208
`
`IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-32, NO. 1, MARCH 1983
`
`Subharmonic Sampling for the Measurement of
`Short-Term Stability of Microwave Oscillators
`
`NEIL D. FAULKNER AND ENRIC VILAR I MESTRE
`
`Abstract-The use of digital techniques for the precise measure-
`ment of medium- and short-term frequency stability of oscillators
`requires good resolution of the frequency counter. In the case of mi-
`crowave oscillators, this is often preceded by some form of down-
`conversion. This paper is concerned with the theoretical principles,
`description, operation, and performance of a down-conversion system
`based on the subharmonic sampling technique. The system is intended
`for use on microwave oscillators with good frequency-stability per-
`formance and has as main features the use of a single frequency-stable
`reference oscillator and a pulse generator-sampler which allows power
`optimization of the subharmonic selected. The complete experimental
`system is presented and the limitations are discussed.
`
`SAMPLING
`GATE
`
`NARROW
`BAND
`RECEIVER
`
`ANALYSIS
`
`PULSE
`REFERENCE
`OSCILLATOR
`GENERATOR
`Basic principle of sampling, recovery, and analysis of an
`Fig. 1.
`oscillator.
`
`bandwidth tuned receiver or a phase-locked tracking re-
`ceiver can recover that signal with sufficiently good signal-
`to-noise ratio for further analysis. Having recovered the signal
`at a low-frequency range, counter resolution increases and
`standard digital techniques and numerical algorithms can be
`used to full advantage, e.g., [4], inclusive of standard Fourier
`analysis. It is perhaps interesting to note that subharmonic
`sampling is sometimes used to phase lock microwave oscillators
`to a reference of lower frequency, a process known as sampling
`phase locking [5], [6]. The ratio carrier-reference frequencies
`usually do not exceed a factor of 50; in fact, they are often less.
`In the technique described in this paper, the ratio of the
`frequencies may theoretically be up to about 104 and the ex-
`perimental system described has allowed up to a factor of
`600.
`
`I. INTRODUCTION
`pRECISE MEASUREMENT of the degree of spectral
`purity of oscillators using digital techniques and numerical
`algorithms leads to the requirement of frequency counters with
`good resolution [1] in order to resolve the small frequency
`variations of the signal. These counters are of the reciprocal
`type [2] and allow, in general, the use of a variable averaging
`time or observation time T. Fixed values of r can also be used
`but this requires correction for the dead time [3] if it is sig-
`nificant.
`Good resolution becomes increasingly difficult as the fre-
`quency of the oscillator increases, and this is particularly so
`in the microwave region. In that region, mixing with a mi-
`crowave source of higher degree of spectral purity than the one
`under investigation is a standard procedure. Alternatively, one
`can use a similar statistically equivalent microwave source. In
`both cases, however, one is usually limited by the need to
`synthesize another microwave source. This may be neither easy
`nor convenient.
`This paper describes theoretical and experimental work
`which has been carried out so as to use a single oscillator,
`usually a 5- or 10-MHz crystal oscillator with excellent per-
`formance of short- and long-term stability, which drives a
`subharmonic sampler. This is then followed by a tracking re-
`ceiver to recover the down-converted microwave signal. The
`system concept is illustrated in Fig. 1. It will be shown that
`although the subharmonic or down-converted replica of the
`oscillator microwave spectrum can be at various tens of deci-
`bels below the original level of the signal, a moderate noise
`
`II. MEASUREMENT PRINCIPLE
`A. Subharmonic Sampling
`The operation of the measurement system outlined in Fig.
`is based on a process known as subharmonic sampling,
`1
`whereby the oscillator microwave signal of interest is sampled
`at a rate lower than the nominal carrier frequency. This results
`in replicas of the signal spectrum symmetrically positioned
`around the various harmonics of the sampling function. The
`net result is a form of down-conversion and the recovery of the
`original microwave signal spectrum can be carried out by
`bandpass filtering one of the replicas, followed by suitable
`Manuscript received July 23, 1982. This work was supported in part by the
`narrow-band amplification.
