Article pubs.acs.org/ac
Rethinking Data Collection and Signal Processing. 1. Real-Time Oversampling Filter for Chemical Measurements Nicholas D. Laude, Christopher W. Atcherley, and Michael L. Heien* Department of Chemistry and Biochemistry, University of Arizona, 1306 East University Boulevard, Tucson, Arizona 85721, United States S Supporting Information *
ABSTRACT: Minimizing noise in chemical measurements is critical to achieve low limits of detection and accurate measurements. We describe a real-time oversampling filter that offers a method to reduce stochastic noise in a time-dependent chemical measurement. The power of this technique is demonstrated in its application to the separation of dopamine and serotonin by micellar electrokinetic chromatography with amperometric detection. Signal-to-noise ratios were increased by almost an order of magnitude, allowing for limits of detection of 100 and 120 amol, respectively. Real-time oversampling filters can be implemented using simple software algorithms and require no change to existing experimental apparatus. The application is not limited to analytical separations, and this technique can be used to improve the signal-to-noise ratio in any experiment where the necessary sampling rate is less than the maximum sampling rate of the analog-to-digital converter. Theory, implementation, and the performance of this filter are described. We propose that this technique should be the default mode of operation for an analog-to-digital converter.
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recorded is much less than that rate at which the analog-todigital converter used is capable of sampling.11−13 One could use the full potential of the analog-to-digital converter by collecting data at the maximum rate possible, averaging a set number of these data points together, and then recording the data at a lower rate set by the experimenter. For practical reasons, scientists sample at a rate necessary to adequately reproduce the information contained in the output of the experiment. For example, chromatographic peaks that have widths of a few seconds can be adequately represented from data acquired using 10 Hz sampling.14 If one were to sample a 30-min chromatographic separation at 1 MHz, the recorded data file would, at a minimum, require 3.6 GB of hard drive space (stored as a binary file, 5−10 times larger for an ASCII text file). Sampling at megahertz frequencies for this amount of time makes little sense from a practical standpoint. Data storage can accommodate files this large, but they are cumbersome to manipulate. However, collecting data at this high rate would allow one to process it using a method known as “oversampling and averaging”.15 This is sometimes referred to as boxcar averaging; however, this should be distinguished from the moving-average filters, which often receive the same name.6,16 Oversampling and averaging has seen some use by electrical engineers to increase the effective number of bits of an analogto-digital converter. It has also been used by some measurement scientists to reduce noise.15−19 Briefly, many data points are collected over a specified time interval and averaged into a single data point, which then represents the measured voltage
xperiments result in a time-varying output composed of a chemical signal with noise superimposed. We extract information from the signal that can be used to quantify a substance or physical property. The portion of the output that does not contain relevant information is noise, regardless of its source. Researchers strive to minimize the noise without distorting the information contained in the signal. Indeed, many methods (e.g., signal averaging, analog and digital filtering, smoothing, convolution) have been developed to minimize noise.1−10 Experiments (e.g., liquid chromatography, capillary electrophoresis, spectroscopy, mass spectrometry, voltammetry) produce chemical information that must be recorded. The physical or chemical phenomenon is converted to a measurable electrical output by transducers (e.g., electrochemical sensor, photomultiplier tube, Faraday cup, photodiode, thermistor). If necessary, the electrical output from the transducer is converted to a voltage prior to quantization, in which a continuous analog voltage output is converted to discrete time and voltage values. In the modern laboratory, digitization occurs with the aid of a personal computer (PC). The PC has revolutionized the way in which data are collected, stored, and analyzed. Commonly, a PC is interfaced with an analog-to-digital converter that takes a time-varying analog voltage output and digitizes it for display and storage. Continuous improvements in the CPU clock speed, bus bit rate, amount of physical memory in the computer, and interface devices open up new possibilities for data processing. These improvements are of great importance, because it allows data processing methods to be applied in real time. A simple method to collect and process time-varying data to minimize noise by taking advantage of high-speed digitizers is presented here. In many experiments, the rate at which data are © 2012 American Chemical Society
Received: July 29, 2012 Accepted: September 3, 2012 Published: September 14, 2012 8422
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over that time interval. This serves to diminish stochastic noise in inverse proportion to the square root of the number of data points averaged.19 In this work, we sample data at a high rate (up to 1 MHz) and average it in real time, recoding the average and discarding the numerous data points used to construct the average. We call this a real-time oversampling filter. This implementation eliminates the need for storage and manipulation of large data files and allows for simple software manipulation of the degree of oversampling. This filter is applied to a capillary electrophoresis experiment and is shown to increase the signal-to-noise ratio for the detected analytes. The filter is broadly applicable and easily implemented with existing hardware and software.
