NIST Technical Note 1337: Characterization of Clocks and Oscillators

248 A. L. LANCE, W. D. SEAL, AND F. LABAARofthe mixer. Hence, for If I < v o , and considering only the phase modulationportion of the spectral density of the (square root of) power. we obtainand, using the definition of f.e(f),SW(jfl)/{Vrms)2 ~ 4[SJif'P(vo + f)]/(Ptot) (24)f.e(f) == [S,;;rp(l'o + f)]/(P1oJ ~ 1S6q,(1 f I). (25)Therefore, for the condition that the phase fluctuations occurring at rates(f) and faster are small compared to one radian, a good approximation inradians squared per hertz for one unit isIf the small angle condition is not met, Bessel-function algebra must be usedto relate !E(f) to SM,(f)·The NBS-defined spectral density is usually expressed in decibels relativeto the carrier per hertz and is calculated for one unit asIr is very important to note that the theory, definitions, and equations previouslyset forth relate to a single device.(26)(27)D. MODULATION THEORY AND SPECTRAL DENSITYRELATIONSHIPSApplying a sinusoidal frequency modulation fm to a sinusoidal carrierfrequency l'O produces a wave that is sinusoidally advanced and retardedin phase as a function of times. The instantaneous voltage is expressed as,V(t) = Va sin(2nvot + ~¢ sin 2nfmf), (28)where ~¢ is the peak phase deviation caused by the modulation signal.The first term inside the parentheses represents the linearly progressingphase of the carrier. The second term is the phase variation (advancing andretarded) from the linearly progressing wave. The effects of modulation canbe expressed as residual fm noise or as single-sideband phase noise. Formodulation by a single sinusoidal signal, the peak-frequency deviation ofthe carrier (l'O) is~vo = ~¢'fm'(29)~¢ = ~vo/fm, (30)where Im is the modulation frequency. This ratio of peak frequency deviationto modulation frequency is called modulation index m so that ~¢ = m andm = ~vo/fm' (31 )TN-199