QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS

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1 Metrol. Meas. Syst., Vol. XIX (01), No. 1, pp METROLOGY AND MEASUREMENT SYSTEMS Index , ISSN QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS Łukasz Zaworsk 1,), Darusz Chabersk 1), Marcn Kowalsk 1), Marek Zelńsk 1) 1) Unversty of Ncolaus Coperncus, Faculty of Physcs, Astronomy and Informatcs, Grudządzka 1/5, Toruń, Poland ( zawor@fzyka.umk.pl, daras@fzyka.umk.pl, markow@fzyka.umk.pl, marzel@fzyka.umk.pl) ) Apator S.A, ul. Żółkewskego 1/9, Toruń, Poland Abstract Methods of tme nterval measurement can be dvded nto asynchronous and synchronous approaches. It s well known that n asynchronous methods of tme-nterval measurement, uncertanty can be reduced by usng statstcal averagng. The motvaton of ths paper s an nvestgaton of averagng n tme nterval measurements, especally n a synchronous measurement. In ths artcle, authors are consderng the method of averagng to reduce the nfluence of uantzaton error on measurement uncertanty n synchronous tme-nterval measurement systems, when dsperson of results, caused by nose s present. A mathematcal model of averagng, whch s followed by the results of numercal smulatons of averagng of measurement seres s presented. The analyss of results leads to the concluson that n partcular condtons the nfluence of the uantzaton error on measurement uncertanty can be mnmzed by statstcal averagng, smlar to asynchronous measurements. Keywords: tme nterval measurement, tme-to-dgtal-converter, uantzaton error, averagng. 01 Polsh Academy of Scences. All rghts reserved 1. Introducton The resoluton of modern tme-to-dgtal converters s stll ncreasng and often s better than 50 ps. In such measurement systems the uantzaton error s often neglgble. Despte ths fact, n a wde range of applcatons the use of advanced tme-to-dgtal converters s economcally unjustfable and the uantzaton error s stll an mportant source of measurement uncertanty. The uantzaton error can be decreased by averagng a seres of asynchronous tme nterval measurements. Smlar methods n A/D converson are wdely used and reported [1-5], though one can hardly fnd such analyss n the tme doman. In the paper the authors present an analyss of statstcal averagng n synchronous tme measurement systems, that, n the presence of nose, can also lead to reducng the contrbuton of uantzaton error to the measurement uncertanty.. Tme nterval measurement methods A tme-to-dgtal converter (TDC) s a devce that converts a tme-nterval T, determned by two physcal events, whch specfy the begnnng and end of the T, to ts dgtal representaton [T]. The most common method of tme-nterval measurement s usng a dgtal counter to count perods of a clock sgnal durng the T. The largest advantage of ths method s ts smple mplementaton and an easly-acheved long measurement range, whch can be addtonally doubled by addng another flp-flop to the counter crcut. The resoluton of ths method s eual to the clock perod T 0. Achevng a resoluton better than 1 nanosecond s problematc, snce a clock sgnal over 1 GHz s reured. Thus, n TDCs, the counter method Artcle hstory: receved on Aug. 30, 011; receved n revsed form on Nov., 011; accepted on Feb. 3, 01; avalable onlne on March 1,