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THEORY Assume an instrument produces a normally distributed output with a mean μ and standard deviation σ. If one were to measure this output once within a specified time interval, the measurement of the mean may be quite poor. One can write the following equation: x = μ ± ts
(1)
where, for a single measurement, x is a measured value, μ is the “mean”, s is the standard deviation of the sample, and the tstatistic indicates the degree of deviation from the mean. The root-mean-square (rms) noise associated with this measurement would be the standard deviation of the sample (s). For more-accurate knowledge of the mean and population standard deviation, more samples should be acquired. One can then write s x̅ = μ ± t (2) n
Figure 1. Representation of the real-time oversampling filter and the reduction of stochastic noise. The measurement of a signal at an arbitrary recording rate selects only one possible value among many in a given sampling period (yellow data points against blue trace). However, using modern electronics, one may sample many of the possible values during the same period (expanded inset of the blue trace) and record the average as a single data point (red data points against blue trace). The oversampled data (recorded in red) contain less of the original noise content and more accurately represent the signal.
where x̅ is the measured mean and n is the number of samples taken. The rms noise associated with the measurement would now be the sample standard error of the mean (s/√n). Thus, random noise in the measurement of the output of this instrument is diminished as more samples are acquired and averaged. Current analog-to-digital converters and personal computers (PCs) are capable of high sample rates and have sufficient processing speed to perform averaging in real time. Indeed, a 16-bit card capable of sampling at 1 MHz can be purchased for ∼$1000. Entry-level analog-to-digital converters can be purchased for less than $200 (14-bit, 10 kHz sampling rate). Here, we apply the sampling concepts above to time-varying data (Figure 1). The blue traces in Figure 1 represent the output of a hypothetical transducer in the time domain; a signal consisting of the baseline interrupted by a Gaussian peak is obscured by the noise that one might encounter. Discrete voltage values have been digitally recorded at regular time intervals (top trace, yellow diamonds). It is clear that these recorded data points may not accurately represent the “true” output. Suppose the exact same output were sampled at a much higher rate. This would result in a representative sample that could be used to construct an accurate population distribution of output voltages over a particular time interval. By temporarily storing large amounts of data collected in computer memory (illustrated by the inset) and then averaging the data using a simple software algorithm, an accurate mean voltage is recorded for each time interval (bottom trace, red triangles). The end result is greater accuracy in recording the true signal while maintaining the same data recording rate.
The real-time oversampling filter is analogous to “signal averaging” or “ensemble averaging”, except that only one experiment needs to be performed to achieve enhanced signalto-noise ratios. For many types of experiments (e.g., analytical separations), ensemble averaging is not practical.1,20 The realtime oversampling filter can be applied anytime the requirements of the experiment do not necessitate the operation of the analog-to-digital converter at its maximum sampling rate. Indeed, we can write three rules for data collection: (1) Data should be filtered in the analog domain to obey the Nyquist criterion, with respect to the data recording rate. (2) An analog-to-digital converter should always operate at the fastest collection rate possible. (3) Further data processing can take place in the digital domain. This includes averaging the data in real time, and recording it at a specified rate. To prevent aliasing, the rate at which data are recorded must adequately represent the frequency content of the output data stream. The Nyquist limit must be obeyed such that the recording rate must be at least two times the highest frequency present in the output. In the real-time oversampling filter presented here, the analog-to-digital converter is operated at the fastest rate possible. Several data points from the analog-todigital converter, n (defined by the ratio of the data collection rate to the recording rate), are averaged in real time and written 8423
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to a data file. This scheme minimizes noise the system and preserves the temporal fidelity of the data without introducing artifacts.
