design considerations for near-infrared filter photometry: effects of noise sources and selectivity

Design Considerations for Near-Infrared Filter Photometry:Effects of Noise Sources and Selectivity

TOSHIYASU TARUMI,* AIRAT K. AMEROV,� MARK A. ARNOLD, and GARY W. SMALL�Optical Science and Technology Center and Department of Chemistry, University of Iowa, Iowa City, Iowa 52242

Optimal filter design of two-channel near-infrared filter photometers is

investigated for simulated two-component systems consisting of an analyte

and a spectrally overlapping interferent. The degree of overlap between

the analyte and interferent bands is varied over three levels. The optimal

design is obtained for three cases: a source or background flicker noise

limited case, a shot noise limited case, and a detector noise limited case.

Conventional photometers consist of narrow-band optical filters with their

bands located at discrete wavelengths. However, the use of broadband

optical filters with overlapping responses has been proposed to obtain as

much signal as possible from a weak and broad analyte band typical of

near-infrared absorptions. One question regarding the use of broadband

optical filters with overlapping responses is the selectivity achieved by

such filters. The selectivity of two-channel photometers is evaluated on the

basis of the angle between the analyte and interferent vectors in the space

spanned by the relative change recorded for each of the two detector

channels. This study shows that for the shot noise limited or detector noise

limited cases, the slight decrease in selectivity with the use of broadband

optical filters can be compensated by the higher signal-to-noise ratio

afforded by the use of such filters. For the source noise limited case, the

best quantitative results are obtained with the use of narrow-band non-

overlapping optical filters.

Index Headings: Filter photometry; Near-infrared spectroscopy; NIR

spectroscopy; Noise; Selectivity.

INTRODUCTION

Absorption bands in the near-infrared (NIR) region arecharacterized by broad, weak, and highly overlapping featuresarising from overtones and combinations of the fundamentalvibrations associated with C–H, O–H, and N–H groups. Thepresence of broad bands in this region has made possible thedevelopment of successful quantitative analyses based onspectra with very low resolution. For example, Brown hasreported that the British thermal unit (BTU) content of naturalgas can be measured with spectra collected with a slit width ofabout 600 cm�1 at 1225 nm.1 Bonanno and Griffiths haveshown that discrimination of several types of alcohols can beperformed with spectra with a resolution of 100 cm�1 fullwidth at half-maximum (FWHM).2 Greensill and Walsh havereported that the concentration of sucrose in a water–cellulosematrix can be measured with spectra with a resolution of 190cm�1 FWHM at 912 nm.3

Filter photometers provide a simple method for performinglow-resolution NIR measurements.4–8 For measurements with asingle detector, optical filters can be mounted on a rotatingwheel to obtain the responses from the filters sequentially.7,8

Multichannel detector measurements are also possible byplacing optical filters in front of each detector element, thereby

eliminating moving parts from the instrument. Similarly, solid-state tunable filters provide a flexible implementation of a filter-based spectrometer with no moving parts.9–11 The simpleinstrumentation of the filter photometer, which is free of themechanical complexity associated with continuous scanningspectrometers, is highly desirable for producing a portable andlow-cost device for dedicated sensing applications.

Besides their simple instrumentation, filter photometers areoften superior to conventional high-resolution spectrometers interms of optical throughput. Enhancement of the signal-to-noise (S/N) ratio by increasing optical throughput has beenreported for atomic absorption spectrometry.12,13 Hirschfeldhas proposed a S/N enhancement method based on theintegration of an analyte band over the region where spectralfeatures of interferent species are absent.14 Filter photometersinherently implement this S/N enhancement by opticallyintegrating spectral bands.

In spite of the high S/N ratios achieved by filter photometers,their use has typically been limited to applications that do notdemand high selectivity.4,7,8 Block and co-workers havechallenged this conventional wisdom by suggesting that a simplefilter-based optical system can achieve sufficient selectivity to beused in complex samples in which the analyte signals are highlyoverlapped with those of other matrix constituents.15–17 The keyto their approach is the use of a set of broadband spectrallyoverlapping filters. The filters can be considered to form a basisset similar to that obtained when B-splines are applied to reducethe dimensionality of high-resolution spectra.18,19

In two separate studies, Arnold and co-workers havedemonstrated that Block’s optical system can provide sufficientselectivity to determine glucose in simulated biologicalmatrices.20,21 However, the question still remains whether therealized selectivity can be attributed to spectrally overlappingfilters. As a first step to address this issue, this studyinvestigates the optimal filter design for a two-channelphotometer for the analysis of simulated two-componentsystems consisting of an analyte and a spectrally overlappinginterferent. The selectivity of these simulated measurements isevaluated on the basis of the angle between the analyte andinterferent vectors in the space spanned by the relative changesrecorded for each of the two spectral channels. The optimalfilter design is investigated under three cases: a source orbackground flicker noise limited case, a shot noise limited case,and a detector noise limited case. Experimental data collectedwith a filter-based instrument are used to help place the resultsin context, and the results obtained here for the two-filter caseare used to gain insight into the general case of a filterphotometer with an arbitrary number of measurement channels.