`U.K. Science and Engineering Research Council (SERC).
`The analysis can be carried out by considering real pulses
`The authors are with the Department of Electrical and Electronic Engi-
`of trapezoidal shape with finite rise and decay time T, and
`neering, Portsmouth Polytechnic, Portsmouth, United Kingdom.
`0018-9456/83/0300-0208$01.00 © 1983 IEEE
`
`RPX-Farmwald Ex. 1040, p 1
`
`

`

`FAULKNER AND VILAR I MESTRE: SUBHARMONIC SAMPLING FOR MEASUREMENT OF MICROWAVE OSCILI ATORS
`
`209
`
`-W
`
`-nw
`
`Fig. 2.
`
`-W
`Trapezoidal pulse and example of sampled spectrum ((5), n0p, =
`15, r = 3, and T, = 7rT/n0Ptwo)
`
`w
`
`duration r, as shown in Fig. 2. This model not only closely
`resembles the pulse observed in the sampling oscilloscope, but
`leads to spectral levels which agree well with the experimental
`values. The sampling function PT(t) which represents the train
`of trapezoidal pulses of periodicity T, can be written as
`
`PT(t) = L s(t - nT)
`n =-
`where s(t -nT) is given by
`s(t - nT) = 1,
`
`T
`T
`nT--< t < nT+-
`2
`2
`
`s(t- nT) =-t+ I + 2T- nT--2- Ts < t < nTs-
`2Ts
`s(-T)=T+
`2
`
`tT
`
`T
`
`(1)
`
`T
`
`2
`
`Tr
`s(t-nT)=+lI+
`TS 21Ts
`
`t
`
`Tr
`'- nT+ -< t <nT + -+ T
`
`2
`
`2
`
`(2)
`
`elsewhere.
`s(t - nT) = 0,
`Expanding in Fourier series we obtain
`(2 sin nwo(r + Ts) sin nwoT,5 ejnwot
`2
`TTsw ny-k
`
`PT(t) =
`
`G
`
`2
`
`2
`
`n
`
`with w0 = 27r/ T, from which one readily derives the amplitude
`
`spectrum of the trapezoidal train
`P(jw) - 27(r + T,) |
`
`)
`
`-
`
`sin no(o(T + Ts)/2 sin n.oT,/2
`flnWo(T + T,)/2
`nOoT,/2
`n=
`[8(w - ncw,o) + b(o + n(4)])
`
`(4)
`
`The frequency spectrum F,(jw) resulting from sampling
`a signal of spectrum F(jw) with our train of pulses is then
`obtained by convolving P(jw) with F(jw). Operating and
`rearranging we obtain
`
`F(jw)=
`
`T
`T
`[FF
`L\
`!
`. FU(co
`
`sin nwo(T+ T?)/2 sin ncooTj12
`G+
`ncolo(T + TS)/2
`ncooT,/2
`n=l
`ncoo)] + FU(co + nco)]
`
`-
`
`(5)
`
`It is interesting to compare this result with the familiar ex-
`pression obtained for the case of rectangular sampling pulses
`with zero rise and decay times. For this case
`
`RPX-Farmwald Ex. 1040, p 2
`
`

`

`210
`
`IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-32, NO. 1, MARCH 1983
`
`F,(jw) = -
`T
`* F(jw) + E 'in nwOT/2 [Fj(w - nwo)]
`nfooT/2
`n=1
`
`+ Fj(w + nwo)]IJ.
`
`(6)
`
`Comparison between (5) and (6) shows that apart from a
`slight modification in the common weighting factor r/T,
`which has now become (T + T,)/T, the various spectral rep-
`licas located at the harmonic frequencies w ± nw0 are now
`multiplied by two terms of sin xlx form. This combined de-
`pendence given by (5) is illustrated in Fig. 2.
