TIME DOMAIN FREQUENCY STABILITY ESTIMATION BASED ON FFT MEASUREMENTS

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1 35 t Annual Precise Time and Time Interval (PTTI) Meeting TIME DOMAIN FREQUENCY STABILITY ESTIMATION BASED ON FFT MEASUREMENTS P. C. Cang, H. M. Peng, and S. Y. Lin National Standard Time & Frequenc Lab., TL, Taiwan, Lane 55, Min-Tsu Road, Sec. 5, Yang-Mei, Taouan, Taiwan 36 Tel: ; Fax: ; betrand@ct.com.tw Abstract Te standard caracterizations o requenc stabilit are, in te time domain, te Allan (or two-sample) variance and, in te requenc domain, te spectral densit unction (SDF). Te ormer is matematicall related to te latter b te conversion between time and requenc domain. In tis paper, te biases o te Fast Fourier transorm (FFT) spectral estimate wit Hanning window are cecked and te resulting unbiased spectral densit are used to calculate te Allan variance. Bot te numerical integral and te curve-itting metods are presented to calculate te variances. Te numerical integral is a straigtorward metod to use, and we can get te integral approximation ater eliminating some spike points rom SDF, e.g. noise caused b ac power. In addition, a common model or SDF is linear combinations o powerlaw processes, wic are distinguised b te integer powers in teir unctional dependence on Fourier requenc wit te appropriate coeicients. Fitting a orm o te above model to te resulting SDF using standard regression tecniques can estimate tese coeicients. Cutler s ormula is adopted to calculate te integral approximation using tese coeicients. Te approximations o variances rom tese two metods are compared and analzed. Finall, we discuss te limitations and possible errors rom tese two metods. INTRODUCTION FFT spectrum analzers generall ave several dierent window unctions available or analzing signals. Recent researc as sown tat te Hanning window provides excellent perormance or analzing noise []. Our lab as establised a pase noise measurement sstem including a pase noise standard (,5,, MHz), a single-cannel noise detector, a dela line unit, and one single-cannel FFT spectrum analzer. Te signal reerence is rom a low-noise requenc reerence (LNFR-4) wit a noise level o about 73 dbc/hz (5 MHz PM, at Fourier requenc KHz). For passive devices, te sstem can measure up to 77 dbc/hz. Tis ear we also ave built up a cross-correlation sstem, wic can measure te noise dbc/hz below te above level. Te ver sort-term stabilit (τ<.5 second) b using tis pase noise measurement sstem is a subject o interest to us, since te traditional time interval counter is applicable onl wen τ is about second. In tis paper, we are tring to calculate te Allan variance o te spectral densit estimate rom experimental results, and possible errors in te spectral densit estimate, e.g. biases rom window unctions, noise rom ac power, etc. will be considered. In general, i te spectral densit o te normalized requenc luctuations S ( ) is known, its matematical relation to te Allan variance can be expressed as []: 6

2 Report Documentation Page Form Approved OMB No Public reporting burden or te collection o inormation is estimated to average our per response, including te time or reviewing instructions, searcing existing data sources, gatering and maintaining te data needed, and completing and reviewing te collection o inormation. Send comments regarding tis burden estimate or an oter aspect o tis collection o inormation, including suggestions or reducing tis burden, to Wasington Headquarters Services, Directorate or Inormation Operations and Reports, 5 Jeerson Davis Higwa, Suite 4, Arlington VA -43. Respondents sould be aware tat notwitstanding an oter provision o law, no person sall be subject to a penalt or ailing to compl wit a collection o inormation i it does not displa a currentl valid OMB control number.. REPORT DATE SEP 4. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE Time Domain Frequenc Stabilit Estimation Based On Ft Measure 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) National Standard Time & Frequenc Lab., TL, Taiwan, Lane 55, Min-Tsu Road, Sec. 5, Yang-Mei, Taouan, Taiwan PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES). SPONSOR/MONITOR S ACRONYM(S). DISTRIBUTION/AVAILABILITY STATEMENT Approved or public release, distribution unlimited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also ADM69, Proceedings o te 35t Annual Precise Time and Time Interval (PTTI) Meeting., Te original document contains color images. 4. ABSTRACT 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT UU a. REPORT unclassiied b. ABSTRACT unclassiied c. THIS PAGE unclassiied 8. NUMBER OF PAGES 6 9a. NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev. 8-98) Prescribed b ANSI Std Z39-8

