SPECTRAL DENSITY ANALYSIS:
FREQUENCY DOMAIN SPECIFICATION AND MEASUREMENT OF SIGNAL STABILITY*
Donald Halford, John H. Shoaf, and A. S. Risley National Bureau of Standards Boulder,Colorado 80302 USA
Summary paper is complementary to NBS TN632 and relies espe- cially upon the material in it and in references 1 - 3, Stability in the frequencydomain is commonly speci- fied in terms of spectral densities. The spectral density Stability in the frequency domain is commonly concept is simple, elegant, and very useful, but care specified in terms of spectral densities. The spectral must be exercised in its use. There are several differ- density concept is simple, elegant, and very usef~l,~but ent but closely related spectral densities, which are care must be exercised in its use. There are several relevant to the specification and measurementof stability different, but closely related, spectral densities which of the frequency, phase, period, amplitude, and power of are relevant to the specification and measurement of sta- signals. Concise, tutorial descriptions of useful spectral bility of the frequency, phase, period, amplitude and densities are given in this survey. These include the power of signals. In this paper, we present a tutorial spectral densities of fluctuations of (a) phase, (b) fre- discussion of spectral densities. For background and quency, (c) fractional frequency, (d) amplitude, (e) time additional explanation of the terms, language, and 4 interval, (f) angular frequency, and (g) voltage. Also methods, we encourage reference to NBSTN632 and to included are the spectral densities of radio frequency power references 1 - 3. Other very important measures of and its two normalized components, Script X(f) and Script stability exist; we do not discuss them in this paper. %(f), the phase modulation and amplitude modulation portions, respectively. Some of the simple, often-needed relationships among these various spectral densities are Some Relevant Spectral Densities given. The use of one-sided spectral densities is rec- ommended. The relationship to two-sided spectral densi- Twelve spectral densities which are especially useful ties is explained. The concepts of cross-spectral densities. in the specification and measurement of signal stability spectral densities of time-dependent spectral densities, are listed below. Symbols otherwise undefined will be de- and smoothed spectral densities are discussed. fined in a later section, where we also will define and derive the relationships of the various quantities. In our choice of Key words(for information retrieval): Amplitude concepts and symbols, we try to optimize conformity with Fluctuations, Cross-Spectral Density, Frequency Domain, traditional usage in the field and simultaneously to mini- Frequency Noise, Modulation Noise, Noise Specification mize what we see to be the hazards of vagueness, lack and Measurement, Oscillator Noise, Phase Noise, Radio of completeness, and inconsistency. By fluctuations we Frequency Power Spectral Density, Script x(f), Script mean noise,instability, and modulation. The twelve 4)n (f), Script 32, Sidebands, Signal Stability, Spectral definitions follow, Density.
S6 ~ (f)Spectral density of phasefluctu- (1) ations 6 $. The dimensionality is Introduction radians squared per hertz. The range of Fourier frequency f is The economic importance of highly stable signals, &om zero to infinity. signal sources, and signal-processors is increasing, e. g., in communications systems, navigation systems, Spectraldensity of frequencyfluc- (2) and metrology. It is accompanied by an increasing need tuations 6 V. The dimensionality is for carefully-defined and widely-disseminated terminol- hertz squared per hertz. The range ogy and language for specification and measurement of of f is from zero to infinity. We use signal stability. A significant contribution was made in the relation 2~(6V) = d( 6 +)/dt, where 1964 by the IEEE-NASA Symposium on the Definition and t is the runningtime variable. Measurement of Short-Term Frequency Stability. Its Proceedings' were followed in February 1966 by the very sy(f 1 Spectraldensity of fractionalfre- (3) useful IEEE SpecialIssue on Frequency Stability. In quency fluctuations y. The 1970 the authoritative paper, "Characterization of Fre- dimensionality is per hertz. The range quency Stability", was published b the Subcommittee of f is from zero to infinity. By definition on Frequency Stability of the IEEE. r It is the most defini- y f 6 V/V . The symbol y is defined 0 tive discussion to date of the characterization and mea- and used In reference 3, surement of frequency stability. Spectraldensity of amplitude (4) Recently Shoaf et al. have prepared a tutorial, how- fluctuations 6 E of a signal. The to-do-it technical report (NBS TN632) on specification dimensionality is amplitude (e. g., and measurement of frequency stability. The present volts) squared per hertz. The range of f is from zero to infinity.
