ESTIMATION OF TIME-DOMAIN FREQUENCY STABILITY BASED ON PHASE NOISE MEASUREMENT
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1 ESTIMATION OF TIME-DOMAIN FREQUENCY STABILITY BASED ON PHASE NOISE MEASUREMENT P. C. Cang, H. M. Peng, and S. Y. Lin National Standard Time & Frequenc Laborator, TL, Taiwan, Lane 55, Min-Tsu Road, Sec. 5, Yang-Mei, Taouan, Taiwan 36 Tel: ; Fax: ; betrand@ct.com.tw Abstract Te time domain caracterization of te frequenc fluctuations is usuall expressed in terms of te Allan variance, σ ( ), or te modified Allan variance, Mod σ ( ). Bot variances can be accuratel determined b te integral relations to S ( f ), te power spectral densit of fractional frequenc fluctuations, wic include five tpes of noise: Wite PM, Flicker PM, Wite FM, Flicker FM and Random Walk FM. Tese noise tpes are distinguised b te integer powers ( α ) in teir functional dependence on Fourier frequenc f. Because te noise is inerent to all kinds of oscillators and measurement sstems, specifing teir contributions to te time domain frequenc stabilit is important and meaningful. In tis paper, bot te numerical integral and te curve-fitting metods are presented to estimate te frequenc stabilit from te results of pase noise measurement of oscillators, amplifiers, etc. Te numerical integral is a direct wa to use and we calculate te integral approximation after smooting some spike points. In addition, owing to te properties of power-law noise processes, te weigting coefficient α of eac tpe of noise component could be estimated wen curve-fitting skills are adopted. Cutler s formula is used to calculate te integral approximation using tese coefficients. Te approximations of frequenc stabilit from tese two was are compared and analzed. Lastl, te limitations and possible errors from te estimating metods are also discussed. INTRODUCTION In tis paper, we are tring to use te pase noise measurement results to calculate te time domain frequenc stabilit due te conversion between time and frequenc domain. In general, if te spectral densit of te normalized frequenc fluctuations S ( f ) is known, its matematical relation to te Allan variance can be expressed as: f 4 sin πf σ ( ) = S ( f ) df () ( πf ) were f is te ig frequenc cutoff of a low pass filter. From equation (), te Allan variance can be straigtforwardl calculated wile numerical integration is adopted. Besides, te power-law model is 493
2 frequentl used for describing pase noise from oscillators, amplifiers, etc. It assumes tat te spectral densit of fractional frequenc fluctuations is equal to te sum of terms, eac of wic varies as an integer power of frequenc. Tus, tere are two quantities tat completel specif S ( f ) for a particular power-law process: te slope on a log-log plot for a given range of f and te amplitude. Te slope is α denoted b α and, terefore, f is te straigt line on a log-log plot tat relates S ( f ) to f. Te amplitude is denoted b α and ence: < < = + α α f for f f S ( f ) () α= for f > f Te Allan variance derived b Cutler from equation () and () is as follows [-]: σ ( π ) ln( πf ) ( π ) ( ) = + ln (3) 3 f ( π ) If te value for eac weigting coefficient α is appropriatel determined, ten te Allan variance could be calculated. Tis is carried out b te curve-fitting skills discussed below. THE EXPERIMENTS Te measurement sstem consists of a FSSE pase noise detector, a FSSM noise standard, a FSSA dela line unit, a SDI LNFR-4 low-noise frequenc standard, and a SRS-76 fast Fourier transformer (FFT) wic is used to analze te output signal from te pase noise detector. Te measurement processes and data recording could be automated under software TestStation version 3.. All te measurement sstem including te DUTs sould be warmed up for at least 4 ours before an test. Te first experiment was te sstem noise floor test. We used a power splitter to divide te LNFR- 4 -MHz output into two signals and ten followed te procedure for passive component measurement. In te second experiment, te pase noise of a 5-MHz frequenc output from te drogen maser 765 was measured using te pase-lockable LNFR-4 serving as a reference. Bot experimental results are sown in Figure and Figure. CALCULATION OF EXPERIMENTAL RESULTS In te frequenc domain, L ( f ) is te prevailing measure of pase noise among manufacturers and users of frequenc standards, and it is defined as [3]: L( f ) = Sφ ( f ) (4) ν β Sφ ( f ) = S ( f ) = ν f α f ( β α ) (5) db H C Z = log( L( f )) (6) were S φ ( f ) is te spectral densit of pase fluctuations. In Figure, it is an L ( f ) vs. f plot wit its x- axis in log scale. For f = Hz~ Hz, we see tat wen f increases b one decade, L ( f ) also goes 494
