Uncertainty Management in the Measurements Performed by means of Virtual Instruments

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1 AMUEM 008 Internatonal Workshop on Advanced Methods for Uncertanty Estmaton n Measrement Sardagna, Trento, Italy, - Jly 008 Uncertanty Management n the Measrements Performed by means of Vrtal Instrments Salvatore cco and Cro Spataro, Member, IEEE Department of Electrc, Electronc and Telecommncaton Engneerng - Unversty of Palermo Vale delle Scenze, 908 Palermo, Italy Phone: , Fax: , emal: ncco@npa.t; cro.spataro@npa.t Abstract The ncertanty management s the dscplne of optmsng the cost of a measrement verss the ncertanty target. In partclar, the paper deals wth the measrements performed by sng a generc vrtal nstrment. The task s acheved by sng the PUMA (Procedre for Uncertanty Management) that s an teratve technqe orgnally conceved for geometrcal and mechancal measrements. The approach s completely based on the Gde to the Expresson of Uncertanty n Measrement rles, bt provdes a more engneerng ology, snce t allows to optmze the cost of the measrements verss the ncertanty target, avodng the se of nadeqate or, on the contrary, too expensve resorces for the ncertanty estmaton. Accordng to the PUMA, for dfferent approaches to evalate the ncertantes are presented and they are appled startng from the coarser, bt qcker and cheaper one and keepng on wth the more precse ologes. In the practcal cases, t s very probable to prove that the frst teraton of the PUMA s enogh to optmse the ncertanty bdget. Keywords Uncertanty Estmaton, Uncertanty Management, Vrtal Instrments. I. ITRODUCTIO More and more often, manly n the ndstral envronment and n the test and calbraton laboratores, many knd of measrements are performed sng a general prpose data acqston board (DAQ) connected to a common personal compter (PC) and processng (sccessvely or n real tme) the acqred data by sng the PC processor tself. The man reason s that these nstrments, sally called Vrtal Instrments (VIs), are less expensve and more flexble f compared wth a correspondng tradtonal nstrmentaton, whch owns the same performances. The workng prncple of a VI s very easy to descrbe: the physcal qantty s transdced n electrc sgnal whch s condtoned to be adapted to the sccessve crcts; the sgnal s sampled at a freqency at least twce hs bandwdth and converted n nmercal codes; the acqred samples are processed by the stable measrement dgtal sgnal processng block, sally developed by the ser, to get the measrement reslts whch are dsplayed n a vrtal panel of the PC montor. However, gven that the VIs are sally desgned, assembled and programmed by the sers themselves, the dffcltes n a correct evalaton of the measrement ncertantes have lmted ther spread on the ndstral envronment and n the test and calbraton laboratores. In fact, for a correct employment n a qalty management system, t s essental to characterse all the employed measrement nstrments and to estmate the measrement qalty [,]. Accordng to the ISO Gde to the Expresson of Uncertanty n Measrement (GUM) [3], the ndex whch qantfes ths qalty s the standard ncertanty assocated wth the measrement reslts. Many Athors have dealt wth the ncertanty assessment n the measrement performed by means of VIs and n [4-9] varos nterestng ologes to accomplsh ths task are presented. We handled the topc too and n order to characterse a generc VI we proposed a nmercal whch, by means of an ad hoc developed software tool, estmates the ncertantes sng the Monte Carlo approach [0]. In ths paper, startng from the acqred experence, we deal wth the ncertanty management, whch s the dscplne of optmsng the ncertanty target and the cost of the measrements. Wthot a systematc approach to the ncertanty evalaton, the performed measrements cold be ether not adeqate for the target ncertanty or too expensve. To acheve the objectve, we perform the ncertanty estmaton by means of the PUMA (Procedre for Uncertanty Management) approach. Ths teratve s completely based on the GUM rles bt offers a more engneerng ology. In fact, the basc nspraton s completely dfferent: the GUM prescrbes that, n the ncertanty evalaton, t s necessary to avod overestmatons, snce an overestmaton debases the measrement qalty; accordng to the PUMA approach, on the contrary, when t s necessary to know f an already avalable nstrmentaton and an already defned measrement envronment are approprate for a stated target ncertanty, the ncertanty overestmate can be tolerated, especally when an accrate ncertanty estmaton nvolves hgh expenses /08/$ IEEE