* Contribution of the National Bureau of Standards, not subject to copyright. 421 Spectraldensity of time interval (5) %(f 1 Normalized frequency domain ( 10) fluctuations 6 'c. The dimensionality measure of phase fluctuation side- is seconds squared per hertz. The bands. 6 We have defined Script a(f) range of f is from zero to infinity. to be the ratio of the power in one We use the relation 6 T= 6 $1 (2~1. phase modulation sideband, referred to the input carrier frequency, on a Spectraldensity of time interval (6) spectral density basis, to the total fluctuations. Same as S (f) . Both signal power, at Fourier frequency 6T difference f from the signal's average x and 6 T are equal to 6 @/(2svo) , frequency V , for a single specified See Eq. The is (35). symbol x signal or device. The dimensionality defined and used in reference3. is per hertz. Because here f is a frequency difference, the range of Spectral density of angular fre- f is from minus V to plus infinity. quency fluctuations 6 R. dimensionality is (radls)The /Hz. The range offis from zero to infinity. Normalized frequency domain (11) By definition, Q E 2nv. measure of fractional amplitude fluctuation sidebands of a signal. Spectraldensity of voltagefluctu- (8) We define Script m(f) to be the ratio ations 6v. The dimensionality is of the spectral density of one amplitude volts squared per hertz. The range modulation sideband to the total signal of f is from zero to infinity. Many power, at Fourier frequency difference commercial spectrum analyzers and f from the signal's average frequency V , wave analyzers exist which measure for a single specified signal or device. and display spectral density (or The dimensionality is per hertz. Because square root of spectral density) of here f is a frequency difference, the voltage fluctuations. A metrologist range of f is from minus V to plus infinity. often will choose to transduce the fluctuations of a quantity of interest Script Gf) and Script%f) are similar into analogous voltage fluctuations. functions; the former is a measure of Then the spectral density (corre- phase modulation (PM) sidebands, the sponding to fluctuations of frequency, latter is a corresponding measure of phase, time interval, amplitude, or amplitude modulation (AM) sidebands. We introduce the symbol Scriptx(f) in . . . ) is measured with voltage spectrum analysis equipment. order to have useful terminology for the important concept of normalized AM SF(V) Spectraldensity of (squareroot (9) sideband power. RFP of) theradio frequency power P. The S (f) Spectraldensity of fluctuations (12) power of a signal is dispersed over 6g the frequency spectrum due to noise, of any specified time-dependent instability, and modulation. The dimen- quantity g (t) . The dimensionality is the same as the dimensionalityof sionality is watts per hertz. The range 2 of the Fourier variable V is from zero the ratio g /f. The range of f is to infinity. This concept is similar from zero to infinity. The total to the concept of spectral density of variance of 6 g(t) is the integral of voltage fluctuations, S6 v(f). Typi- S (f)over all f. Thespectral 6g cally the latter, SGv(f), ismore con- density is the'distribution of the total venient for characterizing a base- variance over frequency. band signal where voltage rather than power is relevant. The former, Discussion of SpectralDensities S= (v), typically is more con- One-sided Versus Two-sided Spectral Densities venient for characterizing the dis- persion of the signal power in the Each of the above twelve spectral densities is one- vicinity of the nominal carrier sided and is on a per hertz of bandwidth density basis. frequency v . To relate the two This means that the total mean-square fluctuation (the spectral den%ties, it is necessary total variance) of frequency, for example, is given mathe- to specify the impedance associated matically by the integral of the spectral density over the with the signal. The choice of V total defined range of Fourier frequency f or f for the Fourier frequency vari- m able is somewhat arbitrary. We VarianceTotal = v(f)df, (13) prefer v for carrier-related measures and f for modulation-related measures. See later section on relationships among As another example, since Script X(f) is a normalized spectral densities. density, its integral over the defined range of difference frequency f is equal to unity, i.e.,
422 this paper, we recommend their use for special 5 problems of measurement, df = 1.
