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Spectral Density Analysis 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 NBS TN632 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.
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