`
`B. Optimization Techniques
`Equation (5) reveals that by an appropriate choice of the
`pulsewidth T, one of the maxima of the spectrum envelope can
`be made to coincide with the replica of interest. To do this we
`note that
`
`Fig. 3.
`
`Pulsewidth for maximum subharmonic power.
`
`a)
`
`LO
`
`z 0I
`
`L-
`
`LL
`
`Fig. 4.
`
`PERCENTAGE VARIATION FROM OPTIMUM PULSE WIDTH
`Power sensitivity to variations in the optimum pulsewidth.
`
`(9)
`
`repetition rate, T, must not exceed 14 ps. This is within the
`capabilities of current technology.
`With all the above points taken into consideration, the final
`level of the subharmonic signal related to the original micro-
`wave signal level is
`Ploss(dB) = 20 log1o (n7r) + 20 log0o sn - T/T
`sin (n7rTSIT)
`The second term is usually much less than the first. In ad-
`dition, the sampling of a signal of a few gigahertz with a
`5-MHz train leads to n of about 1000; that is, losses of 70 dB
`or more are to be expected.
`C. Recovery of the Sampled Spectrum
`We have shown that the power in the replica spectra de-
`creases ( I/n r)2 with increasing order number n even in the
`case of an optimized pulsewidth. In order to be able to recover
`the sampled spectrum for use by the rest of the measurement
`system, we must consider the noise power that is also present
`within the noise bandwidth B,, of the filter (receiver) illustrated
`in Fig. 1. The important factor is to ensure that we have a
`
`(7)
`
`(8)
`
`T+ T, =
`
`X
`
`r = 1, 3,5
`
`X
`
`(T + T,) sin nwo(T + T,)/2
`nwo(T + T.)/2
`\T }
`goes through maxima as (T + T.) increases. Thus maximum
`power of the replica is obtained when
`rT
`2n
`and when this condition is fulfilled, the subharmonic power
`spectrum is 20 log10 (nr) decibels below the original micro-
`wave signal level.
`The ability to vary T and achieve optimum power and
`minimize the loss is a crucial aspect of the technique if a wide
`range of microwave frequencies are to be covered. For exam-
`ple, to cover the range 500 MHz to 20 GHz with a sampling
`rate of 5 MHz (T = 0.2 ,us) T must range between 1 ns and 25
`ps if we select r = I in (8). However, this ratio 40 to 1 in pul-
`sewidth can be relaxed by noting that other values of r are also
`possible and this reduces the range of values of r. This multi-
`plicity of "modes" is illustrated in Fig. 3. Unfortunately, as
`r increases, the sensitivity of the optimized power to slight
`variations in pulsewidth increases as well, and this sensitivity
`is shown in Fig. 4. In addition, between successive maxima we
`have nulls of power which occur when in (8) r takes the even
`values (2, 4, 6, etc.). In summary, the subharmonically sam-
`pled spectrum can be optimized by varying the pulsewidth and
`as this width, say, increases, we go through maxima and
`minima. However, as the order r of the maxima increases, the
`sensitivity to variations in T increases as well and good control
`and stability of -r are paramount. All these aspects have been
`experimentally verified and taken into consideration in the
`experimental unit described in Section III.