0 The definite integral between two frequencies of the Time-Dependent Spectral Densities spectral density of the fluctuations of a quantity is the variance of that quantity for the frequency band The spectral density concept as used by experi- defined by the two limit frequencies. We will con- mentalists allows the spectral density to be time de- sider this in more detail later. pendent. This time dependence is an observable, and we may sometimes desire to specify the spectral Occasionally, unnecessary confusion arises density of its fluctuations, as a statistical measureof concerning one-sided versus two-sided spectral the time dependence. Hence, we arrive at the con- densities. Two-sided spectral densities are de- cept of the spectral density of the fluctuations of a fined such that the frequency range of integration spectral density. Although not much use has been is from minus infinity to plus infinity. For specifi- made yet of spectral densities of the fluctuations of cation of signal fluctuations as treated in this paper, spectral densities, in part due to the large quantity our one-sided spectral density is twice as large as of data required for a precise measurement, the the corresponding two-sided spectrd density. For concept can be exploited where confidence measures example, the total variance is are desired for spectral density statistics. It is a quantitative, statistical measure for characterizing I +m the stability of noise statistics. Unless otherwise stated, it is usually understood that the spectral density of the fluctuations of a spectral density is assumed to be white, that is, it is assumed that the fluctuations of the spectral density arerandom and uncorrelated. where f is the Fourier frequency variable. Two-sided spectral densities are usefulmainly in It is commonly found that measured fluctuations pure mathematical analysis involving Fourier trans- have a spectral density of the fluctuations of their formations. We recommend and use one-sided spectral density which is not random and uncorre- spectral densities for experimental work. Refer- lated. Some authors attempt to describe such ences 3 and 4 use one-sided spectral densities. observations by saying that the fluctuations were The terminology for single sideband (upper or measured and were found to be non-stationary. Such lower) signals versus double sideband (upper and lower) signals is totally distinct from the one-sided a conclusion is logically absurd in stalistics, for in statistics the definition of stationarity7 (a measure on spectral density versus two-sided spectral ensembles rather than on a portion of an ensemble) density terminology. They are totally different makes it independent of the observations of the concepts. fluctuations of any particular entity. Note that any Cross-Spectral Densities particular entity is simultaneously a member of stationary ensembles aswell as of non-stationary Another important concept is cross-spectral ensembles. While stationarity is not a physical densities. For the case oftwo fluctuating quantities observable, it can be postulated for a hypothetical ensemble, thereby giving one bit of information 6a(t) and 6 b(t), we can choose the quantity 6 g(t1 such that about the hypothetical ensemble. Some of the fundamental theoretical aspects jsg(t)l2 5 [aa
In the measurement of spectral densities, there is an important operational aspect which occasionally leads *to some unnecessary confusion. A practical meosure- ment actually gives us the (weighted) average of the 1 spectral density over a range of frequency. We measure a smoothed spectral density - it is smoothed by being averaged in a frequency window. To charac- terize the measurement, we may specify an effective lower frequency, f an effective higher.frequency. 1 1’ f --B f , and information about the shape of the window 02 2 function (bandpass response). Alternatively, we may prefer to specify a center frequency f = (f + f )/2, the effective bandwidth B = (f2 - fl), 0 21 and information about the shape of the window function. The first typeof specification typically is more useful when the higher frequency is large compared with A strictly rectangular bandpass is not realizable, but it the lower frequency. The alternative typically is more is a useful approximation for many. practical situations. useful in the narrowband casewhen the bandwidth is small compared to the center frequency. Lorentzian Bandpass Example As possible terminology, we suggest the use of Script % squared as a frequencydomain measure As another useful illustration, consider a window to represent this weighted average variance. This having a Lorentzian bandpass response,l3 with a allows us to reserve the terminology of. spectral center frequency f and a quality factor Q. The nor- density[symbol: S(f) 1 for thedifferential malized filter response for a Lorentzian shape function element of variance between f and f + df, and to is 2 use for the smoothed spectraldensity (averaged 2 over a range of f). (20) = [‘ -‘.)‘l RectangularBandpass Example
To illustrate the relationships, consider the hypo- thetical case of a window having a rectangular bandpass. We can write for the measurable variance, Script % The effective bandwidth of a Lorentzian is squared,
2 df, x= B = (-$)($)
L
Hence, for Script% we may write l21 df where the limits of integration reflect the rectangular response of the filter (the response is unity between f 1
42 4 m Spectral Density Estimation - Fourier Transform Analysis
It is implicit in some of the previous discussion that we are considering measurements of spectral densities which use frequency domain techniques, e.g, , filtering in frequency domain with resonant circ:lits, Another widely used technique is to acquire data which are quantized in the time domain Uniformity Approximation and then to use Fourier transformation to obtain the frequency domain statistics. There are several If S(f) is approximately uniform over the region possible procedures, and a common name for this of frequency f where the response of the bandpass is technique is spectral density estimation. A rather significant, then a useful approximation is readable discussion, with some relevant references, is presented by Richards.l4 The availability of hard-wired digital computation programs for imple- menting the Fast Fourier Transform (FFT), together with their steadily decreasing cost, is making this technique more and more attractive for on-line measurement (estimation) of spectral density.
