IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984
ACKNOWLEDGMENTThe author wishes to thank Dr. I. Yokoshima for his valuable
guidance.
REFERENCES[1] G. E. Schafer and R. R. Bowman, "A modulated sub-carrier tech-
nique of measuring microwave attenuation," Proc. IEE (London),vol. 109, part B, suppl. no. 23, p. 783, May 1962.
[2] R. W. Beatty, "Microwave attenuation measurements and stan-dards," NBS monograph 97, Apr. 3, 1967.
[3] F. L. Warner, D. 0. Watton, P. Herman, and P. Cummings, "Auto-matic calibration of rotary-vane attenuators on a modulated sub-carrier system," IEEE Trans. Instrum. Meas., vol. IM-25, no. 4,p. 409, Dec. 1976.
[4] T. Kawakami, "Precision microwave attenuation measurement usingdouble phase modulation," presented at Conf. on Precision Electro-magnetic Measurement (Ottawa, Ont., Canada), 1978.
Group Delay Measurements at Audio FrequenciesKAMAL JABBOUR, MEMBER, IEEE
Abstract-The paper discusses the measurement of group delay distor-tion at audio frequencies. Loop measurements and line measurementsare discussed separately, and existing meters are reviewed. For bothtypes of measurement, a stored-program approach using microprocessorsis proposed, and laboratory models are described and evaluated. Theuse of microprocessors results in compact and inexpensive, yet accurategroup delay meters.
I. INTRODUCTIONWl TITH THE RAPID increase in computers in recent years,
and the need for communications between different in-stallations, the telephone network was an obvious candidate tocarry the bulk of the data traffic, and telephone lines currentlyconstitute the backbone of most long-haul computer networks.Originally designed for speech communication, the telephonenetwork is a hostile environment to data signals. Group delaydistortion is a major impairment, causing intersymbol interfer-ence, thus resulting in errors and restricting transmission rates,hence the need for accurate measurement techniques.The measurement of the envelope delay of a modulated
carrier, as an approximation to group delay, is the most widelyused technique at audio frequencies, and will be reviewed here.Loop measurements and line measurements of envelope delaywill be discussed separately, with a brief review of existingmeters. For both types of measurement, the author proposes astored-program approach using microprocessors, describes alaboratory model, and evaluates its performance. The proposedapproach results in compact and inexpensive, yet accurate,group delay meters. For the theory and significance of groupdelay, and different measurement techniques, the interestedreader is referred to [1]-[5], which in turn contain usefulreferences on specific aspects of the subject.
Manuscript received September 9, 1983; revised November 3, 1983and January 12, 1984. This work was carried out at the Department ofElectrical Engineering, University of Salford, England, with a researchstudentship from the University of Salford, under the supervision of Dr.B. H. Pardoe.The author is with the Department of Electrical and Computer Engi-
neering, Syracuse University, Syracuse, NY 13210.
II. ENVELOPE DELAY LooP MEASUREMENTSIf an amplitude-modulated signal with carrier frequency fc
and modulation frequency fm is applied to a two-port network,it can be shown that, for 'm << f, the group delay of the net-work is equal to the ratio of the envelope phase shift throughthe network and the modulation frequency, therefore, numeri-cally equal to the envelope delay through the network [4] -[9].This relationship between the group delay of the network andthe envelope delay also holds if the test signal is amplitude-modulated double sideband suppressed (AM-DSB) carrier [7],[10].This way, the problem of measuring group delay, which is a
physical characteristic of the network, simplifies into measur-ing the envelope delay of an AM-DSB test signal passing throughthe network under test. Fig. 1 shows two possible realizationsof an envelope delay meter.
In Fig. l(a), the AM signal is applied to the network undertest, with carrier frequency fc equal to the desired measuringfrequency, and modulation frequency 'm << f, At the out-put of the network under test, the signal is demodulated andfiltered to reproduce fm which is compared to the originalmodulating signal. The phase shift between the two modula-tion signals, appropriately scaled, is displayed as group delay.Alternatively, both AM signals at the input and the output ofthe network are demodulated and filtered to give the two sinewaves at f'm (Fig. l(b)) and their phase shift is measured anddisplayed.