`In reference to the loss due exclusively to the rise time TX,
`there is no optimization criterion. However, from (5) it is clear
`that T, should be as short as possible to ensure that we operate
`within the main lobe of sin x/x, x = nxT /T. This dependence
`is illustrated also in Fig. 2; for example, for a loss not exceeding
`3 dB in the sampling of a 32-GHz signal with a 5-MHz pulse
`
`RPX-Farmwald Ex. 1040, p 3
`
`

`

`FAULKNER AND VILAR I MESTRE: SUBHARMONIC SAMPLING FOR MEASUREMENT OF MICROWAVE OSCILLATORS
`
`211
`
`signal-to-noise ratio that is high enough, and this can be
`achieved with the use of a narrow-band receiver. To determine
`the required bandwidth for this receiver we can use the concept
`of effective noise temperature of a network with noise figure
`F, and compute the available noise power Np which will be
`given by the familiar expression
`N= kTBn = k(F-)TOB,
`(10)
`where Te and To are the effective and reference noise tem-
`peratures in kelvins, and k is the Boltzmann constant. If we
`now consider that the sampling gate has a conversion loss of,
`say, 7 dB and that the rise time T, introduces an irreducible
`loss of 3 dB to the otherwise optimized subharmonic signal
`selected, we can relate that subharmonic power to the thermal
`noise within Bn and derive an engineering expression for the
`signal-to-noise ratio. That is
`S/N power nth replica (in dB)
`
`input signal power (dBm) - 20 log10 nr
`- 10 - kTeBn(dBm).
`1)
`(
`Equation ( 11) shows that with a noise bandwidth of, say,
`30 kHz, an input level of 0 dBm at 40 GHz, and a receiver
`noise figure of 3 dB the S/N is
`- 31 dB. Because a 30-kHz
`noise bandwidth, and even lower, is perfectly achievable in a
`tracking receiver operating between 5 and 7.5 MHz (sampling
`rate of 5 MHz) we conclude that down-conversion with sub-
`harmonic sampling followed by signal recovery with good
`signal-to-noise ratio is achievable well into the millimeter re-
`gion. Clearly, one must use crystal oscillators of a high degree
`of spectral purity which drive wide-band sampling gates.
`Variable pulsewidth T is fundamental to the technique.
`
`11I. EXPERIMENTAL SYSTEM
`A. Description
`Fig. 5 shows the complete block diagram of the experimental
`measurement system which has been built, tested, and is cur-
`rently in use in the laboratory. A view of the rack unit has been
`shown in [7]. Consistent with the concepts discussed in Section
`II, the 5-MHz master oscillator drives a variable pulsewidth
`generator. The oscillator under test is then sampled in the
`sampling gate and the output is low-pass filtered from spectral
`components above 7.5 MHz, as at least one of the subharmonic
`replicas must lie between 5 and 7.5 MHz. The output signal
`spectrum is a replica of the original oscillator several tens of
`decibels below the original level. A tracking receiver follows,
`which is a phase-locked receiver (PLL) with a voltage-con-
`trolled oscillator (VCO) tunable between 5 and 7.5 MHz.
`After phase lock, the VCO output is a band-pass filtered ver-
`sion of the subharmonic under analysis. In addition, any fre-
`quency drifts will be followed by the VCO. A first version of
`the tracking receiver was simply a variable-tuned circuit but
`this was later abandoned in favor of the PLL which can recover
`signals in a poor signal-to-noise ratio environment.
`The VCO signal is now suitable for analog or digital analysis
`and in addition, the voltage driving the VCO contains the in-
`
`Fig. 5.
`
`Measurement system block diagram.
`
`Fig. 6.
`
`Pulse generator and sampling gate.
`
`formation on frequency jitter of the original microwave os-
`cillator. If the frequency jitter is small because the microwave
`oscillator under test has a good degree of spectral purity, the
`reciprocal counter operating between 5 and 7.5 MHz may lack
`resolution. To avoid this, Fig. 5 shows how the VCO signal is
`mixed with the output of a 5- to 7.5-MHz synthesizer driven
`by the master oscillator so as to obtain an even lower frequency
`replica of the microwave oscillator. In the present system, the
`synthesizer can operate in steps of I kHz and therefore, if
`necessary, a low-frequency replica at about l-kHz center
`frequency is possible. For practical reasons, the output of this
`second mixer is low-pass filtered by a conventional variable
`low-pass filter. The final signal is ready for processing and
`analysis.