Instruments, Austin, TX). Samples were electrokinetically injected from an Eppendorf tube for 2 s using a 30 kV bias. Capillaries were interfaced with the PDMS device, as previously described, and the outlet of the capillary was sheathed with the separation buffer at a flow rate of 40 μL/min delivered by a PHD 2000 syringe pump (Harvard Apparatus, Holliston, MA).2223 (See Figures S1 and S2 in the Supporting Information for details on microfluidic device fabrication.) The outlet reservoir of the device contained a 2-cm length of 0.1-in.diameter Pt wire (ground electrode for the separation). In addition, a Ag/AgCl wire was anchored in the same reservoir to serve as a reference electrode in a two-electrode detection format. Carbon-fiber microelectrodes were mounted on a model M3301R micromanipulator (WPI International, Sarasota, FL) and connected to the input HI of a Keithley 428 Prog picoammeter (Keithley Instruments, Cleveland, OH). Electrodes were positioned at a depth of 150 μm within the frustum of the capillary for amperometric recordings. The picoammeter gain was set at 109 V/A, the bandwidth filter at 12 Hz, and a voltage bias of 1.150 V was applied. Data were recorded digitally using a NI PCI 6251 and LabVIEW software written in-house (National Instruments, Austin, TX). The detection occurred inside a grounded Faraday cage. Real-Time Averaging and Noise Recordings. The realtime oversampling filter was created using LabVIEW software. Two key parameters define the real-time averaging: sample rate and recording rate. The sample rate is the rate at which the analog-to-digital converter digitizes analog data. The recording rate is the rate at which the software writes data to a file. The ratio of the sample rate to the recording rate is n, the number of data points averaged into one recorded point. The analog output of a transducer or voltage source is sampled at the specified rate, and the sampled voltage values are fed into a one-dimensional (1D) array. The average of n data points contained within the array is written to a file, along with the corresponding time value of the first data point. Timing for data recording is kept using the onboard sample clock of the NI PCI 6251 device (National Instruments, Austin, TX). In this manner, the output of the transducer is averaged with the effect of reducing stochastic noise. A more-thorough description of the software can be found in the Supporting Information. A custom millivolt potentiometer was used to produce a 100.0 mV signal for studies on μV-scale noise. An ∼5 V peak-to-peak white noise signal was generated and applied using LabVIEW software and a NI PCIe 6341 system (National Instruments, Austin, TX). In all instances, noise magnitude was calculated by taking the rms of 5 s of recorded data.
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EXPERIMENTAL SECTION Reagents and Materials. Sulfuric acid (98%), hydrogen peroxide (30%), hydrochloric acid (37%), 2-propanol (OmniSolve grade), sodium bicarbonate, and sodium borate decahydrate (ACS grade) were purchased from EMD Millipore Chemicals (Billerca, MA). Solid NaOH, hydrofluoric acid (49%), and perchloric acid (71%), were purchased from Mallinckrodt Chemical, Inc. (St. Louis, MO). Serotonin hydrochloride (5-HT) was purchased from Alfa Aesar (Ward Hill, MA). Dopamine hydrochloride (DA), sodium dodecyl sulfate (SDS), trimethylchlorosilane (TMCS), and propylene glycol monomethyl ether acetate (PGMEA) were purchased from Sigma Aldrich Corp. (St. Louis, MO). Dow Corning Sylgard 184 silicone elastomer (PDMS) was purchased through Ellsworth Adhesives (Germantown, WI). All water used was purified to a resistivity of 18.2 MΩ cm using Milli-Q Gradient A10 water purification system (EMD Millipore). Electrode Fabrication. Carbon-fiber microelectrodes were constructed using methods similar to those previously reported.21 Individual type T-40 carbon-fibers (Cytec Thornel, Woodland Park, NJ) were manually isolated and aspirated into a 4-in. length, 0.68 mm I.D. standard glass capillary (A-M Systems, Inc., Sequim, WA). Filled capillaries were heated and pulled to a fine seal around the carbon fiber using a Type PE-2 pipet puller (Narishige, Japan). The resulting electrodes were then cut under a microscope to a length of 250 ± 50 μm using a No. 11 surgical steel blade. Electrical contact was made through the open end of the capillary by inserting a 4-in. length of stripped 28 AWG wire wrap wire (Squires Electronics Corp., Cornelius, OR) that had been coated in conductive silver paint (SPI