and t, = t, + Tt vitk :K. small L


 Frank Mark Ramsey
 9 days ago
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1 Much of the work of the Cov.nders of this meeting has dealt with the c2laracteri~at.icn cf fr~clucficy :;ources by time r!orrtairl nciisurcments. :'ypi.cailyr the sources are offset tc prrjvide a low frequentj beat signal, statyistics of whose perifid are ccllecteel. Phile tkc transformaticn of tylese time domain statistics is available to the user ic manageable fom, tlic actual bea.t measurenc~t technicjue has some unexanined 1irr.itations. Let us look at the relation of errcrs %.r, this measurement tc thcir freaucncy st~hjlity interpretaticn. We wish to express thc difference betw~cri two c~r~secutiv~ frequenc)? measuremerits in several forms. First we have: 1 since f = '. t  and t, = t, + Tt vitk :K. small L 1 then Af = 7 Lt L C Where t is the period measurement and: kt is the random error in this measurement. We must rememher that our measurements are actually averages over the measurement interval; each quantity must he seen as a time average. Iknce  t represents the nominal. beat period, and At represents the timing deviation normalized to a single period. 
2 If the beat and averaginq periods are both one second, the distinction is hard to see. If vre choose a tenaeriod averaqc of the one second beat period, and further assume that only system timing errors cause the deviations, thcn the difference is clear. If our timina error for one measurement is one microsecond, then the normalize3 error over the ten periods is only 1 microseconds, over 100 periods 01 microseconds, For a fixed beat period, the effect of timing errors gces down proportionately to the number of periods averaged. Suppose that the beat period is chanqcd to ten seconds and the same timing error applies. In this case the A t term is not decreased, since only one period is averaqed. The increased period, however, has a quadratic effect. The & for the first case is: for the second: =lo *   f  2 f (LO seconds) or in general: Where c is the beat period N is the number of contiguous periods measured At is the uncertainty of each measurement same averaging time. (Measurement interval) This would lead us to conclude that a longer beat period would yield better results. Unfortunately, for a sine wave beat signal, increasing the beat period proportionately decreases the slew rate which, in turn, proportionately increases timing error for the standard tize our period measurement to slew rate. The timing errors fall into two categories: those of much higher frequency than the L IIz signals of interest, and those of somewhat Lower
3 frequency. The first corresponds to hiyh frequency noise which can be seen simply as thermal noise at the input of the period measuring counter. The effect of this noise is well. kfiown and its effect on low slew rate neasurements is ofterr specifieci in corrnercial counters. Suppose we have geneaatez. a i FY heat signal 1C volts in mlplitudc. Its slew rate at the zero crossi~ys is abc~ut 36 voltsj scc. Fence a voltaye excursion of 3G microvnlts corresponds to an Error in the perioc! measurement of 1 microsecond. Crle comnercial counter cites an uncertaj.nty of 100 us for such an ir?put. Tf two AC FIz sources are bclirlcj compared, this imrlies a noise floor of 1 ~ for ~ the frequency ~ ~  stability measurement (10 periods measured). The second scurce of error corresponds to Ionqer tern changes in the operatinq point of the measurement system. A ~ood example is a tj.mc varying ir;put offset ~~Ltacje of a zlero crossifig detect.0~ or Schmitt trigger. T11i.s offset may cone fron the counter input stage, a preccc?i.nq amplifier or even the. balance2 rnixe.r. It: is oftcri affected hv long term chanqes in the environ~ent of the rrieasuremcnt system. The change i.n this offset corrcsponc!~ to a chancje in triqqer pint which, a:; see11 in F'iq. 1, rcpresents an over Gr under estimatiur, of the period. If we could somehow track a feature of the wave for^^'^ geometry, such as the peak, rather than a Level, we could acquire imunity to such lonq term drl.fts. Proposed Circuit The integrating A/D convertor framework provides three features to achieve this end. It establishes zerocrossings by inteqration, can cancel DC drifts over intervals lonc;er than onehalf the beat period and eliminates the deadtirne associated with traditional counter regimes. Consider the circuit of Fiq. 2 with a 1 17:. sine wave input (1.f 10 volts peak to peak. Suppose that the integrator is skinrted initia1l.y and allowed to operate when the sigrlal Sevul first reaches .6 volts. \"%en the inteqrator output returns to zerc, it is ret~rned to the shorted condition. If the innut is ar, icieal noiseless sine wave, then the integrator output waveform is symmetric about its peak value, which is analogous to the input's smetry about its zerc crossinq. The best estimate of the zero crossing tine Is a period following the .6 volt trigger time and equal to onehalf the duration of the inteqrating interval.. Clearly there is nothinq sacred about the .6 volt level. It merely provides an arbitrary starting point for the svmmctric interval. Jf the input is corruptce by noise, any components with zero mean and period less than the integrating interval will tend to be rej ectcd. These are precisely the hifiher frequency components which cause the yate to flap when a conventi.ona1 zero crossing detector confronts a low slew rate signal. Notice that this technique always estimates the true zero crossing of the input. If there is a residual DC offset to the input, then the trigger point will be skewed relative to the true mean value of