3 35 t Annual Precise Time and Time Interval (PTTI) Meeting () σ ( τ ) = S 4 sin ( πτ ) ( ) d ( πτ ) were is te ig requenc cuto o a low-pass ilter. Te Allan variance can be easil calculated using numerical integration wile experimental data are properl processed. Besides, te power-law model is requentl used or describing pase noise, and it assumes tat te spectral densit o ractional requenc luctuations is equal to te sum o terms, eac o wic varies as an integer power o Fourier requenc. Tus, tere are two quantities tat completel speci S ( ) or a particular power-law process: te slope on a log-log plot or a given range o and te amplitude. Te slope is denoted b and tereore is te straigt line on a log-log plot tat relates S ( ) to. Te amplitude is denoted b and ence [-3]: () S ( ) = + = or < or > <. Cutler derived equation (3) rom equation () and (): (3) σ ( τ ) (π ) ln (π τ ) = τ + ln τ (π ) τ 3 (π ) τ Te value or eac coeicient could be obtained using standard regression tecniques, and ten we ave te Allan variance. Details or practicing tese metods are discussed in te ollowing sections. THE EXPERIMENTS Te measurement sstem consists o a FSSE noise detector, a FSSM pase noise standard, a FSSA dela line unit, a SDI LNFR-4 low-noise requenc reerence, and one SRS-76 FFT spectrum analzer wit a Hanning window adopted. Te measurement procedures and data recording are automated under te sotware TestStation version 3.. Te basic and simple experiment is te sstem noise loor test. Te LNFR-4 5 MHz output is split wit a reactive splitter to provide two input signals. Tese signals wit eiter o tem passing troug te noise standard in advance are connected to te pase noise detector. Te output o te noise detector is ten ed into te FFT spectrum analzer. A composite grap wit bot te calibration line and te corrected pase noise data is sown in Figure. Te noise components o 6 Hz, Hz, 8 Hz, etc. are commonl encountered because ac power is getting into te measurement sstem or te source under test. Te use o a pase noise standard greatl speeds up te process o calibrating te noise measurement sstem, and we can quickl review te calibration o te sstem wile looking at te inal data. 6

4 35 t Annual Precise Time and Time Interval (PTTI) Meeting CALCULATION OF EXPERIMENTAL RESULTS Because biases due to linearit and SDF accurac o FFT spectrum analzers are tpicall less tan.4 db wit 95% conidence limits, we ma temporaril neglect teir inluence in te process o calculating Allan variance. In Figure, te raw data o te above-mentioned noise loor test and te ones wit outliers removed are sown. Notice tat most o te outliers are rom te noise o ac power. In Figure 3, we calculate te Allan deviation σ (square root o Allan variance) o tese two spectral data using te numerical integration tecniques wit sampling time τ =. s ~ s. Note tat te amplitude o Allan deviation o te raw data varies up and down depending on te sampling time, wile te oter doesn t var obviousl. Tis is because in te time domain te sensitivit to a periodic wave varies directl as te sampling interval. Tis eect (wic is an alias eect) is a ver powerul tool or iltering out a periodic wave imposed on a signal source [4]. Also ound is tat altoug te inluence o ac power noise is important in calculating Allan deviation, eliminating tese outliers rom te raw data seems to reveal te true caracteristic o te device under test (DUT) reasonabl. Next, we will calculate te Allan deviation o te spectral data wit outliers removed using te curveitting metods and compare te results wit te ones using numerical integration metod. In te requenc domain, L ( ) is te prevailing measure o pase noise among manuacturers and users o requenc standards, and it is deined as: (4) (5) (6) S ( ) = S v L( ) = φ S ( ) φ ( ) dbc = log(l ( )) Hz In Figure, it is a L ( ) vs. plot wit its x- axis in log scale. For = Hz ~ Hz, we see tat wen increases b one decade, L ( ) also goes down b one decade. Tis noise process can be identiied as licker PM. For = ~ khz, we ave wite PM. We calculate te S ( ) irst and use te unctions S ( ) = and S ( ) = to it te data in licker PM and wite PM region respectivel, and ten get = and =.6-3. Tis regression tecnique is called as te polnomial curve-itting metod. Te power-law processes can also be expressed as: ln S ( ) = ln ln ( =... + ) (7) + We can also it te data in log-log space wit = and = and ten get = and = Tis regression tecnique is called as te log scale curve-itting metod. Figure 4 sows te calculated Allan deviation using Cutler s equation wit tese coeicients, wile te sampling time τ is rom. to s wit an increment o. s. We ind tat te results using bot te curve-itting metods 63