EXPERIMENTAL

Instrumentation. Experimental data were acquired with afour-channel filter-based instrument that has been described

Received 3 October 2008; accepted 30 March 2009.* Present address: Nihon Buchi KK, IMON Bld. 3F, 2-7-17 Ikenohata,

Taito-Ku, Tokyo, 110-0008, Japan.� Present address: AMETEK, 150 Freeport Rd., Pittsburgh, PA 15238.� Author to whom correspondence should be sent. E-mail: [email protected].

700 Volume 63, Number 6, 2009 APPLIED SPECTROSCOPY0003-7028/09/6306-0700$2.00/0

� 2009 Society for Applied Spectroscopy

previously.20,21 The source was a 20 W tungsten-halogen lamp(Gilway Technical Lamp, Woburn, MA) powered by aradiometric power supply (Model 68831, Thermo Oriel,Stratford, CT) with feedback controller (Model 68850, ThermoOriel). Light collected from the source was collimated and afraction of the collimated beam (approximately 10%) wasdirected by a beamsplitter to the feedback controller.

The remaining light was focused into an optical fiber anddirected to a 10 mm path length quartz flow cell (Starna Cells,Atascadero, CA). A second fiber collected the transmitted lightand directed it to the detection module. The optical fiber was aTECS hard-clad multimode fiber with a numerical aperture of0.39 (Type FT-1.0-EMT, 3M Optical Components, Austin,TX).

Light exiting the collection fiber was collimated andintroduced into an optical assembly designed by Block andco-workers15 that consisted of three 50% beamsplitters, fourbandpass optical filters (see below), and four single-elementInGaAs photodiodes (Model G5851-01, Hamamatsu Corp.,Bridgewater, NJ). One filter was placed in front of eachdetector to achieve the desired four-channel spectral responseof the instrument. Signals from each detector were acquiredseparately with four digital multimeters (Model 3458A,Hewlett-Packard, Palo Alto, CA). Data acquisition wasperformed at 50 Hz, initiated by a signal from a functiongenerator (BK Precision Model 4011, Maxtec InternationalCorp., Chicago, IL), and controlled with custom softwarewritten in the LabView development environment (NationalInstruments, Austin, TX).

For this study, optical filters were used with transmittance inthe first overtone region of the NIR spectrum. To allowcharacterization of the bandpass response of each filter, spectrawere acquired with a Nicolet Nexus 670 Fourier transformspectrometer (Thermo Nicolet, Madison, WI) equipped with atungsten-halogen source, CaF2 beamsplitter, and cryogenicallycooled InSb detector. The filters associated with detectorchannel numbers 1, 2, 3, and 4 had bandpass centers of mass at6258, 6093, 5884, and 5714 cm�1, respectively. Thecorresponding bandpass FWHM values were 325.8, 187.1,184.7, and 155.0 cm�1. The peak transmittance (T) values were64%, 69%, 60%, and 66%, respectively.

Procedures. The instrumental setup described above wasused to study water at 25 8C. Reagent-grade water obtainedfrom a Milli-Q water purification system (Millipore Corp.,Bedford, MA) was used. To simulate various signal levels thatmight be encountered in analytical applications, the sourcebeam incident on the sample cell was attenuated at six levelsthrough the use of 1 to 2 sapphire windows (Meller Optics) or 1to 2 neutral density filters (ND03, ND05, ND20, Thorlabs,Newton, NJ, with optical densities of 0.3, 0.5, and 2.0,respectively). Estimated transmittance values of 86.5%, 75%,50%, 30%, 15%, and 1% were achieved. The light source wasturned off to produce a seventh attenuation setting correspond-ing to the dark current signals of the detectors (i.e., 0% T). Toassess instrumental drift, before each configuration was run, ameasurement of the water sample was performed without anyattenuation of the source beam (i.e., 100% T). Thus, a total ofeight intensity levels were evaluated. For each measurement, atotal of 6000 readings were recorded for each channel. Eachattenuation setting was then replicated three times. Theseexperiments produced a total of 39 sets of data ((6 attenuation

levels 3 3 replicates) þ (6 measurements at 100% T 3 3replicates) þ 3 replicates at 0% T).