2 Fg.. Graphcal representaton of the PUMA II. THE 453- STADARD The ISO/TS 453- standard [] was developed by the Techncal Commttee ISO/TC 3. In spte ths standard deals wth the Geometrcal Prodct Specfcatons (GPS), and therefore wth the geometrcal and mechancal measrements, ts extenson to the other branches of measrements s natral and cold reslt sefl. The standards s flly dedcated to the ncertanty management performed by sng the PUMA approach. The PUMA (graphcally presented n fg.) s based on a seqence of ncertanty over-estmatons, startng from the coarser bt qcker ones and carryng on wth more accrate bt more expensve and more tme-consmng technqes. By applyng only the frst teraton of the, t s very probable to prove that the ncertanty of the consdered measrement process s mnor than the target ncertanty and, conseqently, that the measrement process s adeqate. Jst n few practcal cases, there s the need of sng sophstcated mathematcs and statstcs. The ncertanty management s carred ot startng from the descrpton of the measrement prpose, the measrement nstrmentaton, the measrement condtons and from the defnton of the ncertanty target vale U T. On ths bass, from the frst teraton of the PUMA approach, a frst estmate U E of the measrement ncertanty s obtaned. If U E < U T, than the chosen measrement procedre s adeqate to the measrement prpose. If U E << U T, the chosen measrement procedre s obvosly stll techncally adeqate, bt there s the chance to change the measrement nstrmentaton and/or condtons to make the measrement less expensve. If U E > U T, t s necessary to perform another teraton obtanng a more accrate and lower estmate U E of the ncertanty. By comparng ths new vale wth the ncertanty target, t s possble to verfy that the measrement procedre s adeqate or that t s necessary to perform another teraton. After havng sed all the chances to get the more accrate ncertanty estmaton, f U E > U T, than the chosen measrement procedre s really nadeqate to the measrement. The am of the paper s the extenson of the PUMA to the VIs, defnng varos less and less coarse s; obvosly t s necessary startng from a rgoros and precse evalaton ology whch can be sed as reference. For ths prpose we se the already mentoned nmercal approach whch was valdated n [0] and whch s descrbed n the next chapter. III. THE REFERECE APPROACH For the charactersaton of the whole measrement chan of a VI, accordng to [3], the frst steps to perform are: the dentfcaton of the error sorces whch gve a contrbton to the ncertanty of the measrement reslt drng the transdcton of the qanttes, the condtonng of the sgnals, the A/D converson and the dgtal sgnal processng; the qantfcaton of the standard ncertantes assocated wth each error sorce. Wthot any lose of generalty we do not consder the errors generated by transdcers and condtonng accessores. Even f these errors are often predomnant compared to the errors generated n the A/D converson, the transdcers and condtonng accessores varety s so wde, that t s necessary to analyse separately each partclar staton. On the contrary, t s possble to carry ot a general treatment n the case of the A/D converson process. In any case the proposed can be extended to each partclar transdcer and/or condtonng accessory, by dentfyng all the error sorces, evalatng the assocated standard ncertantes and analysng how these ncertantes nflence the ncertanty of each acqred sample. We do not consder ether the errors generated drng the dgtal sgnal processng snce these errors are sally neglgble f the software block of the VI s correctly desgned. So n the followng we consder only the man error sorces generated drng the A/D converson, whch are: offset, gan, ntegral non-lnearty (IL), nose, cross-talk,

3 settlng tme, tmng jtter, qantzaton and dfferental nonlnearty (DL) []. The followng step to perform s the qantfcaton of the ncertantes assocated wth these error sorces. It can be carred ot by means of statstcal s wth a Type A evalaton accordng to the GUM, (bt n order to estmate the ncertantes assocated wth all the sorces t s necessary to test a statstcally sffcent nmber of nstrments of the same knd), or t s also possble to trn to manfactrers specfcatons (Type B evalaton). Of corse the second way s less expensve and less tme consmng, snce t does not reqre any knd of test from the ser. However evalatng the standard ncertantes startng from the manfactrers specfcatons s not a very effortless task, snce each manfactrer frnshes the specfcatons n an arbtrary way, sometmes nventng some new parameter. In any case t s