Supplies, West Chester, PA). The wire lead was then sealed in place using Loctite Hysol 0151 epoxy adhesive (Henkel Corp., Scottsdale, AZ) and allowed to cure overnight. Analytical Separations and Amperometric Detection. Micellar electrokinetic chromatography (MEKC) experiments were conducted using a 45.5-cm, 14.9 ± 0.5 μm inner diameter (ID), 150-μm outer diameter (OD) fused-silica capillary (Polymicro Technologies, Phoenix, AZ). The capillary was prepared for electrochemical detection by purging with helium gas at 150 psi. The cladding was removed and the outlet was etched by submerging ∼1 mm of the capillary in HF solution, so that a stream of bubbles was visible. This etching process continued for 12 min, followed by immersion in dilute NaHCO3 solution for 2 min and then a 5-min immersion in water. This procedure resulted in a frustum geometry 90 ± 5 μm in diameter at the outlet and 300 ± 10 μm in depth.11 The capillary was conditioned by flushing with methanol for 3 min, water for 3 min, 0.5 M HCl for 5 min, water for 3 min, 1.0 M NaOH for 30 min, and water for 3 min (all at 40 psi backpressure). Aqueous running buffer (20 mM borate, pH 9.61, 50 mM SDS, 2% v/v n-propanol) was then flushed at 150 psi for 15 min. All solutions used were filtered through a 0.2μm Nylon syringe filter (VWR International, LLC, Sugarland, TX). The running buffer was then electrokinetically injected at 15 kV for 5 min until the current stabilized. A Glassman MJ30P400 30 kV power supply (Glassman High Voltage, Inc., High Bridge, NJ) was used to apply high voltage and controlled remotely using software written in house (LabVIEW, National
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RESULTS AND DISCUSSION Effect of a Real-Time Oversampling Filter on rms Noise. When digitizing an analog signal, it is critical that the recording rate be sufficient to capture the frequency content of the data. For example, a Gaussian peak with a full-width at half its maximum height (W1/2) of 1 s has >95% of its frequency content below 2 Hz. Therefore, to represent the Gaussian peak, we must record data at a minimum of 4 Hz, obeying the Nyquist criterion. To adequately measure a Gaussian peak and minimize measurement errors of peak characteristics, we typically record data at a higher rate such as 10 Hz.14 The measurements presented herein were recorded at 25 Hz to accurately represent the narrow peaks obtained in MEKC. We investigated the effect of our real-time oversampling filter on white noise (Figure 2). A ∼1 V rms white-noise output was 8424
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catecholamines and does not complex with borate. At the pH used for these separations, 5-HT is near its isoelectric point and partitions into the SDS micelles.24−26 Approximately 1 nL of a 0.1 N HClO4 solution containing 10.0 μM dopamine and 10.0 μM 5-HT was electrokinetically injected for 2 s at 30 kV. Average separation efficiency was 160 000 ± 20 000 and 110 000 ± 3000 for dopamine and 5-HT, respectively (n = 6). Representative electropherograms contrasting traditional sampling with real-time oversampling show that the S/N ratio is increased by nearly 1 order of magnitude for both analytes (see Figure 3). Noise profiles for Figure 2. Root-mean-square (rms) noise decreases with increasing sampling number when the real-time oversampling filter is used. This trend is expected to be linear, with respect to n−1/2 (black trace). Millivolt scale noise produced by a white noise generator follows the ideal trend for over 2 orders of magnitude (red diamonds). Random noise arising from digitization of data is also diminished using the realtime oversampling filter (blue triangles). Deviation from the ideal trend occurs as 1/f and noise become large in comparison with stochastic noise. Figure 3. Separation of 10.0 μM dopamine and 10.0 μM serotonin was performed using micellar electrokinetic chromatography (MEKC) with amperometric detection. Dopamine migrates faster and is the first peak observed in both traces. Without the use of the real-time oversampling filter, the rms noise is 6.0 pA and the limit of detection (LOD) for both analytes is ∼1 fmol (red trace). Application of the real-time oversampling filter with n = 40 000 samples reduces the rms noise to 700 fA and the limit of detection to ∼100 amol (blue trace). Noise histograms for each separation are rotated and plotted to the right of each electropherogram. Noise profiles were Gaussian, and the standard deviation decreases as the number of samples averaged increases.