5 35 t Annual Precise Time and Time Interval (PTTI) Meeting are ver close, but te also keep an obvious bias rom te ones using te numerical integration metod. Te ormer are rougl about twice as large as te latter or te wole sampling interval. For example, te σ ( τ.s) using curve-itting metods is about 4.8-3, wile te one using numerical = integration is.4-3 and, wen τ = s, te related values are about and CONCLUSIONS In tis paper, we calculate and compare te Allan deviation o te experimental spectral results using te numerical integral and curve-itting metods. We ind tat te inluence o ac power noise pla an important role in calculating Allan deviation, so eliminating tese outliers rom te raw data seems to reveal te true caracteristic o te DUT reasonabl. As or te resulting bias between curve-itting and numerical integration metods, we ave tried to sit te spectral estimate in te range o db based on te assumption tat possible errors ma occur in te calibration process and cecked te bias, but it still exists witout improvement. In order to solve tis problem, we will do more researc in te near uture. REFERENCES [] F. L. Walls, D. B. Percival and W. R. Irelan, 989, Biases and Variances o Several FFT Spectral Estimators as a Function o Noise Tpe and Number o Samples, in Proceedings o te 43 rd Annual Smposium on Frequenc Control, 3 Ma- June 989, Denver, Colorado, USA (IEEE Publication 89CH69-6), pp [] Caracterization o Frequenc and Pase Noise, 986, Report 58 o te CCIR, pp [3] D. Allan, H. Hellwig, P. Kartasco, J. Vanier, J. Vig, G. M. R. Winkler, and N. Yannoni, 988, Standard Terminolog or Fundamental Frequenc and Time Metrolog, in Proceedings o te 4 nd Annual Smposium on Frequenc Control, -3 June 988, Baltimore, Marland, USA (IEEE Publication 88CH588-), pp [4] D. A. Howe, D. W. Allan, and J. A. Barnes, 98, Properties o Signal Sources and Measurement Metods, in Proceedings o te 35 t Annual Smposium on Frequenc Control, 7-9 Ma 98, Piladelpia, Pennslvania, USA (NTIS AD-A87), pp. A-A47. 64

6 35 t Annual Precise Time and Time Interval (PTTI) Meeting -5 LNFR-4 5MHz SELF TEST Noise Measured Calibration Line L() (dbc/hz) ,,, S/N 875 Oset Frequenc (Hz) /4/ :5 A Figure. Pase Noise Measurement o Sstem Noise Floor (5 MHz). Figure. Raw Data and Data wit Outliers Removed. 65

7 35 t Annual Precise Time and Time Interval (PTTI) Meeting Figure 3. Calculated Allan Deviation o Two Spectral Data. Figure 4. Calculated Allan Deviation wit Dierent Metods. 66