Some Relationships Amone. Spectral Densities where B is the effective bandwidth, and f is the center frequency of the frequency window function. In this section we further define and describe If the frequency windowcan be usefully approxi- the symbols and terminology we are using, and mated by a rectangular bandpass response function, we derive or explain some of the useful relation- then ships among spectral densities. 1 We use the Greek letter nu, V, in two some- ffo + zB what different senses in this paper. In one usage it is a Fourier frequency variable (index), for example, as the argument in the RF power spectral density 5 ,,-(v). In another usage it is the RFP frequency of a signal, for example, the instanta- neous frequency v(t) or the average (nominal) Combining Eqs. (25a) and (25b), and recalling the frequency v of a signal. Our usage of the 0. approximations made, we may write symbol f is similarly two-fold. The distinction is usually obvious in context in our treatment, and we sometimes state the usage explicitly. We urge caution on 2 S(f = f ) .=xIfo, B, window . this point in general; it is a hazard in this paper and in I other papers on signal stability. We use the operator 6 as the fluctuations operator. This approximation, Eq, (26), is commonly used to For example, by 6 we mean phase, and by 6 4 we mean describe and interpret measurements made by a fluctuations oi phase. The rate of change of phase with wave analyzer or a spectrum analyzer. It is exact time t is defined as angular frequency R. To obtain for white noise for the case of the (unrealizable) cycle frequency v, we normalize by dividing by 2n. rectangularbandpass compare Eq. (19) l . For the fluctuations of these quantities we may write In summary, note that a wave analyzer or a spectrum analyzer actually gives us a measurement of n2or its square root,12. The corresponding spectral density may be inferred by using the It follov~~from transform theory that approximation of Eq. (26). If for a specific case I I Eq. (25a), (25b), or both, is a poor approxi- mation, then Eq. (26) may also be a poor estimate of the spectral density. For such cases, we recommend the use of Script % squared, and the parameters associated with the measurement of The Fourier variable for angular frequency, W, is Script 3 squared should also be quoted. The commonly used. It is related to the Fourier variable critical parameters include at least f , B, and the for cycle frequency, f, by (L' = 2r f. The value of ? shape of the window function or equlvalent 6 V, normalized to the average (nominal) signal information. frequency V , is called y. This usage is similar to the usageoin reference 3.
42 5 Hence we may use provided that
(30) 2 S (fo)df' << 1 rad
The basic relationship between phase 0, fre- quency W,and time interval T is where f prime is a dummy index for integration.
4 = 2nv7 t L$,# For the types of signals under discussion and for [f I < W , we often may use as a good where 4 is anappropriate constant. The fluctua- approximation tions of time interval 6 T are hence related to the fluctuations of phase 6 $ by (39)
We note that for pure phase modulation. the RF power spectral density of the signal with sidebandsis not Hence necessarily symmetrical. An asymmetric RF power spectral density may reflect a mixture of correlated AM and PM, but it can also arise from special cases of pure PM (or FM). There is some confusion in the literature on this point. For pure AM, the RF power spectral density is strictly symmetrical. Combining Eqs. (28) to (33) This symmetry property is used in a later section on Script w(f). See Eq. (55).