A. Analog ApproachAssuming either of the two circuits in Fig. 1 is realized with
analog components, as in many early meters, the followingproblems are encountered:
i) the accuracy of the test frequency: a frequency meter issometimes incorporated in the set,
ii) the stability of the low-frequency modulation signal:any phase jitter or drift reduces the accuracy of themeasurement,
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984
fm demodulotor BPF
Fig. 1. Two possible realizations of envelope delay meters. NUT: net-work under test; BPF: bandpass filter.
iii) the linearity of the phase meter/display arrangement atvarious carrier frequency and envelope magnitudes,
iv) the phase shift introduced by the demodulation chan-nels: mismatches are unavoidable,
v) the temperature sensitivity of the analog circuits used.
From the above, it should be clear why analog group delaymeters have a poor accuracy, are difficult to operate and main-tain, and are always very expensive.
B. Digital ApproachDue to the inherent problems associated with analog com-
ponents, digital circuits have been used wherever possible toimprove the performance of group delay meters:
i) The carrier frequency has been digitally controlled, elim-inating the need for a frequency meter.
ii) The modulating signal has been obtained from a high-frequency oscillator through binary division. The resultantsignal is filtered to give the desired sine wave. One major draw-back is that the modulation frequency can only take certainvalues, dictated by the hardware used, and a separate low-passfilter is needed at each frequency.
iii) The phase meter has been replaced with a high-frequencydigital counter, gated by the zero crossings of the two enve-
lopes, with a digital display to overcome the errors associatedwith moving-coil indicators [6].Even with these improvements, the accuracy of group delay
meters still suffers from the use of analog demodulation chan-nels, and their flexibility is limited by the hardware used.
III. STORED PROGRAM APPROACH
The stored program approach is characterized by a centralprocessing unit (CPU) achieving a desired task through theexecution of a sequence of simple instructions (the stored pro-gram). The processor has the ability to make decisions, allow-ing different actions to be taken, depending on the situation.Another important feature of this approach is that the proces-sor can store information (data) in memory for later usage. Todemonstrate how stored program control can improve groupdelay measurements, a laboratory model of a microprocessor-based group delay meter has been built from commercial inte-grated circuits [101, and is described in the following sections.
Fig. 2. Block diagram of microprocessor-based group delay meter.
A block diagram of the meter is shown in Fig. 2. A dual-processor design was adopted to increase the flexibility of thesystem. As the operator keys in the frequency at which a mea-surement is desired, the frequency synthesizer generates anAM-DSB-suppressed-carrier test signal which is applied to thenetwork under test (NUT). The envelope delay between theinput and the output of the NUT is measured and displayed asan approximation to group delay. The meter can be dividedinto three functional blocks: the frequency synthesizer, theconditioning circuit, and the CPU.
A. The Frequency SynthesizerThe frequency synthesizer consists of a CPU with its soft-
ware, a digital-to-analog converter, and a low-pass filter. Alook-up table of the sine function allows the generation of acontinuous wave by stepping through the table at a determinedrate. At a sampling rate of about 50 kHz, sine waves at fre-quencies of 1-9999 Hz can be generated with an accuracybetter than 0.1 percent. As the processor clock is crystalcontrolled, it results in a very stable synthesized waveformeven at low frequencies. Different frequencies are obtained byusing different step sizes to cycle through the look-up table.This allows the use of constant sampling rate at all frequenciesof the synthesized signal. Thus only one low-pass filter isneeded to reduce harmonic distortion. A fourth-order Butter-worth low-pass filter with a cutoff frequency of 10 kHz hasbeen used, giving second and third harmonics typically 54-dBbelow the fundamental.To generate the test signal for group delay measurements,
two sine waves at frequencies f, and f2 are generated simulta-neously in digital form within the CPU and added point bypoint, in real time, in the processor's accumulator to generatean AM-DSB-SC signal
y(t) = cos 2irf1t + cos 2irf2 t
= 2 cos 'A'm t * cos cOc t
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where wm = 27r((fi - f2)/2) and wc = 27r((f1 +f2)/2). Withthe Z80 processor used with a clock of 2 MHz, a sampling rateof 25 kHz was achieved, allowing the generation of test signalscovering the frequency range of interest in voice grade channels.The frequency synthesizer offers the following advantages
over the classical approach:i) The carrier frequency is software controlled, and therefore
can be made arbitrarily accurate (0.1 percent in our laboratorymodel). The frequency at which a measurement is desired iskeyed into the meter via a keypad and is clearly displayed onseven-segment LED display, so there is no ambiguity about thevalue of the measuring frequency, eliminating the need for afrequency meter.
