Easy Estimation of Spectral Purity of Test Signals for ADC Testing. David Slepička

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1 Sep. -4, 008, lorence, Italy Easy Estimation of Spectral Prity of Test Signals for ADC Testing David Slepička Czech Technical University in Prage, aclty of Electrical Engineering, Dept. of Measrement Technická, CZ-1667 Prage 6, Czech Repblic Phone: , ax: , Abstract ADC testing freqently confronts the problem of spectral prity of harmonic test signal. The majority of dynamic ADC test methods reqire test signal distortion at least 10 db less than that of the tested ADC the condition that cannot be practically flfilled for testing p-to-date ADCs. More sophisticated approach to ADC testing applies the correction for test signal imperfections. In both cases test signal distortion has to be known roghly in the first case and qite accrately in the second case. In this paper an easy method for the testing of signal spectral prity is proposed. Test signal passed by one of two simple analog filters is measred by a common ADC, whose nonlinearity is mathematically post-corrected. The reslts are compared with common approach applying notch filter and ncertainty analysis is performed. I. Introdction ADC dynamic testing is generally based on spectrally pre sine wave stimls. Standards of ADC testing [1], [] assme high-qality test signal and do not consider other case; or they eventally recommend test signal filtering by pass-band filters, which also leads to spectrally pre sine wave. Unfortnately the manfactre of sch filters is neither easy nor cheap. Alternative non-standardized methods mostly assme the same test signal prity althogh it is hardly achievable in practice. There are in principle three approaches to this problem: A signal prodced by common generators is filtered by high-qality pass-band filters. The problem is restricted to harmonic distortion, mltitone signal is sed and intermodlation components are observed. Alternative easily-to-generate test signals are applied. Special methods enabling the correction for test signal imperfections are applied. Test signal filtering is in principle an easy way how to get rid of spectral imprity bt it is apparently the most difficlt one in practice particlarly at lower freqencies. The limiting factor is an extreme demand on components linearity that leads to high mechanical dimensions at lower freqencies (adio band). Moreover, there remains a problem with phase noise that is difficlt to be filtered ot [3]. Ths special signal generators with sccessive filters have to be designed [4]. Disadvantage of sch approaches is the limitation to one freqency at which the generator is tned and high costs of generator s components. The application of mltitone signal sppresses the need of low harmonic distortion of the test sine wave. If the power combiner of all sine waves is linear enogh, it is only the ADC nonlinearity prodcing intermodlation components in the acqired signal spectrm. The first three harmonic components dominate at the most ADCs; in this case the total harmonic distortion can be estimated from the intermodlation distortion [5]. It is also possible to assess the ADC transfer fnction from intermodlation components [6]. owever, only ADC nonlinearity can be covered by this test and neither sprios components nor wideband noise. The noise of mltitone signal cannot also be too high in order not to mask intermodlation components. Another approach to minimize the test signal distortion is to se special test signals that cold be generated with lower distortion than a common sine wave e.g. exponential [7], Gassian noise [8] or small triangles [9]. The common disadvantage of sch approaches is that these signals contain many freqencies and the reslts of these tests are difficltly transferable to standardized test reslts. The last grop of methods mentioned applies common sine wave passed throgh a set of filters to the tested ADC and comptes the actal ADC performance from several measrements [10], [11], [1]. This approach can be applied for the estimation both of harmonic distortion and of wideband noise of the tested ADC as shown in [1]. Limitations of these methods are the need of set of filters [10] or the level of ADC and test signal distortion that shold not be far away [11], [1]. Another problem arises in vicinity of the fndamental where the accracy of noise estimation is low [1]. In several pblications (e.g. [10]), the possibility of correction for test signal imperfections has been

