NIST Technical Note 1337: Characterization of Clocks and Oscillators

ID8second moment measures (such as power spectra andvariances) are not totally adequate.Two common types of equipment used to evaluate theperformance of a frequency source are (analog) spectrumanalyzers (frequency domain) and digital electroniccounters (time domain). On occasion the digital counterdata are converted to power spectra by computers. Onemust realize that any piece of equipment simultaneouslyhas certain aspects most easily described in the timedomain and other aspects most easily described in thefrequency domain. For example, an electronic counterhas a high-frequency limitation, an experimental spectraare determined with finite time averages.Research has established that ordinary oscillatms demonstratenoise, which appears to be a superposition ofcausally generated signals and random nondeterministicnoises. The random noises include thermal noise, shotnoise, noises of undetermined origin (such as flickernoise), and integrals of these noises.One might well expect that for the more general casesone would need to use a nonstationary model (not stationaryeven in the wide sense, i.e., the covariance sense).Nonstationarity would, however, introduce significant difficultiesin the passage between the frequency and timedomains. It is interesting to note that, so far, experimentershave seldom found a nonstationary (covariance)model useful in describing actual oscillators.In what follows, an attempt has been made to separategeneral statements that hold for any noise or perturbationfrom the statements that apply only to specific models.It is important that these distinctions be kept inmind.III. BACKGROUND AND DEFINITIONSTo discuss the concept of frequency stability immediatelyimplies that frequency can change with time andthus one is not considering Fourier frequencies (at leastat this point). 'The conventional definition of instantaneous(angular) frequency is the time rate of change ofphase; that is(1)where cI>(t) is the instantaneous phase of the oscillator.This paper uses the convention that time-dependentfrequencies of oscillators are denoted by vet) (cycle frequency,hertz), and Fourier frequencies are denoted by"" (angular frequency) or f (cycle frequency, hertz) where'" == Zrrf. In order for (1) to have meaning, the phase cI>(t)must be a well-defined function. This restriction immediatelyeliminates some "nonsinusoidal" signals such asa pure random uncorrelated ("white") noise. For mostreal signal generators, the concept of phase is reasonablyamenable to an operational definition and this restrictionis not serious.Of great importance to this paper is the concept ofspectral density, Sg (I). The notation Sp (I) is to representthe one-sided spectral density of the Ipure real Ifunction g (t) on 0. per hertz basis; that is, the tot aI"power" or mean-square value of g (t) is given by1~ S,(f) df·Since the spectral density is such an important conceptto what follows, it is worthwhile to present someimportant references on spectrum estimation. There aremany references on the estimation of spectra from datarecords, but worthy of special note are [2 J- [5 J.IV. DEFINITION OF MEASURES OF FREQUENCY STABILITYA. General(SECOND-MoMENT TYPE)Consider a signal generator whose instantaneou" outputvoltage V (t) may be written asV(t) = [Vo + t(t)J sin [211"vot + 'I'(t)] (2)where V o and I'll are the nominal amplitude and frequency,respectively, of the output and it is assumedthatand(3)I.~(t) I« I (+)1-""'0for substantially all time t. Making use of (l) and (2)one sees thatand4>(t)vet) = Vo + .; .j>(t)._11"Equations (3) and (4) are essential in order that 'I'(t)may be defined conveniently and unambiguously (seemeasurement section).Since (4) must he valid even to speak of an instantaneousfrequency, there is no real need to distinguishstability measures from instability measures. That is,any fractional frequency stability measure will be farfrom unity, and the chance of confusion is slight. It istrue that in a very strict sense people usually measureinstability and speak of stability. Because the chances ofconfusion are so slight, the authors have chosen to continuein the custom of measuring "instability" and speakingof stability (a number always much less than unity).Of significant interest to many people is the radio frequency(RF) spectral density S\.(f). This is of directconcern in spectroscopy and radar. However, this is nota. good primary measure of frrquency stability for tworeasons. First, fluctuations in the :lmplitude d t) contributedirectly to 8 1 , (n ; and second, for many cases '\'hell(5)TN-149