EVALUATION OF SINE WAVE TESTS OF ADC'S FROM WINDOWED DATA

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1 4 th orkshop on DC odellng and Testng - September 9-1, Bordeaux, France EVLUTION OF SINE VE TESTS OF DC'S FRO INDOED DT István Kollár Department of easurement and Informaton Systems, Techncal Unversty of Budapest, H-151 Budapest, Hungary Phone Fax E-mal: kollar@mt.bme.hu : bstract - Recently, much exctement has been caused n the EUPS group by the evaluaton of the IEEE 141 draft standard. Ths draft contans a lot of useful nformaton and several well establshed facts. However, there are also some detals whch are not fully clarfed, or need further examnaton. Ths paper deals wth one of these: the possblty of data processng, based on wndowed data. Keywords: IEEE-STD-141, wndowng, wndowed data, sne wave method, EUPS. 1. INTRODUCTION Fnte record length effects may deterorate the results of measurements wth snusodals sgnfcantly. The draft standard recognzes ths fact, and at several places advocates coherent samplng. Ths means that n a test, an nteger number of perods needs to be taken, that s, m f = f s, (1) where f s the frequency of the sne wave, m s an nteger less than /, s the number of samples n the record, and f s s the samplng frequency (cf ). Fg. 1. DFT result of a coherently sampled snusodal. =18, f = It s rght ndeed that whenever possble, coherent samplng s the proper choce (Fg. 1). However, when ncoherent samplng may occur (see () and () above Eq. ( )), wndowng s almost unavodable. In the followng sectons we are gong to dscuss the consequences of applyng wndows n detal.. ODELING SINE VE The cause of many problems s the fact that theoretcally, the Fourer transform of a sne wave conssts of two Drac delta functons at the approprate frequences: F { cos ( πf t + φ )} e = j j () φ φ δ ( f f ) + e δ ( f + f ). In strct sense, ths does not exst, and n measurements we obtan only a secondary functon based on ths. e collect samples from a fnte-length record. Ths operaton can be modeled by cuttng out the fnte-length record from the nfnte-length snusodal applyng multplcaton by a wndow functon. The equvalent of ths multplcaton s a convoluton n the frequency doman: F { cos ( πf t + φ ) w( t) } = e jφ jφ ( f f ) + e ( f + f ). The consequence of dscrete processng s frst of all that the shape of the above-descrbed wndow functon slghtly changes because of alasng, so the results for contnuoustme wndows only approxmately hold. In other words, the wndow shape becomes slghtly dependent on the number of samples,. Second, n the DFT we see only the samples of the dscrete-tme wndow, taken at the frequency bns of the dscrete Fourer transform: (3) f s fk = k = k f. (4)

2 Now the problem we try to solve s the followng: we have samples n the tme or n the frequency doman, corrupted by some nose and by some dstorton components. From these samples, we would lke to determne the parameters of the sne wave as precsely as possble, then subtract t from the samples, and evaluate the rest for SIND, THD, etc., maybe usng also the ampltude of the snusodal. In the case of no extra wndowng, we have the so-called angular wndow. For the contnuous-tme case: H ( f ) = δ = ( f ) 1 1.5δ f.5δ f + ( f ) T T 1 1 ( f ).5 f.5 f +. T T The formulae are smlar for the dscrete case. (1) 1 f t T w ( t) = (5) otherwse jπft sn( πft) ( f ) = e T. (6) πft hen we apply the DFT to dscrete ponts, we have the followng: 1 for 1 w ( ) = (7) otherwse. The DFT s clearly a delta functon at zero, and zero elsewhere. However, when wrtng the N-pont DFT, we see more:, /N k sn π N k sn π N jπk ( k) = e. The absolute value of,/n (k) s shown n Fg.. (8) Fg. 3. Hannng wndow n the frequency doman, N=18, =16. part from the man lobes, the repeated angular wndows have zeros at the same places. Ths means that n the Fourer transform there s only a large central peak, whle the snusod has no effect on the other bns. Therefore, coherent samplng s usually nsenstve to wndowng. e wll come back to ths later. Let us dscuss now what we are nterested n, and how ths s obtaned n the case of ncoherent samplng. 3. DISCUSSION Usually, we have one or maxmum -3 snusodal sgnals. e are nterested prmarly not n them, but rather n the rest of the spectrum. e usually wsh to remove them as profoundly as t s possble, snce the remanng part s the error we want to measure. The basc steps are as follows: determne the sne parameters from the record, subtract (suppress) the sne(s), analyze the resduals. Fg.. DFT of the angular wndow, N=18, =16. The records obtaned by coherent samplng are easy to use. s long as the tme doman wndow conssts of at most a few low-frequency snusods whose frequences correspond to the DFT bns, the frequency doman equvalent of the wndow contans zeros at dstances n f from the center. s an example, let us consder the Hannng wndow. w 1 t) = 1 cos π t w ( t) (9) T H ( Determnaton of the sne parameters Let us look frst at a typcal DFT result calculated from ncoherently sampled data. In Fg. 4 we see n the ndvdual bns the samples of the wndow functon, postoned around the frequency of the sne wave. The exact frequency of the sne wave s somewhere between the adjacent large peak pars. The true frequency of the sne wave s at most f/ from the place of the maxmum peak, and the maxmum peak s also down from the theoretcal value (64).