ii) There is no low-frequency oscillator. Both componentsof the test signal, the carrier and the modulation, are generateddigitally within the processor. The result is a crystal-stable testsignal, virtually temperature and frequency independent.
iii) There is no need for an analog modulator or a mixer,as the modulation is done digitally, thus reducing distortion.
iv) The modulation frequency can be selected at will, andis independent of the hardware. This allows the modulationfrequency to be kept arbitrarily smaller than the carrierfrequency.Summing up, a very stable and reliable test signal is digitally
synthesized, eliminating the problems associated with analogcircuits.
B. The Conditioning CircuitThe AM-DSB-SC test signal is applied to the network under
test. The signals at the input and the output of the NUT passthrough similar conditioning channels consisting of squarers,bandpass filters, and zero-crossing detectors, which extract thedifference in phase shifts, AO = 02 ol [7], [10]. A 14-bitbinary counter clocked at 1 MHz gives the phase delay betweenthe two envelopes, in microseconds, as a good approximationto group delay.At this stage, a few points need further clarification:i) If an AM-DSB signal with carrier were used instead of the
AM-DSB-SC signal, the squarers would then be replaced bysimple envelope detectors; the operation of the meter remainsotherwise unchanged.
ii) The use of the zero crossings to measure envelope delaymeans that the amplitudes of the envelopes need not be equal.
iii) As will be shown in the next section, the two demodula-tion channels need not be matched as any errors resulting fromthe mismatch are corrected in software.
C. The Central ProcessorThe central processor (CPUI) board is the heart of the sys-
tem. Its housekeeping duties include reading the keyboard,passing the relevent information to CPU2 to generate the testsignal, and displaying results on a four-digit seven-segmentLED display. In addition to that, CPU1 carries the impor-tant task of processing the data obtained from the binarycounter, and displays them as group delay.When a measurement is required, the desired frequency is
keyed into the system. CPU1 instructs CPU2 to generate therequired test signal, and clears the counter. Switch S is closed
number of reodings
80
70
60
501-
40
3C
2C
10
-5 -4 -3 -2 -I 0I Ft-,eror
2 3 4 5 6,usec
Fig. 3. Error distribution in group delay measurements of first-orderall-pass filter.
to bypass the NUT, allowing identical test signals x(t) to passthrough the two channels of the conditioning circuit. Anymismatch between the two channels, under the prevailingoperating conditions, is measured and recorded. Switch S isthen returned to its normal position, enabling the output ofthe NUT to pass through the y channel. Envelope delay ismeasured and corrected by the channel mismatch previouslyrecorded. This way, the "insertion" group delay of the NUTis measured, independent of the conditioning circuit. This isthe reason why we indicated earlier that the two channels neednot be perfectly matched, as any temperature and frequencydependence is automatically accounted for.CPU1 does not display a single reading, but rather takes a set
of readings and displays their average. This increases the tol-erance of the meter to noise, and reduces the effects of phasejitter caused by the nonlinearities of the conditioning circuit.
D. Evaluation of the Meter
To determine the accuracy of the meter, a first-order all-passfilter was built, using 0.05-percent tolerance wirewound resis-tors and a 0.1-percent capacitor, and used in the tests.With a modulation frequency of 20 Hz, the group delay of
the test filter has been measured at different carrier frequenciesand RC values, and compared to the computed group delay.Over the frequency range 500-7500 Hz, the comparison gavevery close agreement, with a random error smaller than 7 psindependent of the value of group delay. Fig. 3 shows theerror distribution for some 400 readings. This set of readingshas a standard deviation of about 2 ,us. The errors in themeasurements are mainly caused by the harmonic contents ofthe 20-Hz signal, generated in the analog conditioning circuit.