3 down b one decade. Tis noise process can be identified as flicker PM. For f = ~99.75 khz, we ave wite PM. In te region f = khz~ khz, it seems tat flicker PM and wite PM coexist and none of tem could surpass eac oter. After smooting some spike points in te raw data, a transformation from L ( f ) to S ( f ) was made. We used te function S ( f ) = f and S ( f ) f = to fit te data in te flicker PM and wite PM region separatel, and ten got = and = To make sure te values of and were appropriatel determined, we calculated teir contributions of S ( f ) in te flicker PM and wite PM region, and verified tat te interactions between tese two coefficients were insignificant. In Figure3, te blue line sows te results from L ( f ) to S ( f ) transformation and te red line is residuals of te former after te contributions of power-law model f + f and some outling points ave been removed. Figure4 sows te Allan deviations σ ( ) calculated from te numerical integration and te power-law metod wit te cutoff frequenc f = khz and averaging period =.~ s. Te Allan deviations from tese two metods are in good agreement wit eac oter (te relative errors are less tan 6 %) except wen is equal to. s and s. Following a similar procedure to deal wit te experimental results in Figure, tree kinds of noise processes including wite FM ( = ), flicker PM ( = ), and wite PM ( = ) could be identified. Te relative diagrams are sown in Figure 5 and Figure 6. We observed tat te growing rate of te relative errors became faster wen increases and tat tere are two abrupt plunges in te numerical integration wen is equal to. s and s. CONCLUSION In tis paper, we calculated and compared te time domain frequenc stabilit using te numerical integral and te curve-fitting metods. Te curve-fitting metod is useful to obtain te value of a weigting coefficient after identifing te Fourier frequenc range for a certain power-law process. As for te numerical integration, it is straigtforward to use, but its generated results cange abruptl for some values of. In order to solve tis problem, more researc will be done in te future. REFERENCES [] S. R. Stein, 985, Frequenc and Time Teir Measurement and Caracterization, Precision Frequenc Control (Academic Press, New York), volume, capter, pp [] Caracterization of Frequenc and Pase Noise, Report 58 of te CCIR, pp. 4-5, 986. [3] D. Allan, H. Hellwig, P. Kartascoff, J. Vanier, J. Vig, G. M. R. Winkler, and N. F. Yannoni, 988, Standard Terminolog for Fundamental Frequenc and Time Metrolog, in Proceedings of te 4 nd Annual Smposium on Frequenc Control, -3 June 988, Baltimore, Marland, USA (IEEE Publication 88CH588-), pp
4 -35 LNFR-4 Self Test L(f) Offset Frequenc (Hz) Figure. Pase noise measurement of LNFR-4 self test ( MHz). - H-Maser 765 Test L(f) Offset Frequenc (Hz) Figure. Pase noise measurement of H-maser 765 (5 MHz). 496
5 x - LNFR-4 Self Test 8 S (f) Offset Frequenc (Hz) x 4 Figure 3. S ( f ) and its residual after te contributions of powerspectral model and outling points ave been removed. - Numerical Integration - Power Law Metod -3 σ -4 σ Relative Error Figure 4. Allan deviations from two different metods (LNFR-4 self test). 497
6 6 x -9 H-Maser 765 Test 5 4 S (f) Offset Frequenc (Hz) x 4 Figure 5. S ( f ) and its residual after te contributions of power- spectral model and outling points ave been removed. - Numerical Integration - Power Law Metod - σ -3 σ Relative Error Figure 6. Allan deviations from two different metods (H-maser 765). 498
7 QUESTIONS AND ANSWERS DAVE HOWE (National Institute of Standards and Tecnolog): One of te problems wit doing a curve fit to someting like L (f) L (f) itself is smooted, tat is, te residuals are not wite. Wat measures did ou take to sow tat te residuals are wite in te curve-fitting process? And wat sort of FFT window function did ou use? PO-CHENG CHANG: We didn t consider man vectors. So it was a ver eas wa to calculate it. 499
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