2 Ł. Zaworsk, et al.: QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS s used only to obtan a coarse uantzaton of the measured tme nterval, and s combned wth another fne method whose purpose s hgh-resoluton measurement of short tme ntervals. That method s mplemented n an nterpolator subcrcut and often bases on subdvson of the clock nterval, for example, by usng tapped delay lnes. A detaled descrpton of tme nterval measurement methods and converters can be found n [6, 7]. In spte of the chosen method, the resoluton of a converter s one of the most sgnfcant sources of the measurement uncertanty, especally n applcatons n whch the use of ultra-hgh resoluton TDCs s economcally unjustfable. General purpose TDCs are desgned to measure the tme nterval between two events whch are ndependent from each other and from the clock. As the begnnng of the measured tme nterval s asynchronous to the actve edge of the clock sgnal (whch s the boundary between two nterpolaton ntervals), the measurement s called asynchronous. The most popular methods of nterpolatng an asynchronous tme measurement are the Nutt method wth a counter and two nterpolator crcuts, and the free-runnng counter method, based on a free-runnng counter combned wth a sngle nterpolator, whch are sampled wthout stoppng them [6]. Another approach s the synchronous tme measurement, whch s characterzed by the synchronzaton of the begnnng of the measured tme-nterval T wth the clock sgnal. It can be realzed by startng the reference clock by a physcal event (where t can be mplemented, for example, n systems wth trggered oscllators), or by synchronzng the begnnng of the T to the clock sgnal (n measurement systems n whch the start of T s determned by the system tself). 3. Quantzaton error of a tme-to-dgtal converter One of the prmary sources of error n tme-to-dgtal conversons s a uantzaton process that occurs when the length of the tme nterval T s represented by a dscrete value. A uantzaton transfer functon s presented n Fg. 1. The uantzaton error of tme-to-dgtal converson s descrbed by: e [ T ] T, = (1) where: T s a real value of the measured tme nterval, and [T] s the measurement result (uantzed value). Fg. 1. A uantzer (uantzaton transfer functon). Fg.. A uantzaton error n functon of the length of the measured tme-nterval T. The uantzaton error cannot be evaluated n a measurement, because T s unknown. Generally, e takes values between / and -/, where s the uantzaton step of the tme- 116

3 Metrol. Meas. Syst., Vol. XIX (01), No. 1, pp to-dgtal converter (Fg. ). All of these values are eually lkely to occur, therefore the nfluence of the uantzaton error on the measurement uncertanty s modelled by an unform dstrbuton. An uncertanty of a sngle measurement can be estmated as the standard T S devaton of the unform dstrbuton of wdth and s gven by: / T S = = = = 0,89. () 3 1 If the measured tme nterval T s asynchronous to the clock, the begnnng and the end of t are both descrbed by uncertanty, thus n ths stuaton the resultant uncertanty of the measurement T A s larger than n the above-mentoned stuaton: T = + = = 0,408. (3) A It s a well known fact that the uantzaton error can be reduced by averagng technues [6, 8], provded that the measured tme nterval s asynchronous to the clock sgnal, as n Fg. 3. For the analyss of such measurement, T can be dvded nto two parts: T a - between the start of the T and the begnnng of the next uantzaton nterval, and T s - between the and the end of the T. Accordng to t, the tme nterval T a takes random value between 0 and, so the length of T s has an unform dstrbuton n the range ( T, T ). In that case, the value obtaned from the uantzaton can be [T] 1 or [T], whch corresponds to the j-th and j-th+1 uantzaton ntervals. In a seres of measurements of sze N these two results are obtaned wth the numbers N 1 and N respectvely. By averagng the measurement results one can obtan the true value of the length of T. To obtan the proper value of T one must add the / - mean value of the offset T a whch s beng omtted by hardware whle measurng: The measurement uncertanty ˆ N N T = [ T] 1 + [ T] +. (4) N N 1 T s gven as follows: 0,89 T = =. (5) N N Fg. 3. Asynchronous measurement of tme-nterval T. 4. Mathematcal model of uantzaton error n the presence of nose In a stuaton when the start of T s synchronous wth the clock, T a = 0 and all measurement results have the same value, thus averagng does not lead to a reducton of the uncertanty caused by the uantzaton error. However, an analyss of the above-mentoned 117