In other tests, the group delay response of a second-order all-pass filter and that of a fourth-order Butterworth low-passfilter were measured and compared to the response obtainedfrom phase measurements and differentiation. Over a range ofgroup delays from 200 ,us to 5 ms, the mean error was lessthan 2 ,s, giving a relative error better than 1 percent. Therelative error worsens for smaller values of group delay. This isnot a severe limitation, however, since the meter is intended for
Uv
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 2, JUNE 1984
ratede modulator
modulation)41.67Hz
TRANSMITTER
phase [surgedetector
v plotter1.8 kHz, (a)
reference carrier measuring carrier120msec 120 nisec changeover*~~~~~~s i g n a~~~~~~1 inl
(b)Fig. 4. Time division method of group delay measurement: (a) block
diagram of meter and (b) test signal.
use on speech-grade communications systems, where group de-lay distortions of several hundred microseconds are common.
IV. LINE MEASUREMENTS
Line measurements are carried out where the two ports of theNUT are physically far apart, and no return path is available.Line group delay meters consist of two parts: the transmitterto generate the test signal, and the receiver to measure thegroup delay through the NUT. Line measurements differ fromloop measurements in that the phase information about thetest signal at the input of the NUT is not readily available atthe output. Unlike attenuation which can be easily measuredfrom a prior knowledge of the amplitude of the test signal,phase shift-and subsequently group delay-cannot be mea-sured unless the two signals are simultaneously available.This is the reason why the absolute value of group delay of atransmission line is difficult to measure, and measurementsare usually taken relative to the group delay at a referencefrequency. In practice, this is not a limitation, as the additionof a constant delay does not distort the transmitted signal.The time-division principle for measuring group delay will
be reviewed in the following section, and a stored programapproach developed by the author will be presented.
A. Time-Division PrincipleTo ensure compatibility between instruments when measure-
ments are made on international circuits, the InternationalTelegraph and Telephone Consultative Committee (CCITT)recommended the requirements for the characteristics of agroup delay measuring set based on the time-division principle,and used for the frequency band 200 Hz-20 kHz [14].
In this method, illustrated in Fig. 4, the modulation frequencyof 41.67 Hz alternately modulates the measuring carrier and areference carrier of 1.8 kHz. The changeover rate from mea-
RECEIVER
|coujnter MHz
Fig. 5. Microprocessor-based line group delay meter.
suring to reference carrier is 4.167 Hz, obtained from themodulation signals through division. An identifying signal of166.6 Hz also modulates the reference carrier during the last24 ms before changeover to the measuring carrier. If the NUThas different group delay at the carrier frequency and at thereference frequency, a phase change occurs which is measuredat the receiving end to give group delay. Swept-frequencymeasurements are also provided, with outputs to drive x-yplotters [9], [13].Building a stored-program meter satisfying these recommenda-
tions can be done in line with the approach put forward inSection III, resulting in simplified hardware, lower cost, higheraccuracy, and more facilities. Further, the values of group de-lay would be obtained in digital form, giving a wider choice ofinterfaces.
V. DEDICATED MPU-BASED METER
Having been able to measure group delay accurately andreliably with the set described in Section III, the same princi-ples were used to build a dedicated line meter [15]. Themeter is expected to give a fast measurement of the groupdelay of a telephone line over the frequency band 800-2400Hz used for data transmission.
A. Principles of OperationA block diagram of the meter is shown in Fig. 5. The trans-
mitter was built on a 7 X 15-cm board, and consists of an 8-bitZ80-A microprocessor, 2K bytes of program memory, an 8-bitdigital-to-analog converter, and a few other TTL support chips.The transmitter generates an AM-DSB-SC test signal. Themodulation frequency used is 39.0625 Hz, and the carrier fre-quency is stepped in increments of 78.125 Hz every 200 msbetween 742 and 2382 Hz, back and forth. The above fre-quencies were selected for convenience without any loss ofgenerality in the approach, and have no particular significance.