A simple derivation of Eq. (37) is possible. We combine the derivation with an example which As in reference 3, x is defined such that illustrates the operationof a double-balanced mixer as a phase detector. Consider two sinusoidal 5-MHz signals (having negligible amplitude modulation) dt feeding the two input ports of a double-balanced mixer. When the two signals are slightly different in frequency, a slow, nearly sinusoidal beat with a period of several seconds at the outputof the mixer is measured to have a peak-to-peak swing of A Hence we see that x and 6 T are the same quantity. Ptp '
Without changing their amplitudes, the two signals Script x(f) are tuned to be at zero beat and in phase quadrature (that is, d2 out of phase with each other), and the Script x(f) is the normalized version of the output of the mixer is a small fluctuating voltage phase modulation (PM) portion of SJ-- RFP (v), with its centered on zero volts. Provided this fluctuating voltage is smallcompared to A 12, the phase frequency parameter f referenced to the signal's PtP average frequency v as the origin such that the quadrature condition is being closely maintained, 0 difference frequency f equals W - W . A complete and the "small angle condition" is being met. definition is given earlier by (10) .OScript zf) and Script %(f) represent concepts which commonly Phase fluctuations 6 $ between the two signals of arise in the languagesof modulation noise and of phases Q2 andrespectively, where stability of signals, 6$ = 6(02 - I (40) Script x(f) can be related in a simple way to S (f) S$ give rise to voltage fluctuations 6A at the output of the but only for the condition that the phase fluctuations mixer occurring at ratesf and faster are small compared to one radian. Otherwise Bessel function algebra must be used to relate Script x(f) to S (f) . Fortunately, 64 the "small angle condition" is often met in random where we use radian measure for phase angles, and noise problems. Specifically we find as a good approxi- we use ma tion sin &$= &$ (42) (37) for small 6 I$ (6 $ << 1 rad). We solve Eq. (41) for 6 4, square both sides, and take a time average
426 and hence Eq. (47) is useful for values off at least as low as (Zm)-' . Eqs. (47) and (48) correspond to Eqs. (37) and (38) respectively.
Script by its definition [see is a The angular brackets indicate averaging over time. In g(f> (lo)] measure of the phase fluctuation sidebands referred practice, the averaging time typically may be ten to the input (rti) of the unit under discussion. to one thousand times the inverse bandwidthof the Sometimes it is more meaningful to quote a measure measurement system. If we interpret the mean-square 11-12, 15-17 fluctuations of 6 4 and of 6 A, respectively, in Eq. (43) which is referred to the output (rto). in a spectral density fashion, we may write For this purpose we find it to be convenient to place a caret (A) over the quantity which is referenced to theoutput. For example, f(f), 6$, andSs$f> (44a) mean, respectively, Script X(f) rto, fluctuations of phase rto, and spectral density of 6 E rto. where we use
A Method of Measurement of Script x(f)
Usin a double-balanced mixer as a phase sensitive 58 detector, we have one of many ways to easily measure which is valid for the sinusoidal beat signal,. Script x(f) with a voltage spectrum analyzer. Eq. (49) is valid for the case where the reference signal has For the types of signals under consideration, the negligible phase fluctuations compared to the test two phase fluctuation sidebands (lower sideband and signal. upper sideband, at -f and +f from U , respectively) of 0 a signal are coherent with each other by definition. As already expressed in Eq. (391, they are, to a good approximation, of equal intensity also. The operation of the mixer when it is driven at quadrature is such that the amplitudesof the two phase side- bands add linearly in the outputof the mixer, resulting in four times as much power in the output as would be present if only one of the phase sidebands were where v is the measured value of (&A) rms on a square allowed to-contribute to the output of the mixer. root spectral density basis. However, Eq. (50) is the valid equation when we have two equally noisy signals Hence for If I < W we obtain (test and reference) driving the mixer. r I I