4 Ł. Zaworsk, et al.: QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS asynchronous measurement may lead to the concluson that the averagng method could be successfully used when a length of the measured tme nterval s dsturbed by nose, even when the begnnng of T s synchronous wth the clock sgnal. In the case when the measured tme-nterval s dsturbed by nose δ, ts length T n can be modelled by: T n =T +δ, (6) where the dstrbuton of δ s gven by ψ(t). One can notce that T n euals T s, as can be seen n Fg. 4. Fg. 4. Synchronous measurement of a tme-nterval dsturbed by nose δ, gven by rectangular dstrbuton R(0,). Converson of the tme nterval T n to the dgtal value [T n ] s modelled by the uantzaton process, analogcal to one presented n Fg. 1. The result of a measurement seres s then averaged. P The probablty p that the length of T n s n the -th uantzaton nterval (between and +1 ) s eual to: + 1 The expected value of the uantzed measurement seres s: E([ T ]) = [ T ] = n n p = ψ ( t T ) dt (7) N p Q = N = 1 = 1 1 An analogcal euaton descrbes the expected value of E([ T ] n ) = N N pq = = 1 = 1 1 ψ ( t T ) dt Q. (8) T : ψ ( t T ) dt Q. (9) On the bass of these relatonshps (7, 8), one can derve the expected value of the uantzaton error e, descrbed by the followng euaton: e N = [ Tn ] T = ( t T ) dt Q T, ψ (10) = 1 1 and the standard devaton T of the seres of measurements: N N ([ ] ) ([ ]) ( ) ( ). 1 1 T = E Tn E Tn = ψ t T dt Q ψ t T dt Q (11) = 1 = 1 118

5 Metrol. Meas. Syst., Vol. XIX (01), No. 1, pp The presented model can be used to research the nfluence of nose, determned by ψ(t), on the average value n a synchronous tme measurement. Although for partcular dstrbutons ψ(t) euatons (10) and (11) do not have analytcal solutons, the values of systematc error and standard devaton can be numercally calculated n all cases. Fg. 5. The uantzaton error e of the expected value of the seres of measurements as a functon of the real value T of the measured tme-nterval for nose dstrbuton ψ(t)=r(0,a). Fg. 6. The uantzaton error e of the mean value of the seres of measurements as a functon of the real value T of the measured tme-nterval for nose dstrbuton ψ(t)=n(0, n,). 5. Analyss and results Basng on (10) and (11), the nfluence of nose on the uncertanty of measurements can be nvestgated. Fg. 5 Fg. 8 present numercal results of modellng of nose nfluence on the uantzaton error and standard devaton of seres of measurement, obtaned from numercal estmaton. To analyze the behavour of the uantzaton error n systems wth synchronous measurements, one shall analyze maxmum and mnmum values of the error for varous T. If the rectangular dstrbuted nose (ψ(t)=r(0,a)) s used, a relaton of the uantzaton error s presented n Fg. 9. One can notce that for a=n, where n s a natural number, mn(e ) = max (e ) = 0. For n=1 the stuaton corresponds to the asynchronous measurement (Fg. 3). 119

6 Ł. Zaworsk, et al.: QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS Fg. 7. Standard devaton T of a seres of measurements as a functon of the real value of the measured tme nterval T for nose dstrbuton ψ(t)=r(0,a). Fg. 8. Value of the uantzaton error T of a seres of measurements as a functon of the real value of the measured tme nterval T for nose dstrbuton ψ(t)=n(0, n,). If the uantzaton error s zero, the uncertanty of measurement depends only on the standard devaton of measurement results. Ths stuaton s presented n Fg. 7. The standard devaton T for a=, a=, a=3, as a functon of T s presented n Fg. 10. One can observe the well known sem-crcular characterstcs for a=, whch was dscussed n [9-11]. 1 The RMS values of T are eual respectvely to : 0,408=, for a= (whch conforms 6 wth E. (3)), 0,6493 for a=, and 0,9165 for a=3. The uantzaton error, where nose s gven by normal dstrbuton N(0, n,), s presented n Fg. 6. The mnmum and maxmum values of the error are presented n Fg. 11. One can notce that for n >0,7 the uantzaton error e does not depend on T, and ts value s neglgble (less than ). In such a case, the uncertanty of measurement depends only on the varance of the measurement seres. 10