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JABBOUR: GROUP DELAY MEASUREMENTS
The more important part of the meter is the receiver. Theanalog conditioning channel in Fig. 5 gives a square wave at78.125 Hz, related in phase to the envelope of the test signal.The leading edge of each cycle of the square wave generates aninterrupt to the processor. On the other hand, a 16-bit counterruns freely at a clock of 1 MHz. Each time the processor isinterrupted, it reads the counters, compares the reading to theprevious reading modulo 1/78.125 (12.8 ms), and deduces thephase change between the two successive readings, directlygiving the relative envelope delay in microseconds. Eachcarrier frequency is transmitted for about 200 ms, giving 16measurements of group delay per frequency.This relative delay, averaged at each frequency, is stored in
memory. The fact that the counters are not reset after a read-ing eliminates the accumulation of errors from reading toreading. The 742-Hz carrier is preceeded by 500 ms of silence(or unmodulated carrier for that purpose) to identify the startof a sweep. Similarly the 2382-Hz carrier is followed by 250ms of silence to signal the end of a rising-frequency sweep, andthe start of a falling-frequency sweep. The dual slope approachcorrects low-frequency phase drifts, allowing transmitters withslightly differing modulation frequencies to be used with thesame receiver. Up to 32 readings per frequency are obtained,and 'are averaged as group delay. Measurements at 22 fre-quencies are obtained over the range 742-2382 Hz, and areoffset relative to the lowest in-band reading to give positivevalues of group delay distortion. Prior to using the meter, alocal synthesizer measures the response of the analog condi-tioning circuit, which is stored in memory and used to correctsubsequent measurements.
B. Evaluation of the MeterThe receiver of the line meter is interfaced to a microcom-
puter which displays the measurements and plots the frequencyresponse. The telephone line simulator described in reference[3] was used as a test network, and its response measured fordifferent connections (Fig. 6). The simulator consists of threeattenuators and two all-pass filters. Group delay being addi-tive, the two all-pass filters can be cascaded to give a thirdgroup delay response. The attenuators and all-pass filters weredesigned to approximate the responses of Fig. 6(b) and (c), sothat the 16 possible combinations simulate the majority ofconnections on the switched telephone network.
In the simulator, AG and DO are straight connections. Dland D2 were built with active tuned all-pass filters. Al andA2 are first- and fourth-order Bessel filters, and A3 is a fourth-order Butterworth filter followed by a second-order all-passfilter to equalize its phase response.The different connections in the simulator were measured
with the line meter. A second set of measurements were alsoobtained using the loop meter, and a third set from spot mea-surements of phase shift and differentiation. All three setsshowed close agreement. Fig. 7 shows the group delay re-sponses of the line simulator for the most severe attenuationconnections A2 and A3, at different group delays.A variation to this measurement technique was also tested,
whereby a continuous sinewave was used, instead of the AM-DSB-SC test signal. The frequency of the sine wave was swept
DO
0
AttenuationdB
AlI
A2
\3
(a)
2
20 .
800 2400 ltzt(b)
o0
Group delaymsec D3
/ D2
300 2400Hz(c)
Fig. 6. Block diagram and response of telephone line simulator.
Fig. 7. Measured group delay response of telephone line simulator.
over the range of interest, and phase shift was measured atspecified frequencies, allowing the computation of group delaythrough differentiation. Although conceptually simpler, thetechnique results in larger errors, because the change in phaseshift between two frequencies is usually comparable to theerror in the phase-shift measurement. Larger frequency stepswould reduce the phase error, but fail to detect sharp changesin group delay.
VI. CONCLUSIONThe measurement of group delay at audio frequencies is
another field where the stored program approach proves itssuperiority over the traditional design techniques. With theadvances in semiconductor technology, faster microprocessorswill allow these measurement techniques to be extended tohigher frequencies. The advantages of using microprocessorsare more compact and inexpensive meters, with high accuracy.
REFERENCES[1] A. Fettweis, "Significance of group delay in communication en-
gineering," Arch. Electronik Ubertragung., vol. 31, pt. 9, p. 342-348, Sept. 1977.
[2] J. 0. Jones and R. C. Adcock, "Group delay in the audio datanetwork," Post Office Electrical Engineers J., vol. 64, p. 9-15,Apr. 1971.
[3] B. H. Pardoe, "Data transmission through the switched telephonenetwork," Ph.D. dissertation, Dept. of Electrical EngineeringScience, University of Essex, England, Jan. 1975.
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[4] H. Nyguist and S. Brand, "Measurement of phase distortion,"Bell Syst. Tech. J., vol. 9, p. 522-549, July 1930.
[5] B. Wardrop, "The measurement of group delay," Marconi Rev.,vol. XXXV, pt. 187, p. 316-336, 4th quarter, 1972.
[61 R.K.P. Galpin, "Versatile and accurate method of group delaymeasurement," Electron. Lett., vol. 4, no. 2, p. 23, 24, Jan. 1968.
[7] W. Saraga and A.K.H. Miller, "A novel precision group delaymeter," presented at Conf. on Electric and Measuring Instrumen-tation, Testmex 79 (London), June 1979, p. 70-73.