where p is thetotal power of thesignal, and, In case the device being measured has frequency total multiplication (synthesis) by the factor n, the using the definition of Script x(f), definition of Script 'b Figures 1 and 2 illustrate this use of double- we find for one fluctuating signal balanced mixers driven in phase quadrature for measurement of z(f). The method of Figure 1. which measures the differential stability between two parallel phase-processing arms, is used for singles or for pairs of phase- rocessing devices other than oscillating signal sources. The method of Figure 2 is used for oscillator provided the phase quadrature condition is approximately pairs. valid. The phase quadrature condition will be met for a time interval at least T long, provided ScriptW(f) Script %(f) is the normalized version of the amplitude modulation (AM) portion of S J%G-(" * with its frequency parameter f referenced to the signal's average frequencyv. taken as the origin 42 7 such that the difference frequencyf equals the double-balanced mixer at a higher level than the V - V . A complete definition is given earlier test signal. A ratio of 10 dB is adequate except for by (11). special cases, e.g., where the utmost linearity is required. Script %(f) can be related in a simple way to S (f) , the spectral density of the amplitude Amplitude fluctuations E of the signal under 6E 6 fluctuations, 6 E of a signal, defined in (4). Speci- test give rise to voltage fluctuations 6A at the output fically we find as an exact relation of the mixer 6A;: (%)+ . 0 and We solve Eq. (56) for 6 E, square both sides, and take a time average where we use Equation (57) is similar to'Eq. (43). If we interpret (53) themean-square fluctuations of 6 E and of 6A, re- spectivelsin Eq. (57) in a spectral density fashion, we may write which is valid for a sinusoidal signal. The signal under consideration is represented by where V(t) is the instantaneous valueof the signal, V where we use 0 is the average (nominal) amplitude of the signal, 6 E(t) represents the fluctuations of the amplitude, t is the (59) running time index, 4 is a constant, and 6 $(t) is the fluctuation of the pfase of the signal. This re- lation is similar to Eq. 2 of reference 3. which is valid for the sinusoidal beat signal. For \f I < v , Script m(f) is symmetrical about For the types of signals under consideration, by definition the two amplitude fluctuation sidebands the signal's average frequency v , 0 (lower sideband and upper sideband, at -f and +f from V , respectively) of a signal are coherent with 0 each other. As already expressed in Eq. (551, they are of equal intensity also. The operation of the A simple derivation of Eqs. (51) and (52) is possible. mixer when it is driven at colinear phase is such We combine the derivation with an example which illu- that the amplitudes of the two AM sidebands add strates the operation of a double-balanced mixer as an linearly in the outputof the mixer, resulting in amplitude detector. Consider two 5-MHz signals (having four times as much power in the output aswould phase fluctuations of much less than one radian) feeding be present if only one of the AM sidebands were the two input ports of a double-balanced mixer. When the allowed to contribute to the output of the mixer. Hence for If 1 < V we obtain two signals are slightly different in frequency, a slow, 0 nearly sinusoidal beat with a period of several seconds at the outputof the mixer is measured to have a peak-to- peak swing of A P*' Without changing their amplitudes, the two signals are tuned to be at zero beat and in colinear phase and, using the definition of Script w(f), (that is, either zero or-+ nsi radians phase angle difference), and the output of the mixer is a fluctuating voltagecentered on A /2 volts. We notethere is PtP no requirement that the output fluctuations be small compared to A /2 in the measurement of Script w(f). P tP To assure linearity of the demodulation, and to make the measurement be sensitive to the AM of only one (the test signal) of the two signals, we cause the other signal (the reference signal) to drive 42 8 or, using the normalized (fractional) fluctuations Summing Up of amplitude 6 E/V , we may write In this tutorial survey we have discussed some practical aspects of the use of spectral densities for the measurement and characterization of stability of signals. We have suggested some terminology which may be use- ful to the practicing metrologist. Some relationships 1 0 ~ I among various types of spectral densities were explained. We note the similarity of Eq. (63) and