The collected data were loaded into the Matlab developmentenvironment (Version 6.5, The MathWorks, Natick, MA)where all calculations reported here were performed. A DellDimension 8200 computer (Dell Computer, Austin, TX)operating under Windows XP (Microsoft, Redmond, WA)was used.

Simulated Data. Generation of simulated responses fromtwo-component systems and the subsequent analysis of thesedata were performed with Matlab (Version 6.1) on a DellDimension 4400 (Dell Computer) computer operating underWindows XP (Microsoft). The three data sets used in this studyare summarized in Table I.

Simulated analyte and interferent bands were generated overthe NIR first overtone region (5000–7000 cm�1) with anominal point spacing of 2 cm�1. For each data set, the position(5800 cm�1) and width (200 cm�1 FWHM) of the analyte bandwere fixed, whereas the corresponding parameters of theinterferent were varied. A Gaussian function was used togenerate the analyte and interferent bands. Both the analyte andinterferent absorbance peak values were varied over the range0.65–14.85 3 10�3 absorbance units (AU). This absorbancerange is in the same order expected for 1–30 mM glucosesolutions when measurements are made using the first overtoneregion with an optical path length of 1 cm.22 The spectralselectivity listed in the table was calculated as the sine of theangle between the two spectral vectors.23 This calculationproduces values from 0 to 1, with a selectivity of 1 indicatingno spectral overlap. By this criterion, data sets 1, 2, and 3correspond to cases of low, medium, and high selectivity,respectively. While this definition of selectivity is commonlyused, it is mathematical in nature and does not consider theimpact of the selectivity on the actual analytical measurement(e.g., selectivity in terms of its impact on the limit of detection).

Each data set consisted of 55 samples. The concentrations ofthe analyte and interferent within each data set were variedover 1.3–29.7 mM. Specific concentrations of each analyte andinterferent pair were assigned by a uniform design.24 Figure 1illustrates the computed design. The correlation coefficientbetween the analyte and interferent concentrations was 0.0394.For both the analyte and interferent, the mean and standarddeviation of the concentrations were 15.5 and 8.4 mM,respectively.

Responses from two detector channels were simulated asfollows. In the first step, the spectral profile of a model light

TABLE I. Simulated two-component systems.a

Data setInterferent

position (cm�1)Interferent

widthb (cm�1) Selectivityc

1 5810 210 0.0882 5850 250 0.3843 6000 500 0.824

a An absorptivity of 5 3 10�5 L�mm�1�mmol�1 and path length of 1 cm wereassumed for both the analyte and interferent. The position and width (FWHM)of the analyte band were fixed at 5800 cm�1 and 200 cm�1, respectively, forall three data sets.

b FWHM of the interferent band.c Computed as the sine of the angle between analyte and interferent spectra

when treated as vectors. Selectivity values of 0 and 1 correspond,respectively, to complete overlap and complete separation of the analyteand interferent spectra.

APPLIED SPECTROSCOPY 701

source was generated by using Planck’s function to computethe radiance from a blackbody at 2200 K. This simulated theoutput from a common NIR source such as a tungsten-halogenlamp. In the second step, this source profile was multiplied bytransmission spectra representing each of the 55 samples in thedata set to generate single-beam spectra. This calculationassumed a linearly additive mixture model (i.e., that theabsorbances of the analyte and interference add withoutinteraction).

Next, the resulting single-beam spectra were multipliedseparately by each of two Gaussian-shaped filter responses,followed by integration of the resulting products. Thiscalculation assumed a flat responsivity profile for the opticaldetector over the spectral region of the filters. A typical NIRdetector such as the Hamamatsu G5851 series used in theinstrumental setup described previously varies in responsivityby approximately 5% over the 6500 to 5500 cm�1 range.Numerical integration of the signal passing through the filterwas performed by use of the trapezoidal rule.25 This yielded anapproximately noise-free response from each detector channel.The final step involved the addition of noise to each response.Noise values were taken as random normal deviates with amean of 0.0. The standard deviation of the noise distributionwas adjusted to achieve desired levels of the S/N ratio and tosimulate different noise models.

RESULTS AND DISCUSSION

Noise Models. The dominant source of instrumentalvariance associated with filter-based optical measurementscan be considered in terms of three different noise models: (1)source or background flicker noise, (2) source and/or detectorshot noise, and (3) detector or blank noise.26 This simplifiedtreatment neglects the effects of digitization noise and does notexplicitly treat the measurement variance associated with thechemical sample itself (e.g., temperature or scattering effects).

