widths were 1 and 4 nm. The maximum data acquisition frequency in the Shimadzu instrument was 40 Hz. 3. Results and discussion 3.1. Misleading experimental data In order to reproduce the effect of sampling frequency on peak width, first we carried out the separation of a toluene–ethylbenzene mixture on an Ascentis Express C18 (50 × 2.1 mm, 2.7 m) column using an Agilent 1100 instrument. The mobile phase was MeOH:H2 O = 80:20 (v/v%). The chromatograms recorded at 80 Hz and 1.25 Hz sampling rates are plotted in Fig. 1. When the sampling rate is 80 Hz, the number of theoretical plates is N = 1078 for toluene and it is N = 1344 for ethylbenzene. The resolution factor for this separation is Rs = 1.60. Note that we did not attempt to optimize the separation efficiency and the extra-column band broadening caused by the instrumentation is significant. Albeit this fast separation is a good demonstration
80 Hz 1.25 Hz
3
2.5
2.3. Chemicals and reagents Toluene (99.5%) was purchased from Sigma–Aldrich (St. Louis, MO, USA). Ethylbenzene (>99%) was from Fluka (Buchs, Switzerland). Methanol (Chromasolv for gradient elution) and water (Chromasolv Plus for HPLC) were obtained from Sigma–Aldrich (St. Louis, MO, USA).
mAU
2
1.5
1
2.4. Instrumentation and conditions The analyses were performed on an Agilent 1100/1200 Series HPLC and a Shimadzu Prominence UFLC XR systems. The Agilent HPLC system consisted of a micro vacuum degasser (Agilent 1100, Agilent, Palo Alto, CA, USA), binary pumps (Agilent 1100), thermostated column compartment (Agilent 1200),
0.5
0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min)
1
The data acquisition or sampling rate (frequency) is defined as 1/t.
Fig. 1. Chromatograms recorded on the Agilent 1100 instrument at 80 Hz and at 1.25 Hz sampling rate.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
example to illustrate the effect of data acquisition rate on signal quality. When the sampling rate is reduced to 1.25 Hz – and otherwise the experimental conditions are unchanged – it is indeed shocking to see that peaks become broader and resolution drops. At a sampling rate of 1.25 Hz, the number of theoretical plates is N = 205 for toluene and it is N = 308 for ethylbenzene. The resolution factor for this separation is now Rs = 0.75. This experimental result clearly contradicts the sampling theorem. The 4–5-fold drop in separation efficiency cannot be attributed to the difference in sampling frequency, even though this observation is in accordance with the claims discussed above [7,8]. The change of peak width with data acquisition frequency is an artifact and this is discussed below. The nature of the noise in an electronic instrumentation may have various power spectra. Most often a pink or 1/f noise is observed in spectrophotometric detectors [5], although brown noise has also been identified [12]. A visual inspection of the baseline in Fig. 1 shows that the noise level also changes when sampling rate is altered. This should not happen, the frequency of data acquisition should not affect the noise level. The chromatograms plotted in Fig. 1 thus show two important features: (i) there is an apparent increase of band broadening and (ii) decrease of noise level as we lower the sampling frequency. This paradox can be resolved when we look at the two phenomena simultaneously and trace what type of undocumented feature is hidden in the data station software provided by the instrument vendor. We repeated the above separation at nine different sampling rates between 0.3 Hz and 80 Hz. The variance of the baseline noise was calculated for every chromatogram. We observed a monotonous decrease of noise level as the sampling frequency was decreased. This result suggests that when fewer data points are requested by setting a lower sampling frequency, the raw digitized data are filtered and thus baseline noise is smoothed by a digital filter. When a window that contains n points is involved in boxcar averaging or digital filtering with a rectangular window, an n-fold decrease of the original variance is observed due to smoothing: n2 =
2 n
2
-4
10-6
-7
1
10
100
number of points in boxcar averaging Fig. 2. Influence of the number of points of digital filtering on the variance of baseline noise.
Band broadening of undersampled signals is caused by the convolutional nature of digital smoothing. It is entirely unrelated with the sampling rate itself. 3.2. The true effect of data acquisition frequency The same experiments were repeated on a Shimadzu Prominence instrument. That instrument is equipped with a detector with a 40 Hz maximum data acquisition rate. Fig. 3 shows the chromatograms recorded at 40 Hz and 1.5 Hz sampling rate, respectively. We can observe that the serious undersampling – only 7 or 8 points per peak – does not introduce any band broadening or efficiency loss. When the sampling rate is 40 Hz, the efficiency is N = 1837 for toluene and it is N = 2322 for ethylbenzene. The resolution factor for this separation is Rs = 1.71. When the sampling rate is reduced to 1.5 Hz. At a sampling rate of 1.5 Hz, peaks are not at 1000
(3)
n S/N
-5
10
10
40 Hz 1.5 Hz 800
(4)
Thus, when instead of 80 Hz, data are digitized at 1.25 Hz, an 8fold improvement of S N−1 would be observed if the peak heights were unaffected. Unfortunately, when data are digitized at 1.25 Hz, the filtering window is so wide that its large time constant will distort the peak. In Fig. 2, the symbols show the change of baseline noise level with the width of the smoothing window. The solid line indicates the theoretical change of variance calculated with Eq. (3). The excellent agreement between the experimental and calculated variances confirms that when lower sampling rate is set, the data are subjected to digital filtering. The time constant of the digital filter depends on the data acquisition frequency. In some analytical instruments when the data acquisition frequency is set to a value lower than the maximum sampling frequency, digitization still happens at the highest possible rate. Depending on the sampling rate, a digital filtering is applied. This is an undocumented feature of the software, which can deceive the users and they will misinterpret the effect.