Eq. (47). Some common pitfalls were identified and discussed. These are two very useful equations to remember, We thank many people whose many excellent insights, as well as some occasional confusion, have provided A Method of Measurement of Script m(f) content for this survey and have given impetusto its preparation. We will appreciate comments and feedback. Using a double-balanced mixer as an amplitude sensitive detector, we have one of many ways to easily measure Script %(f) with a voltage spectrum analyzer. References Equation (64) is valid for the case where the reference signal contributes only a negligible amount to the out- Proceedings of the IEEE-NASA Symposium on the put fluctuations of the mixer as comparedto the amount 1. Definition and Measurement of Short-Term Fre- contributed by the test signal. quency Stability at Goddard Space Flight Center, Greenbelt, Maryland, 23 - 24 November 1964. Prepared by Goddard Space Flight Center (Scientific and Technical Information Division, National Aeronautical and Space Administration, Washington, D. C., 1965). Copies are available for $1.75 from the U. S. Government Printing where v is the measured value of (&A) rms on a Office, Washington, D. C. 20402. square root spectral density basis. The arrange- ments shown in Figures 1 and 2 need to be modified 2. Proc. IEEE, Special Issue on Frequency Stability, slightly in order to do the amplitude sensitive Vol. 54, No. 2, February 1966. measurements. 3. Barnes, J. A., A. R. Chi, L. S. Cutler, et al., The major change is that the two input signals to "Characterization of Frequency Stability", NBS the mixer must be in colinear phase, that is, the out- Technical Note 394, October 1970, 60 pages: put voltage of the mixer is at an extremum rather than also published in IEEE Trans. on I 81 M, Vol. at zero. A more complicated phase lock loop is used IM-20, No. 2, pp. 105-120, May 1971. Note: (although in some cases no loop is required) than The time domain-frequency domain conversion the one shown in Figure 2; the signal under test chart has a few errors. An updated version must be the weaker of the two input signals (hence (1973) of the conversion chart has been the 10-dB pad shown in Figure 2 must be used in published by J. H. Shoaf et al. in NBS Technical the other input for AM sensitive measurements); 4 Note 632 ; copies of the updated 1973 chart and and some type of DC-blocking may have to be used of TN 632 are available upon request. with the amplifiers when measuring the voltage noise of the demodulated AM in order to avoid 4. Shoaf, John H., Donald Halford, and A. S. Risley, overload due to the large DC component. "Frequency Stability Specification and Measure- ment: High Frequency and Microwave Signals", NBS Technical Note 632, January 1973, 70 pages. Suggestions for Further Reading This is a tutorial, how-to-do-it technical note intended to be useful to the practicing metrologist. A measurement system for AM and FM noise It totally supersedes NBS Report 9794 (1971, isthoroughly described by Ondria. Another unpublished). Copies are available for $0.65 from excellent standard reference for PM and FM noise the U, S. Government Printing Office, measurements is by Cutler and Searle,20 but they Washington, D. C. 20402. Order by SD Catalog No. do not consider AM noise explicitly. Compared to C13.46: 632. the number of papers on PM and FM noise measure- ments, there are relatively few publications 5. Blackman, R. B., and J. W. Tukey, The Measure- on AM noise measurements. For an operational ment of Power Spectra. New York: Dover, 1959, approach to the description of electrical noise pp. iii-x. in general, we strongly recommend the excellent book by Bennett.21 For references to additional 6. Glaze, D. J. , "Improvements in Atomic Cesium Beam relevant publications, we recommend selection from Frequency Standards at the National Bureau of the more than one hundred items in 4ppendix H Standards", IEEE Trans. on Instrumentation and and the Bibliography of NBS TN632. Measurement, Vol. IM-19, No. 3, pp. 156-160, August 1970. This paper was presented by J. A. Barnes at the 16th General Assembly of the International Scientific Radio Union (URSI) in 42 9 Ottawa, Canada, August 1969, and is published Copies are available for $6.50 from Electronics in its proceedings: Progress in Radio Science Industries Association, 2001 Eye Street, N. W. , 1966-1969, Vol. 2, pp. 331-341. Washington, D. C. ,20006. 