The controlling noise model can be elucidated by plottingthe logarithm of the S/N ratio with respect to the logarithm ofthe signal. When measurement noise is limited by thefluctuation of source or background signals, the absolute noiselevel is proportional to the signal, and thus the S/N ratio doesnot depend on the magnitude of the integrated filter responses.This produces a log-log slope of zero in the plot describedabove. In this case, a broad filter bandpass does not improve

the S/N ratio. When shot noise dominates, the S/N ratio isproportional to the square root of the magnitude of theintegrated filter responses. In a log-log plot of the S/N ratioversus the signal, this case produces a slope of 1/2. Finally,when detector or blank noise is the limiting measurementnoise, the noise level is independent of the signal and the S/Nratio is proportional to the magnitude of the integrated filterresponse. This corresponds to a log-log slope of unity in theplot described above. O’Haver has used these three noisemodels to investigate the relationship between optical through-put and S/N ratio for dispersive absorption spectrometry.12

Noise Characteristics of Filter-Based Measurements. Toinvestigate the signal and noise characteristics of typical NIRmeasurements performed with a filter-based system, the 39 setsof water measurements were analyzed. Signal values in each ofthe four channels were taken as the mean of the 6000 detectorsamples in each data set. The corresponding noise estimateswere computed as the root mean square (rms) deviation about atwo-parameter linear least-squares fit to the 6000-point timetrace. This corrected the noise estimate for the slight amount ofdrift across the time trace. Values of the S/N ratio werecomputed as the ratio of the mean signal to the estimated rmsnoise.

Calculations were performed to check the integrity of thedata as a whole. First, the linearity of the response in eachchannel with respect to signal level was evaluated byperforming a two-parameter linear least-squares fit of the meanresponse for the 39 data sets to the estimated transmittancevalues (100%, 86.5%, 75%, 50%, 30%, 15%, 1%, and 0%).Values of the coefficient of determination (r2) associated withthese regression models for channels 1, 2, 3, and 4 were0.9994, 0.9993, 0.9995, and 0.9993, respectively. Thisconfirmed that all measurements were taken within the linearresponse range of the detector and associated electronics.

Second, the overall drift across the data sets was estimatedby taking the mean response for each of the 18 runs at 100% Tand performing a two-parameter linear least-squares fit to thesequence number specifying the order of the data collection.The t-value for the significance of the computed slopes wasused to assess whether the overall drift was significant. At the95% level for 18� 2¼ 16 degrees of freedom, the tabulated t-value is 2.120. The corresponding t-values for the computedslopes for channels 1, 2, 3, and 4 were 1.77,�0.97, 0.78, and�6.70, respectively. These results suggest that only channel 4exhibited appreciable drift across the time span of the datacollection (;2 minutes).

Figures 2A and 2B plot the logarithm of the S/N ratio withrespect to the logarithm of the signal (volts) for the data fromdetector channels 1 and 2, respectively. Each of the eightplotted points denotes the mean of the three replicatemeasurements obtained for one of the levels of light intensity.Error bars are drawn as the corresponding 95% confidenceintervals.

Figure 2A yields an approximate linear response across theentire signal range with a least-squares slope of 0.85 6 0.03. Ifthe two rightmost points are deleted, the slope increases to 0.906 0.02. Channel 1 thus appears to be largely detector noiselimited throughout the response range, with some evidence of adecreasing slope at the highest signal levels.

By contrast, Fig. 2B reveals higher signal levels andproduces two distinct linear regions. From lowest to highestsignal intensity (left to right), the first three points in Fig. 2B

FIG. 1. Concentrations of analyte and interferent pairs based on a 55-pointuniform design.

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produce a least-squares slope of 0.94 6 0.02. This indicates adetector noise limited region analogous to that observed in Fig.2A. However, as the signal level increases beyond thatobserved in Fig. 2A, the S/N ratio reaches a plateau. Therightmost six points in the plot can be used to define a secondlinear region with a slope of 0.01 6 0.08. The t-value for thesignificance of this slope is 0.16. When compared to thetabulated t at the 95% level for 6� 4 ¼ 2 degrees of freedom(2.776), it is clear that the computed slope is not statisticallydifferent from zero. This region of the response is clearlysource noise limited. The behavior observed in Fig. 2B fordetector channel 2 is also observed in the analogous plots forchannels 3 and 4.

The differences in the controlling noise sources betweenchannels 1 and 2 arise from the effect of the broad absorptionprofile of water in the first overtone region. Absorptivities forwater published by Palmer and Williams27 can be used withcubic spline interpolation to estimate the transmittance for a 10mm path length at the centers of mass of the two filterbandpasses. For 6258 (channel 1) and 6093 (channel 2) cm�1,respectively, the estimated values are 0.057 and 0.24% T. Thiscorresponds to an increase in sample attenuation by a factor of4.2 for the location of the higher wavenumber filter. Planck’sfunction expressed as radiance per unit wavenumber at 2200 Kpredicts that the radiance of a blackbody source at 6258 and6093 cm�1 will be 0.00496 and 0.00510 W/cm2�sr�cm�1, adifference of ;3%. Thus, given roughly equal detectorresponsivities and source radiance levels across this spectralregion, the water absorbance will control the signal level. As

revealed by Fig. 2, the signal level will in turn determine thelimiting noise source.