4 the input wave shape. Nonetheless this solves one of the difficulties of timing the slow rise time signal, even though it exhihits the sa.me vulnerability to time varying offset as conventional means. It js possible to perform this operation on both rising and falling  edges of the input. Takinq the midpoint of these two zero crossing estimates has the effect af rejecting DC offset altogether. It might be viewed as a best cstimate of the peak of the sine wave input. Whether it actually corresponds to the peak or not is not significant. Wkat this process does in fact accomplish is to establish the timing of a fixed point on the geometry of the wavefom, so that our period mcasurement is actually a measurement of repetition of geometry rather than of some absolute level. While a varying UC offset certainly affects the zero crossings, it would not affect the timinq between two consecutive peaks. This technique is vulnerable to changes in offset level which occur between the two integration intervals, but is able to reject such changes outside them. If we are measuring single periods, this method will reject a linear drift occurring over half the cycle. If we are measuring the length of a ten period sequence, it will reject level changes occurring over 91/2 of those periods. Since systematic drifts are most likely to occur ovcr long measurement intervals, we find that this technique is extremely effective in removing them. This is not the case with a sinple period counter techique, which counts clock pulses over the ten periods and suffers a trigger error equal to the total DC level change times the slew rate of the input. We have actually formed a bandpass filter for the 1 Hz signal. Since the 10 volt peaktopeak signal will slew from .6 volts to +.6 in about 40 milliseconds, we have a high frequency cutoff of 25 Hz. (Actually sln x a response 3 db at 12 Hz with a first null at 25 Hz.) This is X sufficiently higher than the preceding amplifier's bandwidth that the system noise bandwidth is substantially unaffected. The low frequency response rolls off at 6 db/octave below one half the beat frequency. Notice that the information content of the 1 Hz signal is not affected. Its information is an FM modulation. Changes in frequency ovcr 1000 second intervals are present as.001 Hz sidebands of the 1 IIz carrier, not as direct 001 Hz signals. Hence we have prevented systematic offsets from irreversibly mixing with the Longterm frequency effects we seek to measure. Construction Details It was most desirable to implement a synchronous system to preserve resolution and also to minimize the ill effects of attempting to derive logic triggers from slow analog signals. This is, after all, the fuildamental problem for which the, proposed processor promises solace. The entire operation is that of a fourstate machine, whose states may be
5 Before describing the sequencer in detaii, T would first like to cxplai~ the overall means of obtaining z peri~d measurement. Consistent vritll mv qoal of a 2:erfi "dead time" system and my qfials of simy~licity and low cost, an il;terpolati.on technique is used. A counter with a 1 kf:z clock (mv "coarse" resister) is used in nrecisely cycle). Its slow clock rate permits it to be strobed and reset on the fly, so that no pericd goes u~m~asured. This counter j.s rjossesscd of all the flaws previously described. Another reqister, the "fine," is cycles are done, the timing from the somewhat arbitrary, but nonetheless sy~~cbronous, .6 volt point t.o the midpoint of thc positive half cycle of the wave. As rr.entio:~ed earlier, this point is referred to the qeorr, etry c)f the waveform anc iis UC level independent.. The best estimate for a given period is found by adjustinq the coarse Limit,? by the interpola tions heforc and after it. determined nlll per?od to the nest cycle's 17idpoint. Slncf: the output of the system is alternatifiq readouts of tf and t, it is a small matter to C arrive at a runninq read~ut of actu(11 periods. Let us now considcr the operation of the inteqratcr. I State A of Fig. 3 is the primordial state, the preinterpo1ati.n~ mode. The coarse counter i.s n3kinq its measurement and the irlterjrator is held in an autozero mode. This has an advantage over: simply shorting the integrator in that it nulls out the inteyrator's offset voltaqc as well. Since the integrator never works for more than onehalf second after an autozero, we can expect that longterr;, systematic changes in the inteqrator itself are well below the total systerr noise Fl.oor. State B is the estimation of tlie posititre zero crossing time. It is entered when the input first exceeds .G vol.t, sync11ronousj.y with the 1 khz coarse clock. This prevents gross i millisecond quantization error. At this moment the contents of the coarse coucter are strobed into the output latches and, the coarse counter is reset to zero, long before its next L khz clock edge. Thus the entry point of State R has an uncertainty range of k 1 millisecond. At a 30 volt per second slope of the input, this is only a voltaqe band of + 30 millivolts about .6. The absolute time is unimportant as all residues are absorbed into the I
6 interpolation measurement. \?hen the inteqrator output is driven back to zero, the symmetric interval about the true zero crossing is over. This event is determined by a comparator which senses inteqrator output. Notice that the comparator is not included in the autozero loop. Its trigger point is actually biased 1 volts below zero. This prevents the oscillation which usually accompanies attempted linear operation of a comparator. This also provides a well defined comparator output. Tt will always be low during autozero and during positive inteqrator output. Thus it goes hiqh only after the integrator has in fact crossed zero at the end of B. Since the integrator is slewinq approximately as fast as the input here, there is a delay or 5 milliseconds until the comparator trigger point is reached. The comparator has a window of about 2 millivolts around its nominal trigger point within which noise will cause multiple transitions. Since the 5 millisecond delay is uniform from measurement to measurement and is only affected by changes in the slope