3 Fg. 4. Typcal DFT of an ncoherently sampled sne wave, f =.3 Ths naccuracy s far too much for our purposes, especally because the value of the maxmum also may vary down to about 64% of the true value. There are bascally two approaches to tackle these problems. By the nonparametrc approach we can sgnfcantly mprove the approxmaton of the maxmum value by applyng the so-called flat top wndow []. By ths, we modfy the shape of the wndow n order to acheve that the value of the wndow s essentally constant n the frequency doman between [- f/, f/]. In Fg. 5. the maxmum ampltude error s about.% (the dfference between heghts of the lnes, beng at a dstance f from each other, s much larger). However, the determnaton of the frequency s stll not mproved we can only use ths method for our purposes when the frequency of the sne wave s known. If we know both the ampltude and the frequency, the determnaton of the phase s straghtforward. Fg. 5. Incoherent samplng processed wth flat top wndow Parametrc model fttng more useful approach s parametrc model fttng. Ths can be performed n the tme doman or n the frequency doman. Intutvely t s clear that when performed properly, the procedures n the two domans are essentally equvalent. The basc dea n any case s that we assume a certan model of the sgnal (sne wave wth nose, etc), and determne the parameters from the model whch fts the best the measured data. There s however a fundamental problem. e can solve model fttng properly f the model s vald. hen any nonmodeled phenomena occur, even the best method may yeld wrong results. For example, n the case of ncoherent samplng, any other harmoncs (e.g. harmonc dstorton) may dsturb the fttng procedure. Then the consequence s that the parameters are mprecse, and therefore the removal of the sne wave s naccurate. The error we commt may be n the same order of magntude as the resduals we want to evaluate. Ths can be a serous problem n the four parameter method [1, ]. possblty to crcumvent the above dffcultes s to model every mportant harmonc whch may be present, and after the ft, remove the fundamental sne wave only. The dffculty s that n general t s very dffcult to select the frequences where such harmoncs are present. Selectng too many harmoncs can make the algorthm slow and senstve to local mnma. The only reasonable way s to extend the four parameter method by a mechansm whch takes nto account that most such components are harmoncally related to each other. Therefore, whle there are several components, ther frequences are descrbed by one sngle parameter ω and the (fxed) harmonc numbers only. Ths approach s not yet descrbed n the draft, maybe because of complcatons dffcult to tackle n advance. The equvalent of the four parameter method, or of ts several-component extenson, s to ft the DFT result by a scaled verson of the frequency doman form of the dscrete wndow, or by a set of such wndows. Ths s n general more complex than n the tme doman, so we do not dscuss t further. hen we want to avod the above-descrbed too complex modelng, and stll want to determne the parameters of the sne wave properly, the best way s to avod that dfferent harmoncs dsturb the estmates of each other. Ths s the basc dea of wndowng. ndowng From Fg. 4 t s straghtforward that each ncoherently sampled sne wave results n components at the other frequency bns (leakage). The cause s the form of the angular wndow: ts sdelobes are too large. The dea s then to modfy the shape of the wndow functon to have as small sdelobes as t s possble. Here, there are two approaches agan. One s the use of so-called harmonc wndows: wndows whch are the sum of a few low-frequency snusodal functons, whch therefore have a seres of zeros for coherent samplng (see Eq. (1)). Ths gves the so-called Blackman-Harrs wndows [3,4]. However, we can realze that the requrement of the zeros (for coherent samplng) s not necessary. e can speculate that t s enough to prescrbe that the sdelobes do not surpass a certan level. Ths leads to the desgned low-rpple wndows, lke the Kaser (Kaser-Bessel) wndow [3,4,9] or the Dolph-Chebyshev one [3,4,8].