necessary to formlate some arbtrary hypothess on the knd of the dstrbtons. For the offset, gan, temperatre drft and long-term stablty errors, the manfactrers declare an nterval ± a where the error srely les. Accordng to the GUM, provded that there s no contradctory nformaton, each npt qantty devaton s to be consdered eqally probable to le anywhere wthn the nterval gven by specfcaton, that s modeled by a rectanglar probablty dstrbton. As for the non-lnearty errors, the worst-case vales of IL and DL are sally reported n the specfcatons. The standard ncertanty related to nose can be drectly obtaned from the techncal specfcatons, snce t s sally expressed as rms vale. The cross-talk errors are prodced by the nterference n the mlt-channel acqston. Its related ncertanty s expressed as mnmm rato between the sgnal rms vale and the nterference sgnal rms vale. The settlng tme s the amont of tme reqred for a sgnal that s amplfed to reach a stated accracy and stay wthn the specfed range of accracy. The manfactrer declares ths range for the maxmm samplng rate and for the fll-scale step, bt the errors on the measred sgnal depend on the actal samplng rate and on the actal step. Impact of tmng jtter ncertanty s transformed on ampltde ncertanty as a fncton of sgnal dervatves. The manfactrer declares the apertre jtter vale, typcally expressed as rms vale. After the dentfcaton of the ncertanty sorces and the evalaton of the assocated standard ncertantes, to assess the combned standard ncertanty of the measrement reslts, other two steps have to be carred ot: composton of these standard ncertantes to obtan the combned standard ncertanty of each acqred sample; stdy of how the ncertantes of each acqred sample combne and propagate drng the dgtal sgnal processng. To perform smltaneosly these tasks we proposed a nmercal approach based on the Monte Carlo, developng a software tool whch smlates a tre A/D converson and takes nto accont all the ncertanty sorces. The tool s placed between a measrement npt sgnal smlator and the software block of the VI and smlates a set of measrements carred ot by dfferent realsatons of the same nstrment. In the followng, ts workng prncple s descrbed. The npt sgnal smlator generates samples as f they were obtaned from an deal samplng process of the sgnal and samples are sent to the A/D smlaton tool. The core of the tool s a FOR loop exected M tmes. The samples vector, nsde the loop, s modfed n order to smlate the errors generated drng the A/D converson process. To smlate the offset, a constant vale s added to each sample of the sgnal. Ths vale s a random nmber wthn the range declared by the manfactrer. For each smlated measrement, the generated random nmber changes so that t les n the specfcaton range accordng to a rectanglar dstrbton. In the same way, gan errors are smlated. In ths case each sample of the sgnal s mltpled by a constant vale. A whte nose s added to smlate the thermal nose, and to smlate the crosstalk nterference, another sgnal s added. The IL errors are smlated dstortng the transfer fncton wth components of second, thrd, forth and ffth order and wth other two spros components, so that the maxmm devaton from a lnear transfer fncton s always eqal to the maxmm IL vale declared n the specfcatons. As for the settlng tme errors, the software tool calclates the errors for the actal samplng rate and the actal step between each two contgos samples, startng from the settlng tme accracy at the maxmm samplng rate declared by the manfactrer. The tmng jtter errors are smlated by mltplyng a random nmber, wthn the range of apertre jtter declared n the specfcatons, by the dervatve of the sgnal; the so obtaned vales, whch are the ampltde errors cased by the samplng tme errors, are added to each sample. At last, the smlaton of the qantzaton process, whch takes nto accont the DL errors, s performed. The so modfed samples are sent to the software block of the nstrment, whch calclates the measrement reslt. The M measres are collected otsde the loop and the standard devaton of the measrements reslts, that s the combned standard ncertanty, s calclated. The man advantage of ths s that t ntrnscally takes nto accont every possble correlaton between each qantty. However, t s obvos that the effectveness of the descrbed approach s strctly dependng on how the A/D converson process and the ntrodcton of the errors are smlated. So wth the am of valdatng the approach, we appled the nmercal on varos DSP basc blocks, whch are typcal of a measrement chan. The obtaned reslts have been compared wth the ones obtaned by means of expermental tests and the comparson, as descrbed n detal n [0], has postvely valdated the nmercal. IV. THE PUMA METHOD The descrbed nmercal approach leads to an accrate estmaton of the ncertantes of a measrement performed by sng a generc VI; however, t s necessary to trthflly smlate a real A/D converson process. Ths entals a large