2 Sep. -4, 008, lorence, Italy analysed. If the vector representation of test signal harmonic distortion is known, this is a simple task of complex sbtraction. In some cases when the test signal distortion is small enogh, this correction is not needed at all. So, the problem can be redced on how to estimate test signal imperfections. In this paper a modified method based on simple passive filters and mathematical post correction [11], [1] is analysed, practical reslts are shown and ncertainties are estimated. II. Proposed approach The principle of the proposed method comes from the approach [11], [1] and the same measrement setp (see ig. 1) slightly modified for the test signal measrement. Test signal G(jω) is measred twice each time passed throgh one of the filters (jω) or (jω). or the verification of reslts there is also a notch filter N(jω) tned at fndamental freqency of the test signal. N(j ) G(j ) (j ) D(j ) X(j ) R1 generator ADC R R1=R (k) (j ) igre 1. Block diagram of the measrement setp ilter characteristics (jω) and (jω) can be almost random, there are only three reqirements on them. They mst be different, their attenation at fndamental freqency mst be the same and they mst be linear so that they do not prodce harmonic distortion. The measrement of generator freqency spectrm consists of two parts: harmonic components and noise. This comes from the character of generator and ADC distortion. In case of (additive) noise, both generator and ADC freqency spectra are spposed to be independent (ncorrelated); ths otpt noise power consists of the sm of ADC inpt signal noise power and ADC noise power: E { } ( ) { ( ) } jω E G jω D =, E { } ( ) { ( ) } jω E G jω D = (1), () where 1 (jω) 1 is the freqency spectrm measred when filter (jω) is applied and (jω) measre with filter (jω). Generator noise is done by the formla E { } { ( ω) } E E j { G } =. (3) armonic distortion is a correlated distortion and freqency spectral lines have to be treated as vectors: = G D, G D Generator harmonic distortion can be expressed as G = (4), (5) =. (6) or the comptation of (6) filter characteristics have to be known. reqency characteristic (jω) of the filter sed in the measrement setp according to ig. 1 is freqency independent and done by R 1 ( jω ) = = = (7) R R 1 becase R 1 = R = R. If freqency characteristic (jω) is determined from components characteristics, 1 Continos freqency ω is sed as the argment of discrete freqency spectrm for the simplicity.

3 Sep. -4, 008, lorence, Italy parasitic components are not incorporated and reslts cold be biased. So, it is advantageos to determine this characteristic by measrement. An easy way is to se saw signal T(jω), which contains odd and even harmonic components, instead of sine wave G(jω) in ig. 1 and compte (jω) from measrements (jω) with filter (jω) and t (jω) with filter (jω) as t j( t T ) t ( jω) j( t ) t j( t ) = e = e = e. (8) T III. Measrement ncertainties In this section the ncertainty of the generator harmonic distortion determined by (6) is analysed. As the sorce of ncertainties the qantization noise or any wideband noise is assmed. The ncertainty of type B of (6) G is done by ( ) ( ) ( ) ( ) 4 ( ) = (9) 1 G where indices of sign appropriate ncertainties. Since all components in (9) are in the complex form, the ncertainties are in the same form. General spectral component and its ncertainty are given by = e j = e = e, (10) j j rms j j ( j e ) = e ( ) = ( ) = ( ) rms rms ormla (11) decomposes complex ncertainty into rms vale and phase ncertainty [13] ENBW0 rms = q, N. (11) NNPG ( k ) M ( k) N = q (1), (13) where ENBW 0 is eqivalent-noise bandwidth of the window w(n) (the sage of time window for leakage sppression is assmed) and given by [14], N is nmber of samples, q is qantization ncertainty and NNPG is the normalized noise power gain of the sed window w(n): N 1 4 w ( n) n= 0 ENBW 0 = N, N 1 w ( n) n= 0 U S q =, 1 1 NNPG N = w ( n) ENOB 1 N n= 0 (14), (15), (16) where U S is fll-scale range and ENOB is the effective nmber of bits. M(k) is the modle of amplitde freqency spectrm at freqency bin k. Unlike rms vale ncertainty, phase ncertainty is the fnction of freqency (the freqency index is often omitted in this paper for the sake of simplicity). Uncertainties and cold be both determined applying (11). Uncertainty of follows from (7) R R 1 = =, R R 8R 1 4 R ( R R ) ( R R ) 1 R R = (17), (18) 3 where R is the ncertainty of both R 1 and R and done by their tolerance R. Uncertainty of = ( ) comes from (8) sing (11). Uncertainty can be determined from (0) as t, 1 = t (19), (0) = t t = t t 4 t. (1) It follows from (1) that modle ncertainty is independent of freqency; ths Sbstitting (1), (17) and (7) into (19), ncertainty of is t = =. All argments (jω) were omitted in the rest of this paper for the sake of notation simplicity.