4 In general, we may observe when we prescrbe the frequency doman behavor, and wsh to desgn a symmetrc tme seres to ths, we just desgn a lnear phase FIR flter. Consequently, any good FIR flter desgn algorthm, e.g. the well-known REEZ algorthm can be used to obtan the best wndow we can have at all. It s true for each wndow that sdelobe suppresson s at the cost of wdenng the man lobe. Therefore, wndowng somewhat decreases selectvty: non-overlappng components may not be closer than a few bns (a few tmes f). Therefore, the frequency of the sne wave must be larger than a few tmes f=f s / for proper processng. In the lght of the above dscusson, we can probably refne the statement n the draft ([1, Secton 4.1.5]: "The wndow functons are chosen n a trade off between the effectve nose bandwdth (ENB), or resultant DFT bns, and mnmum stopband response of the wndow flter functon as dscussed n the followng clause." In the above lght, ENB s not very mportant. The proper selecton of the wndow functon can be as follows: select the one wth the allowed stopband response, wth as small manlobe bandwdth 1 as t s possble. Now the value of the ENB wll be determned by the wndow: t can be approprately calculated from the wndow samples. Its defnton may however deserve a lttle explanaton. Frst of all, let us observe that t s ampltude scalng ndependent, and ts value (see Eq ) s exactly 1 for the angular wndow. Let us frst dscuss the scalng of the wndows. There are a few strateges for ths. e can a) keep the scalng of the frequency doman peaks of snusodals constant (that s, mantan the same value of the tme doman ntegral), or b) keep the varance of a whte nose sequence constant. In ths paper we follow the frst strategy. By ths the heght of the peak of the Fourer transform of a sne wave s the same as wthout wndowng. For random sgnals, we cannot gve a measure n terms of the ampltude, only n terms of the standard devaton or the varance. The ntegral of the wndow functon remans constant (the denomnator n the expresson of the ENB), whle the varance changes for whte nose n the followng way: var ch 1 1 w 1 1 = = w 1 1 w = ENB. (11) Because of Parseval's theorem, ths s also true n the frequency doman: 1 Keepng sne waves n mnd, let us defne the wdth of the manlobe as the dstance between the two stopbands, because ths determnes the mnmum necessary dstance between non-overlappng harmoncs. Ths s at least theoretcally true for non-coherent samplng the maxmum value of the spectrum can be smaller, dependng on the frequency msmatch and the shape of the wndow. The smaller the frequency msmatch, the closer the maxmum to the theoretcal value s. var ch 1 ( k) 1 k= 1 = = w 1 ( k) k= = ENB. (1) Therefore, the ENB accounts for the change n the nose varance. It s equal to the multplcatve term, and ts square root corresponds to the multplcatve term n the standard devaton. e know now how the ampltude of the transform of a wndowed sne wave changes, and we also know how the varance or the standard devaton of the nose peaks change. The last thng we have to dscuss s whether the power of a sne wave can also be calculated from the spectrum, wthout the necessty of calculatng the ampltudes. Here the precse answer s unfortunately no. The sum of squares of the frequency doman samples gves the power of the sne n the wndow, and ths can be dfferent from the power of the contnuous-tme sgnal. However, the devaton s usually not large. If we need the approxmate power of the harmonc (as for the total harmonc dstorton), the sum of the squares of the frequency doman samples, dvded by ENB and by, gves an approxmate value of the power. The result s not fully accurate (see Fg. 6), snce t slghtly depends on the frequency of the sne, but wll stll yeld a good enough approxmate value of the power we are lookng for. Fg. 6. Value of the power calculated from the adjacent lnes, as a functon of the frequency. Kaser wndow wth beta=3, =18. Now that we successfully separated the effect of the dfferent harmoncs, we can turn back agan to the determnaton of the parameters. Now our problem s smplfed to the followng one: gven a few frequency doman samples (around the sne frequency), determne the parameters. The standard way for ths s the so-called Interpolated FFT [5]. Ths s a set of algorthms whch, based on some approxmaton, determne the parameters of the sne wave from the samples around the maxmum. Ths gves a good estmate of the ampltude and of the frequency. nother possblty s as follows. In testng we usually have some tme for off-lne calculatons. There s a smple algorthm for ths purpose. If we select all the samples from