7 Metrol. Meas. Syst., Vol. XIX (01), No. 1, pp Fg 9. Mnmum and maxmum values of uantzaton error as a functon of wdth of rectangular dstrbuton of nose. Fg 10. Characterstcs of standard devaton of a measurement seres for dfferent wdths of rectangular dstrbuton. In Fg. 1 the mnmum and maxmum values of standard devaton T of a measurement seres as a functon of standard devaton of normal dstrbuton n are presented. If n >0,7, T s ndependent of T and the uncertanty of measurement can be estmated as a standard error of the mean: N ([ T ] [ T ]) = 1 n n S =. (1) T N ( N 1) Fg. 11 The maxmum and mnmum values of the uantzaton error, for nose dstrbuton ψ(t)=n(0, n,) as a functon of n. Fg 1. The maxmum and mnmum values of the standard devaton, T, for nose dstrbuton ψ(t)=n(0, n,) as a functon of n. 6. Summary The am of ths artcle was to dscuss the averagng of the results of tme nterval measurements. The mathematcal model of averagng for both synchronous and asynchronous measurements s presented. The results of smulatons for asynchronous measurements correspond to theoretcal approaches presented n [9-1], but are sgnfcantly more general. In case of synchronous measurements, the model allows to research the nfluence of dfferent types of nose on results of averagng. 11

8 Ł. Zaworsk, et al.: QUANTIZATION ERROR IN TIME-TO-DIGITAL CONVERTERS The most nterestng case dscussed n ths paper s synchronous tme-nterval measurement n the presence of nose wth a normal dstrbuton N(0, n,). When the standard devaton n s greater than 0,7, the uantzaton error s nsgnfcant, and the measurement uncertanty depends only on the dsperson of the measurement seres. The presented results of studes can be appled n error analyss of synchronous measurement systems. Acknowledgements Ths work was supported by the Polsh Mnstry of Scence and Hgher Educaton N N References [1] Waanenaker, R.A., Lpshtz, S.P., Vanderkooy, J., Wrght, J.N. (000). A theory of nonsubtractve dther. IEEE Transactons on Sgnal Processng, 48 (), [] Krause, L. (006). Effectve uantzaton by averagng and dtherng. Measurement, 39, [3] Wdrow, B., Kollar, I. (008). Quantzaton nose: roundoff error n dgtal computaton, sgnal processng, control, and communcatons. Cambrdge Unversty Press. [4] Alegra, F.C. (009). Study of the random nose test of analog-to-dgtal converters. Metrology and measurement systems, 16(4), [5] Lal-Jadzak, J., Senkowsk, S. (009). Varance of random sgnal mean suare value dgtal estmator. Metrology and measurement systems, 16(), [6] Kalsz, J. (004). Revew of methods for tme nterval measurements wth pcosecond resoluton. Metrologa, 41, [7] Henzler, S. (010). Tme-to-Dgtal Converters. Sprnger Seres n Advanced Mcroelectroncs Seres, #9. Sprnger-Verlag New York, LLC. [8] Fundamentals of Tme Interval Measurement (1997). Hewlett-Packard App. Note [9] Baront, F., Fanucc, L., Lunardn, D., Roncella, R., Salett, R. (001). On the dfferental nonlnearty of tme-to-dgtal converters based on delay-locked-loop delay lnes. Nuclear Scence, IEEE Transactons on, 48(6), [10] Szymanowsk, R. (006). Quantzaton error nfluence on measurement uncertanty of nterpolaton tme counter. Measurement, Automaton, Control, 9 bs, (n Polsh) [11] Zelńsk, M., Kowalsk, M., Chabersk, D., Grzelak, S., Frankowsk, R. (008). Measurement system of oscllators' phase fluctuaton and ts applcatons. Electrcal Revew, 5(84), (n Polsh) [1] Chabersk, D., Zelnsk, M., Grzelak, S. (009). The new method of calculaton sum and dfference hstogram for uantzed data. Measurement, 4,