`B. Pulse Generator and Sampling Gate
`The circuit diagram is shown in Fig. 6 and the technology
`employed throughout is conventional microstrip and 50-u
`transmission-line impedances. The pulse generator uses
`standard TTL logic driving bipolar transistors into and out of
`their avalanche region [8]. The collector of T1 is biased close
`to breakdown voltage and when TF
`is triggered into avalanche
`conduction a charged delay line, Z0, To, is rapidly discharged
`through the load resistor RL. This results in a positive pulse
`of around 20-V peak amplitude with 200-ps rise time (00-90
`
`RPX-Farmwald Ex. 1040, p 4
`
`

`

`212
`
`IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-32, NO. 1, MARCH 1983
`
`Lii
`<
`
`In
`00
`
`5
`
`4-
`
`3
`
`SAMPLED 1GHZ OSCILLATOR PHASE LOCKED To REFERENCE +
`
`5MHZ REFERENCE OSCILLATOR *
`
`-3
`
`2--
`4606
`H~~~~~~~~~~~~~
`0-
`
`en
`
`zi
`
`i
`
`_
`10
`1
`.1
`OFFSET FREQUENCY FROM CARRIER (HZ)
`Phase noise measurements verifying the principles of the
`measuring technique (see text).
`
`100
`
`-2
`. -3-
`In
`wi -4-
`U,
`I -5
`I-
`-6
`.01
`
`U a
`
`L
`
`Fig. 7.
`
`of the features of a commercial counter are unnecessary. The
`synthesizer and further mixing should allow the use of a more
`simple purpose-built counter with a 5-MHz clock signal given
`by the system master. In this way, it will be possible to resolve
`small frequency variations of a stable microwave oscillator in
`a very cost-effective way. The synthesizer introduces some
`inevitable degradation due to the jitter in the logic circuitry.
`To measure this, in the present system, the synthesizer output
`at 5.001 MHz was mixed with the master at 5 MHz and the
`two-sample Allan variance fractional frequency stability
`analysis gave a value of 10-9 at a gate time of I s with a -I
`slope in the sigma-tau plot (white phase noise). It is important
`to note, however, that this one part in 109 represents 5 mHz
`at the frequency of analysis, -5 MHz. When a microwave
`oscillator is down-converted, this 5-mHz imprecision is added.
`This implies for example that a l-GHz oscillator can be as-
`sessed with the ultimate fractional frequency limitation of 5
`X 10-2 = 10-9/(1000/5) = 10-9/200, where 200 is the
`subharmonic number n.
`Further sources of imprecision include additive noise and
`the master-clock jitter. The thermal noise of the receiver is
`uncorrelated with the phase noise of the signal under test. The
`quadrature component of that noise, assumed to be spectrally
`flat, can be estimated within Il-Hz bandwidth (kTe,) and it can
`be related to the subharmonic level. It is reasonable to identify
`this ratio, in decibels, as the function L(f) or single-side
`band-to-carrier ratio expressed in units known as dBc/Hz. For
`example, because in the present system kTe is about -174
`dBm, the sampling of a 0-dBm 20-GHz signal (n = 4000)
`leads to the floor -80 dBc/lHz. Because the sampling process
`introduces multiplication of the phase noise of the master
`reference oscillator by the factor n, this noise will appear at
`the replica of the sampled microwave signal. However, it re-
`mains to be further investigated whether some or all of these
`frequency-jitter variations are effectively canceled due to the
`coherence of the counter clock and the sampling rate (both are
`derived from the 5-MHz master).