The experimental results described above indicate that for agiven instrument, more than one noise model may control thecharacteristics of the measurement. In many applications inwhich NIR spectroscopy is attractive, significant variability inthe background absorbance of the sample may be encountered.This situation will produce behavior similar to that encounteredhere with water. It is clear from these results that the impact ofthe controlling noise model must be considered whenattempting to optimize a filter-based measurement.

Noise Characteristics of Simulated Data. The three noisemodels described above were used to construct simulated datafor the purpose of exploring optimal filter designs. Even thoughnone of the experimental measurements appear to be limited byshot noise, this model was also evaluated with the simulateddata for completeness. In constructing the simulated data, fourlevels of the S/N ratio, 2600, 5200, 13000, and 52000, wereused. The integrated filter response with no sample present(i.e., with only the blackbody source in the field of view of thedetector) was divided by the target S/N ratio to obtain thestandard deviation of the appropriate noise distribution. TheseS/N ratios bracket the maximum of ;7000 observed with theexperimental measurements of water in the first overtoneregion at a path length of 10 mm. For clarity, it should also bestated that in comparison to measurements in conventionalscanning spectroscopy, these S/N values correspond to the S/Nratio of a single-beam spectrum rather than the S/N valueassociated with an analyte absorbance reading.

The noise level was calculated so that the simulated filterresponse positioned at 5800 cm�1 with the largest FWHM (260cm�1) had the target S/N ratio of 2600, 5200, 13000, or 52000.As predicted by Planck’s function expressed as radiance perunit wavenumber, 5800 cm�1 corresponded to the maximumsource intensity among the filter bandpass positions studied.Other filters were then assigned the S/N ratio expected on thebasis of the noise model. For example, in the source noiselimited case, the ratio of the integrated responses of the widestand narrower filters was used to adjust the amount of noiseadded to the response from the narrower filter in order to obeythe noise model. Similarly, the square root of the ratio was usedto scale the noise for the shot noise limited case. Table IIsummarizes the range of S/N ratios across the filter positionsand widths studied. Once the noise scaling factors weredetermined, random normal deviates were generated and addedto the responses for each of the 55 mixture samples.

TABLE II. Range of S/N ratios in simulated data sets.a

Noise modelb

Filter position

5530 5800 6205

Shot noise limited [515, 2626] [510, 2600] [500, 2548]Detector noise limited [102, 2653] [100, 2600] [96, 2497]

a Minimum and maximum values correspond to the narrowest and widest filterswith FWHM values of 10 and 260 cm�1, respectively, when a S/N ratio of2600 was assigned to a background (i.e., blackbody) measurement with afilter having FWHM of 260 cm�1 at 5800 cm�1. Values become 2, 5, and 20times larger when the assigned S/N ratio for the blackbody measurement is5200, 13000, and 52000, respectively.

b Because of the linear relationship between signal and noise in the sourcenoise limited case, the S/N ratio was always the specified value of 2600,5200, 13000, or 52000.

FIG. 2. Plots of the logarithm of the S/N ratio with respect to the logarithm ofthe signal (volts) for detector channels (A) 1 and (B) 2. The points plottedrepresent mean values computed from the three replicate measurements at eachlevel of signal attenuation. The corresponding error bars denote the 95%confidence intervals. Dashed lines in each plot indicate linear least-squares linescomputed from the data.


Simulated Calibration Models and Filter Optimization.Once the integrated filter responses with noise were generatedfor each sample in the synthetic data sets, they were convertedto relative values, Irel, with respect to the integrated filterresponses for the blackbody source:

Irel;k ¼ 1000Isamp;k � Ibkg;k

Ibkg;k

� �ð1Þ

In Eq. 1, for filter (i.e., detector channel) k, Isamp is theintegrated response for the sample and Ibkg is the responsecorresponding to the background source with no samplepresent. The relative responses were scaled up to parts perthousand (ppt) to make subsequent calculations numericallybetter conditioned.