[81 B.N.S. Allen, "Development of group delay measuring sets,"Post Office Electrical Engineers J., vol. 66, pt. 1, p. 47-52, Apr.1973.
[91 L. Matuka, "Measurement of amplitude and phase characteristicsof transmission paths," Budavox Telecommun. Rev. (Hungary),no. 3-4, p. 23-27, 1972.
[101 K. Jabbour and B. H. Pardoe, "A microprocessor-based group
delay meter," Proc. Inst. Elec. Eng., vol. 128, pt. A, no. 5, pp.358-361, July 1981.
[111 F. Coenning, "Progress in the technique of group delay measure-ments," NTZ Commun. J., vol. 5, p. 256-264, 1966.
[121 P. N. Ridout and P. Rolfe, "Transmission measurements of con-nections in the switched telephone network," Post Office Electri-cal Engineers J., vol. 63, p. 97-104, 1970-1971.
[131 N. Montefusco, "Test set for group delay and attenuation distor-tions," Siemens Rev. (Germany), vol. 44, no. 5, p. 221-224, May1977.
[151 K. Jabbour and B. H. Pardoe, "Mircroprocessors in group delaymeasurements," presented at the Conf. on the Influence of Micro-electronics on Measurements, Instrumentation and TransducerDesign, UMIST (Manchester, U.K.), June 29-July 1, 1982.
Limitations of Hydrogen Maser FrequencyStandards Due to Receiver Noise
LEON PROST, MEMBER, IEEE, GIOVANNI BUSCA, AND FRED E. GARDIOL, SENIOR MEMBER, IEEE
Abstract-In this paper we give a description of the different noisesources occurring in the receiving electronics of an active hydrogen maserand their influence on overall stability. It is shown that the most signif-icant noise sources in the PLL locking a VCXO to an atomic oscillatorare the preamplifier and the frequency multiplier. A measuring methodyielding the noise behavior of the entire receiver is presented and com-pared to the calculated result obtained by taking into account the dif-ferent relevant singular noise sources in the loop.
INTRODUCTION
IN THE FIELD of time and frequency standards, the hydro-gen maser now has the best frequency stability for averaging
times between 1 and 10 000 s. Performance of the masernow reaches the level of I X 10-15. However, the signal de-livered by the maser is at a frequency and a power level notdirectly usable. This implies that a receiver is needed to trans-fer the stability of the maser by means of a phase-locked loopto a quartz crystal oscillator (VCXO) working usually between5 and 100 MHz and delivering output power at a level of about10 dBm [1] - [4]. The noise contribution of the receiver elec-tronics degrades the frequency stability of the original masersignal. Usually only the additive thermal noise of the first RFamplifier is taken into account [5], [6]. But other types ofnoise, which can play a major role for the longer averaging
Manuscript received March 24, 1983; revised December 6, 1983.Financial support of this work by the Commission pour l'Encourage-ment de la Recherche Scientifique is gratefully acknowledged.
L. Prost and F. E. Gardiol are with the Electromagnetism and Acous-tics Laboratory of the Federal Institute of Technology, Lausanne,Switzerland.G. Busca is with Oscilloquartz SA., Neuchatel, Switzerland.
times, mainly flicker of phase, are known to exist. In spite ofthat, the flicker or similar noise contributions of the entirereceiver have not been considered theoretically or evaluatedfrom experiments. Another important practical reason forsuch an evaluation is the necessity, in case of poor stability,to discriminate between physical package and the receiver topermit a diagnosis of the cause of the problem.Here we present a measurement system which is able to
directly extract the receiver overall noise limitation. The sys-tem includes one physical package and two receivers of a similardesign. We shall also give a theoretical analysis of the receivernoise limitation measurement method. However we want tomention an alternative method using two physical packagesand one receiver. The output signals of the two masers aremixed and treated by one receiver which is of a different typethan the one described in this paper.
A. Fundamental Concepts ofFrequency StabilityThe signal of a high-quality oscillator is usually written as
E = [Eo + e(t)] cos [27ro + ¢(t)] (1)
where vJo is the nominal oscillator frequency, e(t) expressesthe amplitude instability, which is supposed to be negligible,and b(t) expresses the phase instability or phase noise.
It has become current use to define the following quantities[7] :