`Various tests with a variety of oscillators have been carried
`out. Fig. 7 is an illustrative example of both the principles of
`the system and the type of numerical analysis. In this case,
`
`percent). The maximum pulsewidth of twice To occurs when
`T, is brought out of the avalanche mode by the pulse reflected
`at the other end of the delay line. In order to achieve a variable
`pulsewidth, a second transistor T2 is used at the end of the
`delay line which also operates in the avalanche mode. This is
`triggered at a different time from T1 and generates a second
`pulse of opposite polarity which travels towards T1 and results
`in an overall shorter pulsewidth. Continuously variable pul-
`sewidth is obtained by altering the delay Td between the
`triggering of T, and T2. The resulting pulses observed have
`a duration T which can be varied between 0.4 and 1.5 ns with
`full 20-V amplitude on a 50-Q load (sampling oscilloscope).
`The measured edge speed is about 10 ps/V, and therefore the
`present unit could, in theory, operate a sampling gate up to
`about 40 GHz depending upon the sampling gate and the
`diodes used.
`The circuit configuration for the sampling gate is shown in
`Fig. 5 and requires the use of a single-polarity sampling pulse
`for operation, as produced by the pulse generator. The diodes
`used within the gate are normally reverse biased ensuring
`minimum signal transfer during the period of no sampling
`pulse. By using the reverse-bias (Vbias) level, the switch-on
`point of the sampling gate can be set where the rise slope of the
`pulse edge is steepest. Schottky-barrier diodes have been used
`as switching elements since it is necessary to have a fast re-
`sponse to the sampling pulse. The transformer arrangement
`provides the necessary balanced signal required by the two-
`diode configuration in addition to providing isolation between
`the sampling gate ports. This transformer limits the frequency
`of operation of the sampling gate to about 3 GHz at present,
`due to the ferrite-core construction; but higher operating
`frequencies are plausible using other techniques. The con-
`nection between the pulse generator and sampling gate is made
`via a semi-rigid coaxial link to facilitate the use of alternative
`sampling gates.
`C. Implementation of the Measuring System Tests and
`Limitations
`The construction of the experimental unit of Fig. 5 is mod-
`ular and the circuits are mounted in a 19-in rack [7]. The pulse
`generator is allowed to settle in temperature and it is conve-
`nient to use a power spectrum analyzer to check levels of both
`microwave oscillator under test and harmonics of the pulse
`generator. The output of the sampling gate is then monitored
`by the analyzer and the subharmonic in the range 5-7.5 MHz
`is identified. If necessary, the spurious feedthrough 5-MHz
`signal is canceled or minimized and the signal is then tracked
`by the PLL receiver. The output of the VCO can then be sent
`to the reciprocal counter which is interfaced with the computer
`via the IEEE-488 bus. Currently an HP 5342A and Commo-
`dore minicomputer are used and the frequency counts, aver-
`aged over the variable gate time, are stored for processing.
`The master oscillator of the overall system is itself the internal
`clock of the counter and thus complete synchronism in the
`system is achieved.
`As outlined before, the VCO output frequency variations
`may be such that even the resolution due to the high clock rate
`(500 MHz) of the counter is not sufficient. In addition, many
`
`RPX-Farmwald Ex. 1040, p 5
`
`

`

`FAULKNER AND VILAR I MESTRE: SUBHARMONIC SAMPLING FOR MEASUREMENT OF MICROWAVE OSCILLATORS
`
`21 3
`
`-30DBM
`
`OUTPU
`SAMPL
`
`SUB-HARMONIC
`REPLICA
`
`TO RECOVERY
`RECEIVER
`I
`SUB-HARMONIC
`REPLICA
`
`5MHZ
`
`5MHZ
`COMB
`LINE
`
`L
`
`5MHZ
`REFERENCE
`SIGNAL
`
`Fig. 8.
`
`Feedthrough cancellation technique.
`
`Fourier analysis of the frequency counts was carried out; the
`upper spectrum was obtained with the experimental system
`so as to analyze a -1-GHz microwave signal derived from
`phase locking a Ferranti cavity oscillator to a 5-MHz Vectron
`crystal oscillator. On the other hand, the lower spectrum shown
`in the figure is the direct analysis of the -kHz beat between
`the previously mentioned 5-MHz Vectron and another one of
`the same production series (next serial number). It is clear that,
`as theoretically expected, both spectra differ by 20 log1I n =
`46dB, n = 1000/5.