of the input, and since the noise at the inteqrator output is quite small. we lose nothing to talre the first comparator transition as the integrator output zero crossing. The net delay is subtracted out in the period determination algorithm and is uniform from period to period. Thus the comparator causes the system to enter State C, which is almost identical to A. The integrator is autozeroed, and the coarse counter is still running, but the duration of State C is beinq timed in the fine counter. State D is the exact analague of State n, only it functions on the falling slope edge of the input. It is triggered by the input signal reachinq +.6 volts (with no synchronization required this time). his state defines a time interval symmetric about the zero crossing and passes to State A when this is done, with the same convention on the comparator as State G. To construct an interpolation interval from the durations of States B, C, and D is a simple matter. Suppose we assign the entry into State B as our time origin. This is also the timing edge of the coars: register, so it is the logical place to begin an interpolation. Clearly the time to the best estimate of the zero crossing (rising edge) is half the duration of State B. Similarly, the time to the best estimate of the zero crossing of the falling edge is the sum of the durations of B and C and half of D. Simple algebra yields a time to the midpoint of the zero crossings and shows that a counter clocked with relative rates of 3:2:1 during B, C, and D respectively will yield this same interval. (See Appendix 1). Specifically, rates of 15, 10, and 5 MHz will allow interpolation to 100 nanoseconds. CMOS clocking rates Limited me to one quarter of this resolution. (See Fiq. 4) Thus we see how the interpolation is actually generated for the 1 Hz signal with a measurement of 1 secunti. Multiple period measurements 490
7 I required adding a period countcr to thc triqqcr GE State E. This way an interpolation will he t3erf~~~)ez ilnil:iallv and on the final rxriod nt the rlumbcr seiect~ci, an? not intersrcninu neriocls. Preli.rr~narv results Time permittc? or:ly a crud evtiluation of this teclmique, but the test signal was derived from a ' YE quartz ~scillator locked to a hydroger! maser. T t was shaped to approximately sine form with 10 volt peaktopeak aml~litude and some white noise added. Three cases were qai.n 1 imiter, and the circuit of this paper. Only sinqlc oeriod mcasurenents were taken. '?he results are expressed as a Fractional "rcqucncv referre2 to 10 MHz and arc the root of an Allen variance for 100 trials, and mav be compared to results of Illan [l] ant? Ycznhardt 121. A slow rate of 30 volts/sec applied to an HI' 5327R c,lose to thc nr'edicted error The HP 5327B preceded by a qain of looc limiter, effcctivcl.,r i~creasinq slew rates to 30,000 volts/sec Although the proposed circuit clcdr.l?i does have a lower noise floor, it: is hard to say by how much. These results renresent. a. noise floor approximately 5 times hiqher than Fillan and 5i( ttimcs higher than Reinhardt. Three factors come to mind. F!C mav be nearing the ncise level of the test signal itse1.f. Elere snnc furthc? measurements would be useful. Secondly, the clockin? rate userl 53 i~ this "  rr,c~del yiel.ds a minimum relative quantization error of 2 :< 1.0. Thirdly, good low noise circuit. techniques, as exempl.if ied i.n both Rl La E2 and Reinhardt, may provide lower noise floors. Nonetheless, the TO level is quite respcctablc for a low cost a1ternat:ive to a dedicated frequency counter
8 Response to Svstematic Error This is a fairly coarse test also, introducing about a 1 volt offset Tor two out of every ten periods of the 1 Hz 10 volt peaktopeak input. Care was taken to make the transitions far from zero crossings so as to avoid changing trigger conditions. For a signal with a 20 volt per second slew rate we would expect a 50 millisecond timinq error bases purely cn the zero crossing detector technique. Only the hiyh gain limiter alternative was measured, as it would yield the same data as for the counter alone, but with less noise. Actual values of 42 milliseconds were observed. They occurred symmetrically since the longterm average frequency did not change. Figure 7 shows some results for a step change representing dri.ft during the integration period. Note that the counter's "dead time" yields only half as much data. It gi.ves three normal periods, followed by a period 42 milliseconds too lonq, followed by one 42 milliseconds too short, then back to three normal periods. Looking at the processcr output, we see seven normal periods, one 20 milliseconds too lonq, one normal, and finally one 2G millisecvnds too short. Notice how the full error is present i,n the coarse register and half is compensated by changes in the interpolation register. These and other data show a factor of 25 suppression of DC offsets occurring outside the integration period. Actual performance may be considerably better with the much smaller perturbations one would expect in a practical system. Note that in Fig. 7, due to the lowered clack rates, the period is determined by: Conclusion Even in coarse testing, the circuit has achieved the goa.ls of no.ise floor reduction, reduced sensitivity to systematic drifts, and no dead time. This Level of performance was achieved by a prototype wherein little attention was paid to Lownoise techniques. Standard CMOS and BIF'ET parts were used. No shielding or separation of analog and digital grounds were attempted. The output of the prototype was 7 decades of BCD information from three registers which could have easily been transmitted through the IEEE bus through an appropriate interface. All subsequent calculations and choices of averaging time could be left to the minicomputer or calculator controlling the bus. It is interesting to note that the net cost of this technique is considerably below the counter it outperformed.