7. Middleton, D. , An Introduction to Statistical 16. Meyer, Donald G., "A Test Set for Measurement of Communication Theory. New York: McGraw-Hill, Phase Noise on High-Quality Signal Sources", IEEE 1960, pp. 33-35. Equivalent statistical definitions of Trans. on Instrumentation and Measurement, Vol. stationarity are given in many other treatises on IM-19,No. 4, pp. 215-227, November 1970 (Proc. probability, statistics, and noise. of the 1970 Conference on Precision Electromagnetic Measurements, Boulder, Colorado, 2-5 June 1970). 8. Fano, R. M. , "Short-Time Autocorrelation Functions He gives a detailed discussion of a highly accurate, and Power Spectra", Jour. Acoustical Soc. America, high resolution measurement system. Vol. 22, No. 5, pp.546-550, September 1950. 17. Baugh, Richard A., "Low Noise Frequency 9. Turner, C. H. M., "On the Concept of an Instantaneous Multiplication", Proc. 26th Annual Symposium on Power Spectrum, and its Relationship to the Auto- ' Frequency Control, U. S. Army Electronics Command, correlation Function" , Jour, Applied Physics, Vol. 25, Fort Monmouth, N. J., 1972, pp. 50-54. Copies are No. 11, pp. 1347-1351, November 1954. available for $6.50 from Electronic Industries Association, 2001 Eye Street, N. W., Washington, D.C. 10. Rothschild, Donald R. , "A Note on Instantaneous 20006. Spectrum", Proc. IRE, Vol. 49, No. 2, p. 649, March 1961. 18. Van Duzer , V. , "Short-Term Stability Measurements", Proc. of the IEEE-NASA Symposium on the Definition 11. Halford, Donald, "Infrared-Microwave Frequency and Measurement of Short-Term Frequency Stability, Synthesis Design: Some Relevant Conceptual Noise NASA SP-80, 1965, pp. 269-272. See reference 1. Aspects", Proc. of the Frequency Standards and The basic method described by Van Duzer is simple, Metrology Seminar, Quebec, Canada, 30 August - elegant, easily instrumented, easily modified, ex- 1 September, 1971, pp. 431-466. Copies of the tremely versatile, and capable of the best resolution Proceedings are avajlable for$10.00 from the which is attained at today's state-of-the-art. See Quantum Electronics Laboratory, Lava1 University, also references 4, 15-17, and 19 for additional Quebec,Canada. descriptions of the use of double-balanced mixers. 12. Wells, J. S., and Donald Halford, "Frequency and 19. Ondria, J. G., "A Microwave System for Measure- Phase Stabilization of an HCN Laser by Locking to ments of AM and FM Noise Spectra", IEEE Trans. on a Synthesized Reference", NBS Technical Note 620, Microwave Theory and Techniques, Vol. MTT-16, May 1973, 56 pages. Copies are available for $0.60 No. 9, pp. 767-781, September 1968. He gives a from the U. S. Government Printing Office, very detailed discussion of a particular measurement Washington, D. C. 20402. Order by Catalog No. system. C 13.46: 620. 20. Cutler, L. S., and C. L. Searle, "Some Aspects of 13 Harvey, A. F. , Microwave Engineering. New York: the Theory and Measurement of Frequency Fluctuations Academic Press, 1963, p. 284. in Frequency Standards", Proc. IEEE, Vol. 54, No. 2, pp. 136-154, February 1966. Thisis a useful treat- 14I Richards, Paul I., "Computing Reliable Power ment of some of the theory, mathematics, and measure- Spectra", IEEE Spectrum, Vol. 4, No. 1, pp. 83-90, ment techniques - with physical insight into the January 1967. noise processes of practical concern. 15. Meyer, Donald G., "An Ultra Low Noise Direct 21. Bennett, W. R., ElectricalNoise. New York: Frequency Synthesizer", Proc . 24th Annual Symposium McGraw-Hill , 1960. on Frequency Control, U. S. Army Electronics Command, Fort Monmouth, N. J., 1970, pp. 209-232. 430 PHASE NOISE MEASUREMENT Foil AMPLIFIERS. CAMCITORS. CABLES, PADS, TRANSMISSION PATHS, FILTERS, ETC. ALSO PAIRS OF FREOUENCY MULTIPLIERS Figure 1. To measure Script x(f>,the double-balanced mixer is driven in phase quadrature as shown. By driving the two input ports in phase-parallel instead, this system measures Script %(f> of the device under test. See text for additional details for measuring Script %(f>. Note that time domain measurements can be made concurrently. --0 ltCQlOtl fltQUtlC1 OSCILLATOR STABILITY MEASUREMENT L PHASESENSITIVE MODE ISHORT TERM1 Figure 2. To measure Script x(f>,the double-balanced mixer is driven in phase quadrature as shown. By driving the two input ports in phase-parallel instead, this system measures Scriptwf) of the weaker of the two driving signals. See text for additional details. Note that time domain measurements can be made concurrently. 431