`Finally, it was mentioned at the beginning of the section that
`the output of the sampling gate contains a spurious 5 MHz and
`its harmonics. Even after low-pass filtering all components
`above 7.5 MHz, that 5-MHz signal is present with a level of
`about -30 dBm. If we consider that the level of the subhar-
`monic may be about -60 dBm or less, and in addition it can
`be close to 5 MHz, it is clear that the PLL tracking receiver
`will lock onto the spurious 5-MHz feedthrough. To avoid this,
`the output of the sampling gate is fed to a power combiner and
`at the other port a fraction of the 5-MHz master signal is in-
`jected (Fig. 8). Simple vector calculations show that by careful
`control of amplitude and phase of this injected signal, the 5
`MHz can be canceled. This has been verified experimentally
`and the measurement system contains a monostable pulse
`generator triggered by the reference with variable delay.
`Control of both delay and amplitude of the 5-MHz pulse train
`injected allows good cancellation of the spurious signal and the
`PLL receiver can track with no difficulty.
`
`IV. CONCLUSIONS
`This paper has described the theoretical principles, design,
`testing, and limitations of a novel system of instrumentation
`guided to help the measurement of short-term frequency sta-
`bility of microwave oscillators. The technique does not avoid
`the use of mixing and a second oscillator; however, by using
`the subharmonic sampling principles and a theoretical analysis
`leading to power optimization criteria of the subharmonic
`selected, only one low-frequency master oscillator is required.
`The traditional microwave mixer becomes in this technique
`a wide-band sampling gate. Finally, as technology in pulse
`generation, measurement, and synthesis advance [7], and
`multioctave wide-band techniques are becoming widespread,
`the technique described in this paper may be attractive and
`used in a compact measuring unit. It has proved its usefulness
`in quick measurement of L-band cavity oscillators and SAW
`oscillators at the authors' laboratory.
`
`ACKNOWLEDG MENT
`One of the authors is grateful for a SERC CASE student-
`ship with Ferranti Ltd. The early enthusiastic work of G.
`Koutsavakis and later support of J. McA. Steele are warmly
`acknowledged.
`
`REFERENCES
`
`[3]
`
`[ ] "Fundamentals of the electronic counters," Hewlett-Packard Co., Ap-
`plication Note 200.
`[2] R. D. Ely, "Frequency measurements by frequency meter or timer
`counter?" IEE Electron. Power, Jan. 1978.
`P. Lesage and C. Audoin, "Effect of dead-time on the estimation of the
`two-sample variance," IEEE Trans. Instrum. Meas., vol IM-28, no. 1,
`pp. 6-1 1, Mar. 1979.
`[4] D. W. Allan, "Statistics of atomic frequency standards," Proc. IEEE,
`vol. 54, no. 2, pp. 221-230, Feb. 1966.
`[51 R. Russel and D. L. Hoare, "Milimetre wave phase locked oscillators,"
`presented at the Military Microwaves Conf., Brighton U.K., 1978.
`[6] G. P. Koutsavakis, "Sampling phase lock loop voltage controlled oscil-
`lator," M.Sc. thesis, Dep. Elec. Electron. Eng., Portsmouth Polytechnic,
`Portsmouth, U.K., July 1978.
`[71 CPEM 82 Conference Proc., W. A. Alspach Ed., NBS, Boulder, CO,
`IEEE Cat. No. 82CH 1737-6.
`[8] H. M. Rein and M. Zahn, "Subnanosecond pulse generator with variable
`pulse width using avalanche transistors," Electron. Lett., vol. I 1, no. 1,
`Jan. 9, 1975.
`
`RPX-Farmwald Ex. 1040, p 6
`
`