4 Sep. -4, 008, lorence, Italy = 1 ( t ) R R ) t. () Similarly as in (11), modles or rms vales can be sed in (). IV. Experimental reslts Practical applicability of the proposed method was verified by experimental measrements on highqality instrments. As signal sorce, ltra-low distortion generator Stanford Research DS360 at the signal freqency of 0.9 kz was applied and as ADC, high-qality NI PXI-59 digitizer, which has the highest dynamic range of any digitizer on the market, was sed at the sampling freqency of 50 kz. ilters and were manfactred sing common metal resistors 3.9 kω with 1% accracy and high-voltage foil capacitor 3.3 n/650 V that proved to be sfficiently linear. Measrements and comptations were performed as described in section II. In each measrement 1 Msa data record was acqired, divided into 63 segments with 50% overlapping, Blackmann-arris 7 term window applied to each segment and amplitde freqency spectrm was compted applying Welch method of averaging. Rms vales of harmonic components were compted from the spectra according to well-known formla [15] and relative phases of higher harmonic components were estimated sing a simple method [16]. or the verification of the proposed method, generators freqency spectrm was measred throgh notch filter that decreased signal dynamic range. Notch filter freqency characteristic was also measred and sed for the correction of the measred freqency spectrm N ; ths, the inflence of notch filter on measred reslts is minimal. Reslts are shown in ig. and Tab. 1. rom the spectrm of it is not clear if harmonic distortion and noise come from the generator or digitizer becase their performance is roghly comparable. The correction revealed that test signal harmonic components dominate only p to 4 th and its wideband noise is below ADC s one. Note that signal and noise levels are shifted p in ig. b, c becase they are expressed in dbfs nits and they correspond to signal level before filtering (and attenating). The reslts of correction are in agreement with the reslts N measred sing notch filter. Only the second harmonic component was estimated with low accracy. The reason was dc offset that was slightly different in each measrement and conseqently the signal occpied slightly different part of ADC transfer fnction. Another sorce of this error was lower difference in filters characteristics and conseqently higher ncertainty (9). This also led to increased noise in the vicinity of the fndamental in the corrected amplitde freqency spectrm. The estimate of some higher particlarly even harmonic components was also biased becase of high noise level relatively to their level. The accracy of reslts was estimated by compting ncertainties according to section III. The ENOB was estimated from the SNR (Signal to non-harmonic ratio) [1] of signal (ig. a) as db) 1.76 ENOB = SNR( (3) 6.0 becase noise spectral density approaches to niform. Uncertainty G was compted in the complex form, the range of modles of generator s harmonic components was compted with the coverage factor of and phase expanded ncertainty with the same coverage factor was determined (see Tab. ). Assming vales of corrected N correspond to generator s characteristics, the compted vales of G with expanded ncertainties are in agreement with the correct generator s vales. The only errors appear at nd harmonic component, which was already discssed, and some weak harmonic components close to noise floor. V. Conclsion In this paper an easy-to-implement method for the assessment of harmonic signal s qality was proposed. It is based on several measrements of this signal passed throgh simple passive filters and posterior comptation of generator s characteristics. The ncertainties of this method were determined for the practical case of incoherent sampling. Practical measrement and their verification proved the applicability of the proposed approach and compted ncertainties of measrements were in agreement with the real vales. owever, not only wideband noise, from which the ncertainties were determined, is a significant sorce of ncertainties bt also instabilities and e.g. dc shift, which cased an increased error in experimental measrements. Practical applicability of this method is obvios for fast and easy estimation of harmonic distortion and noise of generators bt also of ADCs as proposed in [1]. The advantage of this approach is also that harmonic distortion is expressed in the complex form and conseqently can be sed for the correction of generator nonlinearity, too.