4 sage of resorces and tme both for the software developng and for performng the M smlatons. In order to save resorces, accordng the PUMA ology, t s necessary to fnd a coarser, bt faster, approach whch leads to an overestmate of the ncertanty. For ths prpose, t s possble to se a smplfed verson of the nmercal approach. The offset, gan, IL, nose and cross-talk errors are smlated as prevosly descrbed; as for settlng tme, tmng jtter, qantzaton and DL, the correspondng rms vale s calclated for each error sorce startng from the worst case manfactrer specfcatons (n partclar for the tme jtter worst case, t s possble to se the followng expresson [3]: jtter = log f 3π xτ α V range where τ α s the rms apertre jtter and f x s the maxmm freqency component of the sgnal. Provded that wth good approxmaton these ncertanty sorces can be consdered not correlated, the root sm sqare of the rms vales s calclated, obtanng an eqvalent random nose whch s added to the npt sgnal by the software tool. The advantage of the smplfed verson of the nmercal approach s a hge redcton of the software tool complexty and of ts execton tme; n fact by sng ths verson, the smlaton of settlng tme, tmng jtter and qantzaton (pls DL) s performed jst addng an eqvalent nose to the npt sgnal; obvosly the sage of the worst cases specfcatons to smlate settlng tme, tmng jtter, qantzaton and DL leads to an overestmate of the ncertantes. Another approach to the ncertanty estmaton s the se of a theoretcal applyng the ncertanty propagaton law of the GUM. When a measrand estmate y s determned from other samples x, x,, x, throgh a fnctonal relaton y = f(x, x,, x ), the combned standard ncertanty estmate c (y) of the measrement reslt s the postve sqare root of the estmated varance c(y), obtaned from: ( y) = c f f ( x ) x + = = j=, j x x j r( x, x ) ( x ) ( x ) where (x ) s the estmated standard ncertanty assocated wth the sample estmate x and r(x,x j ) s the estmated correlaton coeffcent assocated wth the samples x and x j. Ths s based on a frst-order Taylor seres approxmaton of y; therefore there s an mplct error whch leads to a less precse ncertanty estmaton than the correct sage of the Monte Carlo approach. Besdes ths, the evalaton of the correlaton coeffcents s a very hard task, also becase they are strctly dependng on the npt sgnal. On the other hand to gnore the correlatons cases a heavy j j nderestmate of the ncertantes. So the theoretcal approach s actally napplcable becase of dffcltes n the exact dentfcaton of correlaton coeffcents. Bt f we consder separately each ncertanty sorce, we can observe that as for the offset and gan, the correlaton coeffcents are approxmately eqal to, whle wth regards to the other ncertanty sorce the correlaton coeffcents can be spposed eqal to 0. Moreover, n case of errors de to gan, the relatve standard ncertanty r (x) = (x)/ x has to be consdered constant on each npt sample. In all other cases t s the absolte standard ncertanty, to be consdered constant on each npt sample. Therefore, all the ncertanty sorces can be dvded approxmately n three classes: I. completely correlated npt qanttes and I = const; II. completely correlated npt qanttes and rii = const ( r = relatve ncertanty); III. not correlated npt qanttes and III = const. In ths way, t s possble to overcome the dffcltes of the exact evalaton coeffcents, snce, by means of the proposed classfcaton, the ncertanty sorces are dvded n three classes wth spposed correlaton coeffcent exactly eqal to or 0. Moreover the ncertanty propagaton law becomes easer to apply, that s respectvely for the three classes: ci cii ciii ( y) = ( y) = ( y) = I = rii = III x x = x x Startng from the above consderatons, the dea of an approxmated theoretcal has arsen. It s based on the followng steps: to sbdvde the ncertanty sorces n the three classes; to carry ot the root sm sqare of the ncertantes of each class, obtanng three vales ( ci, cii and ciii ) of ncertanty for each acqred sample; to apply the propagaton law separately for each sorce class, gettng three standard ncertanty vales ci, cii, ciii. to carry ot the root sm sqare of these three vales obtanng the combned standard ncertanty of the measrement reslt.