600
signal
√
σ /n
10
√ When peak height is unchanged after digital filtering, an n-fold improvement of the signal-to-noise ratio will be observed: S/Nn =
2
σn
variance of baseline noise
180
400
200
0
-200 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time (min) Fig. 3. Chromatograms recorded on the Shimadzu Prominence instrument at 40 Hz and at 1.5 Hz sampling rate.
A. Felinger et al. / Analytica Chimica Acta 854 (2015) 178–182
181
the 40-Hz signal is merely 7.7% larger than that of the 1.5-Hz signal.
900 800
3.3. Reconstruction of undersampled signals 700 600
signal
500 400 300 200 100 0 -100 0.3
0.35
0.4
0.45
0.5
0.55
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time (min) Fig. 4. Chromatogram recorded on the Shimadzu Prominence instrument. Original sampling rate was 4 Hz (black symbols) and a ten-fold numerical interpolation to 40 Hz (solid line) was calculated.
Trigonometric interpolation can be used via Fourier transform, to improve the properties of undersampled signals. As it is illustrated in Fig. 4, when the sampling rate is 4 Hz, too few data are collected and the peak apices will be cut off. The peak apices (location and amplitude as well) can, however, be restored by calculating the Fourier transform of the digitized signal, applying the zerofilling method and calculating the inverse Fourier transform. As Fig. 4 demonstrates, not only the slightly irregular peak shape, but also the slowly varying baseline in the 0.3–0.35 min range is reconstructed. With this calculation we restored the signal to data point density of 40 Hz. Of course the reconstructed, interpolated signal does not contain information components beyond 2 Hz (see Eq. 1), nevertheless, the location of the peak apex and the peak amplitude can accurately be determined from the reconstructed signal. When we repeat the same measurement with 40 Hz sampling rate (see the solid line in Fig. 5), the signal recorded at 40 Hz, and the one that was recorded at 4 Hz but was interpolated to 40 Hz (see the red symbols in Fig. 5) cannot be distinguished. This is an additional confirmation that undersampling does not affect the peak shape. 4. Conclusions
all broader; the number of theoretical plates is N = 1931 for toluene and it is N = 2780 for ethylbenzene. The resolution factor for this separation is only slightly affected and definitely did not decrease: Rs = 1.78. Fig. 3 illustrates the true effect of undersampling. When the data acquisition frequency is too low, only a few points per peak are recorded. We will miss the peak apices, numerical integration of the signal will be highly inaccurate, but peaks remain as sharp as they were at the highest sampling rate. The noise level of the signals sampled at 40 Hz or 1.5 Hz are practically the same: the standard deviation of the baseline of 900 4 Hz interpolated to 40 Hz 4 Hz 40 Hz
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This research was realized in the frames of TÁMOP 4.2.4.A/211-1-2012-0001 “National Excellence Program – Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and co-financed by the European Social Fund. The research infrastructure was supported by the Hungarian Scientific Research Fund (OTKA K 106044).
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Data acquisition rate has no influence at all on band broadening and hence resolution. What some researchers observe is not the effect of sampling rate but it is the consequence of undocumented software features. Loss of efficiency or resolution may indeed be observed when a modern detector is set to a slow sampling rate. The loss of efficiency, however, occurs when instrument manufacturers think that a good signal-to-noise ratio is more important than faithful representation of the original signal and they sacrifice peak shape by coupling a digital filtering to data acquisition. Another important result is that a modern fast detector cannot always be used to emulate the performance of slow detectors, since when a fast detector is run at slow data acquisition rate, the inner clock still digitalizes the analog signal at a fast rate, but the digital manipulation or signal processing step alters the shape of the signal.
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Fig. 5. Chromatogram recorded on the Shimadzu Prominence instrument. Original sampling rate was 4 Hz (black symbols) or 40 Hz (solid line). Ten-fold interpolation of the 4 Hz data to 40 Hz (red symbols) coincides with the signal that was indeed sampled at 40 Hz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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