5 Sep. -4, 008, lorence, Italy References [1] IEEE Std : IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters, The Institte of Electrical and Electronics Engineers, Inc., New ork, 000. [] DNAD: Methods and draft standards for the dynamic characterization and testing of Analoge to Digital converters, Eropean project SMTC-CT98-14, [3] M. Komárek, V. Papež, J. Roztočil, P. Schánek, Sine-Wave Signal Sorces for Dynamic Testing igh-resoltion igh-speed ADCs, XVIII-th IMEKO World Congress, CD-ROM, 006. [4] V. Papež, S. Papežová, ighly Pre Sine-Wave Signal Sorces for ADC Testing, 15th IMEKO TC 4 International Symposim, Iasi, vol. 1, s , 007. [5] M. Benkais, S. Masson, P. Marchegay, A/D Converter Characterization by Spectral Analysis in Dal-Tone Mode, IEEE Transactions on Instrmentation and Measrement, vol. 44, no. 5, pp , [6] D. Slepička, D. Dallet, V. Shitikov,. Barbara: ADC Characterization in the reqency Domain by Dal Tone Testing, XVIII-th IMEKO World Congress, CD-ROM, 006. [7] R. olčer, L. Michaeli: Testing DNL and INL of ADC by the exponential shaped voltage, ADDA & EWADC 00, pp , 00. [8] J. olb, J. Vedral, J. Kbín: Improvement of Stepp-Gass ADC Stochastic Test Method, ADDA & EWADC 00, pp , 00. [9]. Alegria, P. Arpaia, A. C. Serra, P. Daponte: Performance analysis of an ADC histogram test sing small trianglar waves, IEEE Transactions on Instrmentation and Measrement, vol. 51, no. 4, pp , 00. [10] V. aasz, D. Slepička, reqency Spectrm Correction Method for the ADC Testing, IEEE IMTC 004, pp , 004. [11]. Zhongjn, C. Degang; R. Geiger, Accrate testing of ADC's spectral performance sing imprecise sinsoidal excitations, 004 International Symposim on Circits and Systems ISCAS '04, vol. 1, pp , 004. [1] D. Slepička, D. Dallet, V. Shitikov,. Barbara, Posteriori reqency Spectrm Correction for Test Signal Imperfections in ADC Testing, IEEE IMTC 007, CD-ROM, 007. [13] M. Novotný, D. Slepička: Uncertainty Analysis of the Phase and RMS Vale by Non-Coherent Sampling in the reqency Domain, IEEE IMTC 005, pp , 005. [14] P. Carbone, E. Nnzi, D. Petri: Windows for ADC dynamic testing via freqency-domain analysis, IEEE Transactions on Instrmentation and Measrement, vol. 50, no. 6, pp , 001. [15] O. M. Salomon: The Use of DT Windows on SNR and armonic Distortion Comptations, IEEE Transactions on Instrmentation and Measrement, vol. 43, no., pp , [16] D. Slepička: Phase Differences Determination between the ndamental and igher armonic Components at Non-coherent Sampling, 13th IMEKO TC-4 Symposim, pp. 4 47, 004. Table 1. Reslt comparison harmonic component signal parameter nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th M (dbc) Δ (rad) G M (dbc) compted Δ (rad) N M (dbc) correction Δ (rad) Table. Reslt ncertainties harmonic component parameter nd 3 rd 4 th 5 th 6 th 7 th 8 th 9 th 10 th 11 th G G (dbc) G G (dbc) angle( G ) (rad)

6 Sep. -4, 008, lorence, Italy a) signal (jω): harmonic generator G(jω) filter (jω) ADC D(jω) b) inpt test sine-wave compted by the correction method c) signal N (jω) with reconstrcted fndamental: harmonic generator G(jω) filter N(jω) ADC D(jω) igre. Amplitde freqency spectra (Welch average of 63 spectra)