5 There are some approxmatons n ths : the frst one conssts of combnng the ncertantes after they are propagated, whereas actally the ncertantes frst are combned n each acqred sample and then propagate throgh the software block; the other approxmaton s the sbdvson of the ncertanty sorces n the three classes wth spposed correlaton coeffcent exactly eqal to or 0. These approxmatons lead to ncertanty overestmatons slghtly heaver than the ones obtaned by sng the smplfed nmercal approach. An nterestng advantage of the theoretcal approach s that t can be appled drng the nstrment desgn stage when the software block of the nstrment s not already developed. The last, faster and coarser approach s an even smpler verson of the theoretcal ; n ths case the correlaton coeffcents of all ncertanty sorces are set eqal to ; as for the gan errors, t s always consdered the worst case that s settng gan = rgan x Fll Range; n the propagaton law, the absolte vales of the partal dervatves are consdered: c( y) = = x Therefore the applcaton of the theoretcal becomes qte fast and smple, snce t s enogh jst performng the followng steps: to carry ot the root sm sqare of all the ncertantes, obtanng a sngle vale () of ncertanty for each acqred sample; to apply the smplfed propagaton law, gettng the combned standard ncertanty of the measrement reslt. Obvosly to consder the ncertanty of each acqred sample totally correlated and to consder the worst case of the gan error lead to a qte heavy overestmate of the measrement ncertantes. V. PRACTICAL CASES Wth the am of verfyng the applcaton of the PUMA, we appled all the proposed ncertanty estmaton approaches on varos DSP basc blocks that are typcal of a measrement chan. As example n the followng, the procedre for the measrement of the DC vale, the RMS vale and the ampltde of a sngle tone by means of FFT s reported. The VI s consttted of the atonal Instrments AT-MIO-6E0 data acqston board (6 sngle-ended or 8 dfferental channels, sccessve approxmaton bt ADC, 00 ks/s max samplng rate, ± 0 V maxmm npt sgnal range) and a PC wth an ITEL 870 MHz processor; the LabVew 7.0 s the programmng langage sed to drve the acqston board, to process the acqred samples and to realze the ser nterface. The npt sgnal s a KHz snsod wth a 4 V peak vale pls a 5 V DC sgnal. The sed samplng rate s 0 KS/s, the tme wndow s 0 ms and the samplng s coherent wth the npt sgnal. We consder a Type B evalaton of standard ncertantes, based on manfactrer s specfcatons, assme rectanglar dstrbtons and sppose to operate wthn ± K of the data acqston board self-calbraton temperatre, wthn ± 0 K of factory calbraton temperatre, after one year of the factory calbraton and to set the gan eqal to 0.5. In Table I the consdered ncertanty sorces, the manfactrer specfcatons and the standard ncertanty vales are reported: Table I - Uncertanty sorces of bt DAQ model Uncertanty sorce Manfactrer specfcaton Standard ncertanty vales offset ± 09 μv 640 μv gan ± 0.05 % 90 ppm IL ± LSB 89 μv DL ± 0.5 LSB 40 μv qantzaton ± 0.5 LSB 40 μv nose 0.07 LSB rms 34 μv settlng tme for ± 0. LSB n fll scale step 00 μs 8 μv tme jtter ± 5 ps 40 μv cross talk - 80 db 707 μv Startng from these vales, by the applcaton of the for ncertanty estmaton approaches, the reslt of table II are obtaned: Table II - Uncertanty vales [mv] Measrand DC vale RMS vale FFT Smplfed theoretcal Theoretcal Smplfed nmercal mercal We appled the PUMA approach to a 6 bt DAQ, namely the atonal Instrments PCI-MIO-6XE0 model board (6 sngle-ended or 8 dfferental channels, sccessve approxmaton 6 bt ADC, 00 ks/s max samplng rate, ± 0 V maxmm npt sgnal range), whose manfactrer specfcatons (for gan set to ) and the standard ncertanty vales are reported n table III:

6 Table III - Uncertanty sorces of 6 bt DAQ model Uncertanty sorce Manfactrer specfcaton Standard ncertanty vales offset ± 4 μv 8 μv gan ± 36 ppm ppm IL ± LSB 76 μv DL ± LSB 76 μv qantzaton ± 0.5 LSB 88 μv nose 0.33 LSB rms 0 μv settlng tme for ± 0.5 LSB n fll scale step 00 μs 88 μv tme jtter ± 5 ps 40 μv cross talk - 80 db 707 μv Startng from these vales and sng the same npt sgnal employed for the bt DAQ, by the applcaton of the for ncertanty estmaton approaches, the reslt of table IV are obtaned: Table IV - Uncertanty vales [mv] Measrand DC vale RMS vale FFT Smplfed theoretcal Theoretcal Smplfed nmercal mercal Analyzng the reslts, t s possble to notce that n some cases the coarser approach lead to an ncertanty estmaton roghly sx tmes greater than the one obtaned by sng the more precse ology. However, t mst be nderlned that the applcaton of the smplfed theoretcal approach reqres a neglgble amont of tme and of resorces, f compared wth the tme and the resorces nvolved by the nmercal. We obtaned smlar reslts consderng other data acqston boards, other npt sgnal and other measrement algorthms. VI. COCLUSIO In the paper, for approaches for the ncertanty assessment of a measrement carred ot by sng a generc VI were presented. The approaches mply dfferent levels of estmate accracy, bt at the same tme dfferent amont of reqred resorces. Therefore, the for ologes are perfectly adeqate for the mplementaton of the PUMA and for a correct management of the ncertanty bdget. Often, n fact, t s not necessary to obtan a very accrate ncertanty evalaton, bt t s enogh to know f an already avalable nstrmentaton and an already defned measrement envronment are approprate for a stated target ncertanty. All the proposed ncertanty estmaton s can be extended to transdcers and sgnal condtonng accessores. After the dentfcaton of the error sorces that arse drng the qanttes transdcton and the sgnals condtonng and after the evalaton of the related standard ncertantes, the obtaned vales have to be dvded n the three classes n order to carry ot the theoretcal. As for the nmercal s, the software tool has to be modfed n order to smlate, besde the A/D converson, the whole measrement process. REFERECES [] ISO 900 Qalty management system- Reqrements, 000. [] ISO/IEC 705: General reqrements for the competence of testng and calbraton laboratores, 999. [3] ISO/IEC 3005, Gde to the expresson of ncertanty n measrement, 995. [4] G. Betta, C. Lgor, A. Petrosanto: Uncertanty analyss n Fast Forer Transform algorthms, IMEKO TC-4 Internatonal Symposm, aples, Italy, September 998, pp [5] G. Betta, C. Lgor, A. Petrosanto: Strctred approach to estmate the measrement ncertanty n dgtal sgnal elaboraton algorthms, IEE Proc. Sc. Meas. Technol., Vol. 46, o., Janary 999, pp. -6. [6] Ghan, E.; Locc,.; Mscas, C.: Ato-evalaton of the ncertanty n vrtal nstrments Instrmentaton and Measrement, IEEE Transactons on, Volme 53, Isse 3, Jne 004 Page(s): [7] Korczynsk, M.J.; Hetman, A.: A Calclaton of Uncertantes n Vrtal Instrment, IMTC 005. Proceedngs of the IEEE, Volme 3, 6-9 May 005 Page(s): [8] Lampas, D.A.; Podesta, L.: A practcal approach to evalate the measrement ncertanty of vrtal nstrments, IMTC 004. Proceedngs of, 8-0 May 004 Page(s):46-5 Vol.. [9] Ferrero, A.; Gamba, R.; Salcone, S.: A based on randomfzzy varables for onlne estmaton of the measrement ncertanty of DSP-based nstrments, Instrmentaton and Measrement, IEEE Transactons on, Volme 53, Isse 5, Oct. 004 Page(s): [0] S. cco, C. Spataro: A Monte Carlo Method for the Ato- Evalaton of the Uncertantes n the Analog-to-Dgtal Converson- Based Measrements, COMPEL, The Internatonal Jornal for Comptaton and Mathematcs n Electrcal and Electronc Engneerng, Vol. 3,., 004, pp [] ISO/TS 453-, Geometrcal Prodct Specfcatons (GPS) - Inspecton by measrement of workpeces and measrng eqpment - Part : Gde to the estmaton of ncertanty n GPS measrement, n calbraton of measrng eqpment and n prodct verfcaton, (999). [] IEEE Std 4, 00, Standard for Termnology and Test Methods for Analog-to-Dgtal Converters. [3] R. H. Walden: Analog-to-Dgtal Converter Srvey and Analyss, IEEE Jornal on selected areas n commncaton, Vol.7,. 4, Aprl 999.