In the simulation studies described here, an instrument withtwo detector channels was assumed. The values of Irel,k from thetwo filters were used to build linear calibration models of the form

cj ¼ b0 þ b1Irel;1;i þ b2Irel;2;i þ ei ð2Þ

In Eq. 2, i denotes the sample number, ci is the correspondingtrue analyte concentration, the bi are coefficients estimatedfrom a multiple linear regression analysis, and ei is theconcentration error or residual. The Irel,k in Eqs. 1 and 2 areanalogous to transmittance values, which are logarithmicallyrelated to concentration through the Beer–Lambert law. In theexample studied here, the large transmittance values (i.e., smallvalues of absorbance) and limited dynamic range of transmit-tance made the logarithmic transform unnecessary (i.e., noconversion of transmittance to absorbance was required inorder to obtain a linear response with concentration).

For each of the 55 mixture samples, four replicatemeasurements were generated by sampling the random numbergenerator four times in assigning the noise values. Threereplicates were used in assembling a calibration set incomputing the regression model, while the fourth replicatewas placed in a prediction set. The calibration and predictionset combinations were generated 100 times to obtain anaverage estimate of the prediction errors, computed as theaverage percent relative standard error of prediction (%RSEP).The equation below specifies the calculation of the jth of 100

values of %RSEP:

%RSEPj ¼100

c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnp

i¼1

ðci � ci;jÞ2

np

vuuuut ð3Þ

In Eq. 3, c denotes the mean of the true analyte concentrations,ci is the actual concentration for sample i, ci,j is thecorresponding concentration predicted by the jth regressionmodel, and np indicates the number of observations in theprediction set (np ¼ 55 in the current example). An optimalvalue of %RSEP approaches zero. As calibration performancedegrades, %RSEP increases to a limiting value of approxi-mately 54.2%. This limiting value is determined by substitutingthe standard deviation of the analyte concentrations (8.4 mM)as the standard error (rightmost) term in Eq. 3. In this limitingcase, the calibration model explains none of the concentrationvariance in the prediction samples.

The regression models computed here are highly over-determined, given the ratio of observations to independentvariables of (3355):2. In this case, there is no issue of over-fitting the model to the calibration data and either calibration orprediction statistics could be used to evaluate the modelperformance. The use of replicates of the calibration samples inthe prediction set provides the most optimistic measure ofperformance for the filter-based measurement, as no issues ofinstrumental drift or mismatch between calibration andprediction samples are present.

To investigate the issues regarding optimal filter design foreach of the three data sets, three noise models, and four levelsof the S/N ratio, the bandpass FWHM values of the Gaussian-shaped filters were varied over eight levels (10, 40, 70, 100,140, 180, 220, and 260 cm�1). The positions of the bandpasscenter were varied from 5530 to 6130 cm�1 for detectorchannel 1 and from 5545 to 6205 cm�1 for channel 2. Theposition of each filter was varied over these ranges in steps of30 cm�1. All combinations of these position and widthparameters were investigated, yielding 30912 values of%RSEP. The filter pair giving the lowest value of %RSEPand those producing values of %RSEP that were judged

TABLE III. Results for source noise limited case.

Data set S/N ratioa %RSEP (stdb) No. of best filter pairsc Filter widthd,e (cm�1) Filter overlapd,f Angled,g (degrees)

1 2600 25.4 (7.9) 40 20 3.18 6.301 5200 14.7 (4.9) 23 20 3.88 6.451 13000 6.26 (2.29) 28 20 3.18 6.391 52000 1.76 (0.59) 29 26 2.87 6.712 2600 7.4 (2.7) 40 20 3.89 27.22 5200 3.7 (1.4) 32 20 3.18 26.42 13000 1.50 (0.59) 15 20 4.59 29.32 52000 0.53 (0.15) 58 26 3.31 31.13 2600 3.9 (1.3) 70 36 3.82 54.83 5200 2.0 (0.7) 133 26 3.75 55.63 13000 0.86 (0.23) 27 26 4.19 56.83 52000 0.39 (0.06) 61 32 3.75 61.0

a S/N ratio corresponding to a blackbody measurement with a filter having FWHM of 260 cm�1 positioned at 5800 cm�1.b The standard deviation of %RSEP across the 100 prediction sets for the filter combination that produced the best results.c The number of filter pairs whose %RSEP was found not statistically different from the lowest %RSEP on the basis of a two-sided t-test at the 95% level.d Median of the values corresponding to the filter pairs producing statistically equivalent values of %RSEP.e Geometric mean of the filter widths (FWHM) comprising the filter pair.f Defined by the bandpass overlap of the two filters as described by Eq. 4.g Angle between the analyte and interferent vectors in the space spanned by the two filter responses.


statistically not different from the lowest value by a two-sidedt-test at the 95% level were considered as the optimal filterdesigns for each data set, noise model, and S/N ratio.

