NIST Technical Note 1337: Characterization of Clocks and Oscillators

194SAMUEL R. STEIN)----1BANDPASS FILTER v'm MEAN - SQUAREMet - '0)METERFIG. 12-1 An rf speclrum analyzer. The device produces an output proponional 10 themean-square value of the signal passing through a tunable narrow-band filter centered atfrequency10'in Fig. 12-1. The spectrum of the filtered voltage V'(t) is equal to the squareof the magnitude of the filter transfer function H(f - fo) multiplied by thespectrum of the input signal (Cutler and Searle, 1966). The variance of thefiltered voltage is obtained from Parseval's theorem:Cl~·(fO) = f~ IH(f - fo)f S~(f) df. (12-2)If the bandpass filter is sufficiently narrow, so that S!i'(j) changes negligiblyover its bandwidth, then Eq. (12-2) may be inverted. With this assumption,the power spectrum is estimated from the measurement using Eq. (12·3):S!i'(fo) = (1~,(fo)/B, (12-3)where B = J~ '" IH(f' - 10)1 2 df' is the noise bandwidth of the filter andfo itscenter frequency. Figure 12-2 shows a typical two-sided rf spectrum. Formany oscillators the spectrum has a Lorentzian shape, that is,STS = 2< y2>/7r AfJdBIi (f) 1 + U/(A!3dB'''l))2 . (12-4)The Lorentzian lineshape is completely described by the mean square voltage and the full width at half maximum A!JdB'-f e 0 f eFIG. 12-2 The rf spectrum of a signal. It is orten useful 10 divide Ihe speclrum inlo thecarrier and the noise pedeslal. The speclral densily of the carrier exceeds that of Ihe noisepedestal for Fourier rrequencies smaller in magnitude IhanJ;.IN-64