5 the man lobe, and set the rest to zero, we have wth good approxmaton the Fourer transform of a wndowed sne wave. Takng the IDFT of the two selected lne groups, we obtan the tme samples of the wndowed sne wave. hen dvdng by the wndow functon (and dscardng those values whch are dvded by a very small number where the wndow functon s close to zero), we can use the four parameter method to determne the desred parameters. Ths farly smple procedure has not been descrbed n the draft yet. Proper removal of the sne components s we already mentoned above, removal of the sne s only effectve f the sne parameters are precsely known. Inaccurate estmaton of the parameters s usually a problem because the sdes of the sdelobes n Fg. are steep, so a small error n the removal may cause a large resdual. Ths s the man ssue for wndowng: we have to accept that the values of the sne parameters are not exact, but f coron s necessary whch s comparable to the quanttes to measure (harmonc dstorton components, nose), accurate characterzaton of the DC s llusory. The removal s farly easy when there are no sgnfcant sdelobes. e smply dscard the group of the few large samples. e can even subtract the sne from the samples, take the ordnary DFT wthout wndowng (or wth some wndowng agan for the determnaton of the largest peaks, lke needed for SFDR), and evaluate the result. nalyss of the resduals s we have seen above, we can have both the wndowed and the non-wndowed verson of the resduals, as we lke. The only queston s whch method s senstve to ncoherent samplng, and what s the result of wndowng. THD [1, Secton 4.1.5] The draft standard says about the Total Harmonc Dstorton test: "The test descrbed above s based on DFT analyss (e.g., va DFT) on unwndowed sample sets." "ndowng s not recommended for these procedures due to the wdenng of resoluton bandwdth." Based on the above dscussons, we can state the followng. If we talk about the effect of perodc components, lke n the case of the THD, t s logcal to use a close-to-flat-top wndow after the removal of the sne wave. The precson of the flatness depends on the requrement for the measurement of the ampltudes: ths s usually not very strct. Read the ampltudes, and sum up the power accordngly. The prce we pay s that the nose floor ncreases, and we must not have harmoncs too close to each other. These dsadvantages are usually tolerable. nother possblty s to select the GROUPS of lnes belongng to the harmonc peaks (the number depends on the wndow), add the squared absolute values together, and dvde by the ENB and. Ths wll gve a good estmate of the power of the harmonc, wth lower nose floor. SFDR [1, Secton ] The standard does not dscuss the effect of wndowng to the Spurous-Free Dynamc Range. However, the calculaton s straghtforward: take the average of the wndowed DFT's calculated wth an approxmate flat top wndow (wth possbly small bandwdth to prevent alasng), and look for the largest peak. SIND [1, Secton 4.5.1] Here we want to calculate the power of the resduals to have the Sgnal to Nose and Dstorton Rato. Ths s straghtforward agan: use the nonwndowed resduals, and add the squared absolute values, or do the same wth wndowed data, and dvde by the ENB value. 4. SUGGESTIONS In the prevous sectons, we have formulated suggestons to extend and mprove the draft standard IEEE 141. e feel however that the desgn of wndow functons, dsplayng ther propertes, evaluatng complex algorthms wth no programmng bugs etc. s a dffcult task for most users. On the other hand, nobody can undertake the job for developng all algorthms for all mportant computers, ready-to-use but also flexble enough for specal purposes. The suggeston s that EUPS, maybe jontly wth the draft commttee, make model algorthms publcly avalable va a EB ste, n order to allow that people can compare ther algorthms to standard ones, have quck solutons for a few cases. The author s ready to work on such an undertakng based on atlab tself and on functon -fles. 5. CKNOLEDGENT The author expresses hs thanks to József Németh and László Sujbert for crtcal remarks and suggestons concernng the paper. REFERENCES [1] IEEE Standard 141 Draft Standard for Termnology and Test ethods for nalog-to-dgtal Converters. ay 5, [] L. Schnell, ed., "Technology of Electrcal easurements". pp John ley & Sons, Chchester, [3] Nuttall,. H., "Some ndows wth Very Good Sdelobe Behavor." IEEE Trans. on SSP, Vol. 9, No. 1, pp , Feb [4] Harrs, F. J., "On the Use of ndows for Harmonc nalyss wth the Dscrete Fourer Transform." Proc. IEEE, Vol. 66, No. 1, pp , Jan [5] Schoukens, J., R. Pntelon and H. Van hamme, "The Interpolated Fast Fourer Transform: Comparatve Study." IEEE Trans. on Instrumentaton and easurement, I-41, No., prl 199, pp [6] Kollár, I., "Dynamc Range of Dgtal Spectrum nalyzers". Proc. 5th IEKO TC7 Symposum "Intellgent easurement", Jena (GDR), June 1-14, Nova Scence Publshers, Commack, NY, pp [7] Prestley,. B., "Spectral nalyss and Tme Seres." cademc Press, London, [8] Gecknl, N. C. and Yavuz, D., "Dscrete Fourer transformaton and ts applcatons to power spectra estmaton." Elsever, msterdam, [9] Rabner, L. R. and B. Gold, "Theory and pplcaton of Dgtal Sgnal Processng." Prentce Hall, Englewood Clffs, 1975.