Two additional calculations were performed to characterizethe optimal filter combinations. Overlap of the filter bandpassfunctions was characterized by use of a resolution parameter, r,

analogous to that employed in chromatography:

r ¼ qjp1 � p2jðw1 þ w2Þ

ð4Þ

FIG. 3. Response of each filter of the pairs in the absence of noise plottedagainst each other for data set 1. (A) The response of the pair giving the lowest%RSEP when the S/N ratio was 13000 in a source noise limited case. Theangle between the analyte and interferent vectors was 5.038. (B) The responseof the pair giving the largest angle (20.68) between the analyte and interferentvectors. Solid and open diamonds represent the analyte and interferentresponses, respectively.

FIG. 4. Best filter pairs in a source noise limited case with S/N ratio of 5200 ?1

for data sets (A) 1, (B) 2, and (C) 3. Solid lines represent responses of the twofilters, while short dashed and long dashed lines represent analyte andinterferent peaks, respectively. Filter overlap, computed with Eq. 4, was (A)11.5, (B) 11.5, and (C) 2.40.

TABLE IV. Results for shot noise limited case.

Data set S/N ratioa %RSEP (stdb) No. of best filter pairsc Filter widthd,e (cm�1) Filter overlapd,f Angled,g (degrees)

1 2600 34.9 (8.2) 138 134 0.83 4.861 5200 26.0 (8.0) 132 118 0.93 5.361 13000 12.1 (4.0) 50 118 0.95 5.421 52000 3.18 (1.07) 36 126 0.94 5.442 2600 14.0 (4.5) 101 134 0.94 23.12 5200 6.8 (2.5) 21 134 0.95 23.52 13000 2.85 (1.03) 52 137 0.95 23.92 52000 0.83 (0.24) 79 140 0.94 24.23 2600 6.6 (2.5) 126 180 1.03 51.43 5200 3.3 (1.2) 122 180 0.98 51.43 13000 1.38 (0.48) 156 161 0.94 50.53 52000 0.50 (0.10) 113 191 1.08 54.5



In Eq. 4, p1 and p2 denote the wavenumber positions of filters 1and 2, respectively, w1 and w2 are the corresponding bandpassFWHM values, and q is a constant (q¼1.17741) that scales theresult such that r¼ 1.5 for two Gaussians that are just baseline

resolved. Thus, r increases with increasing distance betweenthe two filter positions, and values of r , 1.5 denote filter pairswhose bandpass functions overlap. Complete overlap isindicated as r approaches 0.

FIG. 5. Best filter pairs in a shot noise limited case for data sets (A) 1, (B) 2,and (C) 3 when a blackbody measurement with a filter having FWHM of 260cm�1 at 5800 cm�1 had a S/N ratio of 5200. Solid lines represent responses ofthe two filters, while short dashed and long dashed lines represent analyte andinterferent peaks, respectively. Filter overlap, computed with Eq. 4, was (A)1.14, (B) 0.83, and (C) 0.93.

TABLE V. Results for detector noise limited case.

Data set S/N ratioa %RSEP (std b) No. of best filter pairsc Filter widthd,e (cm�1) Filter overlapd,f Angled,g (degrees)

1 2600 36.4 (6.3) 1900 124 0.49 0.311 5200 29.5 (9.2) 96 199 0.68 4.301 13000 15.1 (5.0) 50 199 0.69 4.391 52000 4.03 (1.42) 53 199 0.72 4.452 2600 16.9 (6.0) 109 216 0.72 20.42 5200 8.6 (3.3) 46 216 0.68 19.82 13000 3.53 (1.16) 49 216 0.70 20.52 52000 0.97 (0.29) 79 220 0.70 20.53 2600 7.1 (2.7) 46 239 0.78 50.03 5200 3.6 (1.0) 35 239 0.85 50.03 13000 1.50 (0.54) 73 239 0.85 50.53 52000 0.51 (0.11) 67 239 0.92 52.8


FIG. 6. Best filter pairs in a detector noise limited case for data sets (A) 1, (B)2, and (C) 3 when a blackbody measurement with a filter having FWHM of 260cm�1 at 5800 cm�1 had a S/N ratio of 5200. Solid lines represent responses ofthe two filters, while short dashed and long dashed lines represent analyte andinterferent peaks, respectively. Filter overlap, computed with Eq. 4, was (A)0.64, (B) 0.68, and (C) 0.70.


In addition, the angles were computed between the analyteand interferent responses in the data space spanned by thefilters. In this calculation, each filter bandpass function is usedas a basis vector in a two-dimensional space. The values of Irel,k

are analogous to projections onto these basis vectors and canthus be plotted in a two-dimensional coordinate system. If thefilter set provides selectivity for the analyte with respect to theinterferent, changes in analyte and interferent concentrationswill yield responses that proscribe vectors in this space withdifferent directions. The angle between these vectors is thus ameasure of the selectivity provided by the filters.