9 kcknowledycmer.t This work was performed while the author was wit11 the Hvdroqen!laser Lab of the Smithsonian Pstrophysical nlqervatory and als~ rcyresents a portion of a blaster's Thesis submittea to the Departmefit cf Electrical. Lnyinecring an5 Computer Science at tr.i.t. (June 1977). The initial concept of using the A/' convcrtcr' tc?~loloyy is due to Professrr James Foborge of F.I.T. P.eferenccs [1] Allan, David W., "Pcport on yt3f "ual Vlxer "ire niffercnce Zvstc~", NBSZE 75F27. [2] Relnhardt, Vl ctor dnd Lonaho~, "hcxcsa, "A Slm~le Vechnlrpe for PI ql1 Resolution Time Lorwin Phase yhols~ ~qeasurernent". Pr~cerdinas of thc 8th Annual PTT1 Yeeting. NASIT, GSF
10 P OVERESTIMATED PER l OD L PLRlOD ESTIMATE UNAFFECTED Figure 1. ITFECT OF BC OFFSET Figure 2. SWITCHED INTEGRATOR 494
11 STAT : D 5MHz STATE C 1oWi~ STATE B FINE RE=STE.R 7 DEr4'DES I5 MHz FINE STROBE COARSE STROBE l KHz i COARSE. REGIST E:!? 7 DLC'mES Figure 4. GATING SCIIEFX
12 Figure 5. PfiIILOG SECTION 496
13 '"1 0 * 1/ lv COMPARATOR OUT AUTO ZLRO Figure 6. TNTEGRATOK SCHEMATIC
14 COUNTER WITFi LIFIITCR DATA At from 1.0 second in 1,s T x lo' sec ' (? PROPOSED CIRCUIT (with 1 volt drift during one period) T in 2 x,10 sec T: in sec G OOlOO Fiqure 7. EF'FECT OF SYSTEMATIC OFI?SI?TS
15 I APPENCIX: Interplation Interval t mark the cntrar>ce of Ftate r 1 t? mark the entrance of State r t, mark the end of D Note that the system is synchronized so t alsc marlrs an cyc,ch of tile coarse r' This is the best estimatc of the time of the ~ositive zerc crossing (referred to t ) ) Identically State D gives the best estjmate of the neriative rart? 4) The "peak" is estin~tea by the mi$.pcint of the twc zero crossinqs
16 substituting: t i = tr 6) Substituting into the result of 4) hence the ease of computation by measurj.ng the state durat~ons wltn clock sates of 3:2: 1.
17  QUESTIONS AND ANSWERS DR. VICTOR REINHARDT, NASA Goddard Space Flight Center: I think it is a very interesting approach, especially the sumr~iation of the positive and negativc eclqc. We had some later data from our systeman equivalent dualmixer system, In short terrr we did beat you, but we did notice diurnal variations equivalent to 20 picoseconds due to terriperature effects. We traced it down to the mixer itself and not any of the componentsthe DC offsets in the mixer. So if Bob Vessot is going to give us a oscillator, we certainly need approaches similar to yours to get out the drifts in the measurement sys teiii. DR. ELOMBERG : Thank you. MR, HERMAN DAMS, National Research Council, Canada You mentioned the probleni of dead time in the dual balanced mixer system. In the system we built at NRC about three years ago, which was similar to Dave Allan's, we overcame that problem by simply having a counter with a dual front end so that you count continuously, You take your readout, of course, at the end of the onesecond period from the front end counter, which at that time has stopped, The other one was still countinq. DR. BLOMBERG: I niostly brought that up as d practical issue for systems that would be irnpleniented with co~iiniercia1 ly avai lable counters. I didn't niean to connect this w ~ t q that effort.,;o 1