generated (250 kHz bandpass). The output was then recorded at 25 Hz while the collection rate of the analog-to-digital converter was varied from 25 Hz to 1 MHz (Figure 2, red diamonds). This allows for up to n = 40 000; meaning that 40 000 sampled data points are averaged into one recorded data point (collection rate = 1 MHz, recording rate = 25 Hz). As expected from eq 2 (vide supra), the measured random noise is proportional to n−1/2 (R2 = 0.99). The power of the real-time oversampling filter is clear. The measured rms noise decreases from 1.09 V to 7.73 mV. The real-world limitations of analog-to-digital converters introduce noise in low-voltage measurements. Because the analog-to-digital converter employed here has 16-bit resolution (±10 V input range), the theoretical bit-noise limitation is ∼300 μV (20 V/216 bits), and any variation in the voltage signal smaller than this limiting value will have some probability of being rounded up or down to the nearest discrete value allowed. This effect is observed when measuring the output of a precision DC voltage source (set to 100.0 mV DC). If a bitnoise-limited signal is acquired and the real-time oversampling filter is applied, the rms noise is reduced by over an order of magnitude, well under the theoretical bit noise limit (see blue triangles in Figure 2). Deviation from the ideal trend occurs when the stochastic noise has been effectively averaged out of the measurement and other noise sources, notably 1/f noise or any environmental noise, become larger in magnitude. This deviation was not unexpected but, more importantly, it was not observed until the rms noise was ∼10 μV, much less than the resolution of the analog-to-digital converter. This increases the resolution of the analog-to-digital converter effectively making it a 21-bit device.15,17,18 The effect of the real-time oversampling filter is to reduce stochastic noise in an analytical measurement, regardless of the source. Increased S/N Ratio in MEKC with Amperometric Detection. The real-time oversampling filter was applied to the separation of dopamine and serotonin (5-HT) by MEKC. In this separation technique, a borate−SDS buffer is utilized to achieve separation between these two analytes. Dopamine migrates to the detector rapidly as the dopamine−borate complex weakly partitions into the SDS micelles and is poorly retained. Serotonin lacks the vicinal hydroxyl groups of
both datasets with and without real-time oversampling were fit using a Gaussian nonlinear least-squares regression with an R2 value of 0.99 (see the Supporting Information). Real-time oversampling decreases the rms noise of data recorded, thus reducing the width of the Gaussian noise profile (eq 2). The S/ N ratio for both dopamine and 5-HT increases with the square root of n. In the case of dopamine, the S/N ratio increases from 34 ± 1.0 for n = 1 to 300 ± 50 for n = 40 000. Similarly, the S/ N ratio for 5-HT increases from 32 ± 1.0 to 290 ± 50 (Figure 4). S/N ratios were calculated using the peak current amplitudes and the rms noise over a 5-s interval of the baseline prior to each peak. These intervals were chosen based on the average W1/2 value for both dopamine and 5-HT. While the increasing trend for S/N is not linear with n1/2, this is expected in low-frequency amperometric recordings such as these. As the power for random noise is diminished through oversampling and averaging, 1/f noise becomes predominant.27−29 A representative noise power spectrum for an amperometric recording under these experimental conditions is available in the Supporting Information. Without the real-time oversampling filter, the detection limit for dopamine and 5-HT was 870 amol and 1.02 fmol, respectively (3σ). With the real-time oversampling filter and n = 40 000, the limit of detection for dopamine and 5-HT was reduced to 100 and 120 amol, respectively. This was achieved without any change to the bandwidth filter on the picoammeter, and the data underwent no subsequent signal processing in software. The decrease in the noise shows that 8425
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Facility and the support of Paul Lee. The authors thank the University of Arizona for supporting this work.
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Figure 4. Signal-to-noise (S/N) ratios for dopamine and serotonin increase with the square root of the number of samples averaged into one recorded datum. Overall improvement in S/N was slightly less than 1 order of magnitude for both analytes, achieved solely by application of the real-time oversampling filter. Error bars are plotted as standard deviation (n = 3). Only the upper error bar for dopamine and lower error bar for serotonin are plotted, for the sake of clarity.
the real-time oversampling filter can be easily implemented and used to improve the S/N ratio in analytical measurements.
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CONCLUSIONS We have presented the real-time oversampling filter as a new implementation of the technique known as oversampling and averaging. In this work, it has been applied to achieve nearly an order-of-magnitude increase in the signal-to-noise ratio in the separation of dopamine and serotonin by MEKC with amperometric detection. The real-time oversampling filter makes use of the capabilities of modern analog-to-digital converters and a simple software algorithm reduces stochastic noise, regardless of its source. The technique compliments analog filtering and results in manageable datasets that can undergo subsequent signal processing (e.g., smoothing, filtering) in the digital domain. It is not constrained in its application to steady state or highly reproducible rapid measurement, as is ensemble averaging. The use of a real-time oversampling filter is appropriate whenever a measurement does not require that the analog-to-digital converter be operated at its maximum sampling rate. We submit that this powerful yet simple technique should be a part of every chemist’s toolbox.
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ASSOCIATED CONTENT
S Supporting Information *
Additional materials including microfluidic device fabrication procedures, noise spectrum for the amperometric detector, noise histogram analysis, and software description and instruction. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*Fax: 520-621-8407. E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Eric B. Monroe for useful comments. The syringe pump used in the separation experiments was generously loaned by Craig A. Aspinwall. Device fabrication was made possible through the use the CBC Clean Room 8426
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