Optimal Filter Design in the Source Noise Limited Case.Table III summarizes the lowest values of %RSEP along withthe characteristics of the optimal filter pairs in the source noiselimited case. In this case, filter pairs with narrow bandpassFWHM values and non-overlapping responses produce thelowest prediction errors. Angles between the analyte andinterferent vectors in the space spanned by the two filterresponses were slightly larger than the angles between the purecomponent spectra of analyte and interferent bands. The anglesdirectly calculated from the pure component spectra were5.058, 22.68, and 55.48 for data sets 1, 2, and 3, respectively.

Though filter pairs producing large angles between theanalyte and interferent are desired for selective measurements,the filter pair giving the largest angle did not produce thelowest %RSEP values. As an example, Fig. 3 shows analyteand interferent responses for data set 1 in the absence of noiseplotted against each other for two pairs of filters. Figure 3Acorresponds to the filter pair that gave the lowest %RSEP whenthe S/N ratio was 13000 and Fig. 3B corresponds to the filterpair that produced the largest angle (20.68). As shown by theaxis scales, the filter pair that gave the largest angle hadresponses approximately 100 times smaller than the responsesproduced by the filter pair corresponding to the lowest %RSEP.The %RSEP value corresponding to the filter pair that gave thelargest angle was 51.1%, significantly larger than the lowest%RSEP (6.26%). The optimal filter pair is thus determined bythe tradeoff between the sensitivity and selectivity.

Figure 4 shows the best filter pairs when the S/N ratio was5200 with panels A, B, and C corresponding to the best pairsfor data sets 1, 2, and 3, respectively. The increased separationbetween the analyte and interferent peaks in data set 3 allowswider filters to be more effective.

Optimal Filter Design in the Shot Noise Limited Case.Table IV summarizes the lowest %RSEP values along with thecharacteristics of the filter pairs in the shot noise limited case.Figure 5 shows the best filter pairs when the S/N ratio was5200 at 5800 cm�1 for the filter with FWHM of 260 cm�1.Panels A, B, and C correspond to the best filter pairs for datasets 1, 2, and 3, respectively. Compared with the source noiselimited case, the filter pairs giving the lowest %RSEP valueshad wider FWHM and slightly smaller angles between theanalyte and interferent responses. The decrease in mathematicalselectivity (i.e., increase in spectral overlap) evident from thesmaller angles was compensated by increases in the S/N ratioof the wider filters.

Optimal Filter Design in the Detector Noise LimitedCase. Table V summarizes the lowest %RSEP values alongwith the characteristics of the filter pairs in the detector noiselimited case. Figure 6 shows the best filter pairs when the S/Nratio was 5200 at 5800 cm�1 for the filter with FWHM of 260cm�1. As before, panels A, B, and C correspond to the best

filter pairs for data sets 1, 2, and 3, respectively. The filter pairsgiving the lowest %RSEP values in the detector noise limitedcase had the widest FWHM values and smallest angles betweenthe analyte and interferent responses. The decrease inselectivity evident from the smaller angles was againcompensated by increases in the S/N ratio afforded by thewider filters.

CONCLUSION

When the measurement noise was limited by shot noise ordetector noise, the optimal filter pairs had wide andoverlapping responses in spite of decreases in spectralselectivity evident from the angles between the analyte andinterferent vectors. For these two noise models, the decrease inspectral selectivity was compensated by the increase in S/Nratio afforded by the use of filters with wider FWHM values. Itis anticipated that these results will extend to instrumentaldesigns based on larger numbers of filters and sample matricescontaining more than two components.

These results underscore that the design of an optimal filterphotometer for a given application is non-trivial. The noisemodel that governs the application must be taken into account,as well as the number of matrix constituents, the degree ofspectral overlap of those species, and the strength of the analyteabsorption information. These factors dictate that the filterdesign must be tailored specifically to the target application. Inall cases explored here, the optimal filter set did not simplymimic a low-resolution spectrometer. A filter optimizationprocedure of the type used here must be performed in order toachieve the best analytical performance.

To be practical, an optimization procedure of the typedescribed in this work requires the ability to implement anarbitrary filter bandpass in the spectrometer hardware. In thisregard, advances in tunable filters9–11 or other optoelectronicdevices such as digital micromirror arrays28 are particularlysignificant and applicable.

ACKNOWLEDGMENTS

Some of the instrumentation used in the experimental work was provided byOptix, LP (Cambridge, MA). This research was supported by the NationalInstitutes of Health under grant DK067445.

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design considerations for near-infrared filter photometry: effects of noise sources and selectivity

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