POWER LAW NOISE IDENTIFICATION USING THE LAG 1 AUTOCORRELATION

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1 Preprnt of paper to be presented at the 8th European Frequency and Tme Forum, Unversty of Surrey, Guldford, UK. 5-7 Aprl 4 POWER LAW OISE IDETIFICATIO USIG THE LAG AUTOCORRELATIO W.J. Rley* and C.A. Greenhall * Symmetrcom, Inc., 34 Tozer Road, Beverly, MA 95 USA, wrley@symmetrcom.com Jet Propulson Laboratory, 48 Oa Grove Drve, Pasadena, CA 99 USA, Charles.Greenhall@jpl.nasa.gov Keywords: Power Law ose, Autocorrelaton, Frequency Stablty Analyss. Abstract Ths paper descrbes a new method for power law nose dentfcaton, based on the lag autocorrelaton functon, that can determne the domnant nose type for all common nose processes, from phase or frequency data, for all averagng factors, n a consstent and analytc manner. Bacground It s often necessary to dentfy the domnant power law nose process (WPM, FPM, WFM,, RWFM, FWFM or RRFM) of the spectral densty of the fractonal frequency fluctuatons, S y (f) = h f ( = to 4), to perform a frequency stablty analyss. For example, nowledge of the nose type s necessary to determne the equvalent number of ch-squared degrees of freedom (edf) for settng confdence ntervals and error bars, and t s essental to now the domnant nose type to correct for bas n the newer Total and Thêo varances. Whle the nose type may be nown a pror or estmated manually, t s desrable to have an analytc method for power law nose dentfcaton that can be used automatcally as part of a stablty analyss algorthm. 3 The Autocorrelaton Functon The autocorrelaton functon (ACF) s a fundamental way to descrbe a tme seres by multplyng t by a delayed verson of tself, thereby showng the degree by whch ts value at one tme s smlar to ts value at a certan later tme. More specfcally, the autocorrelaton at lag s defned as ρ = E[( zt µ )( zt+ µ )] σ z () where z t s the tme seres, µ s ts mean value, σ z s ts varance, and E denotes the expected value. The autocorrelaton s usually estmated by the expresson r = t= ( z z)( z z) t t= t+ ( z z) t where z s the mean value of the tme seres and s the number of data ponts [3]. 4 The Lag Autocorrelaton Pror Art The lag autocorrelaton s smply the value of r as gven by the expresson above. For frequency data, the lag autocorrelaton s able to easly dentfy whte and flcer PM nose, and whte (uncorrelated) FM nose, for whch There s lttle lterature on the subject of power-law nose dentfcaton. The most common method for power law nose dentfcaton s smply to observe the slope of a loglog plot of the Allan or modfed Allan devaton versus averagng tme, ether manually or by fttng a lne to t. Ths obvously requres at least two stablty ponts. Durng a stablty calculaton, t s desrable (or necessary) to automatcally dentfy the power law nose type at each pont, partcularly f bas correctons and/or error bars must be appled. Prevous methods for power law nose dentfcaton [], based on the Barnes B and R(n) bas ratos [], have been ad hoc, have not used a consstent methodology for all nose types, and have not handled all cases (e.g. resolvng whte and flcer PM at unty averagng factor). the expected values are /, /3 and zero respectvely. The more dvergent noses have postve r values that depend on the number of samples, and tend to be larger (approachng ). For those more dvergent noses, the data are dfferenced untl they become statonary, and the same crtera as for WPM, FPM and WFM are then used, corrected for the dfferencng. The results can be rounded to determne the domnant nose type or used drectly to estmate the nose mxture. 5 ose Identfcaton Usng r An effectve method for dentfyng power law noses usng the lag autocorrelaton s based on the propertes of dscrete-tme fractonally ntegrated noses havng spectral ()

2 denstes of the form ( sn π f ) -δ. For δ < ½, the process s statonary and has a lag autocorrelaton equal to ρ = δ / (-δ) [4], and the nose type can therefore be estmated from δ = r / (+r ). For frequency data, whte PM nose has ρ = -/, flcer PM nose has ρ = -/3, and whte FM nose has ρ =. For the more dvergent noses, frst dfferences of the data are taen untl a statonary process s obtaned as determned by the crteron δ <.5. The nose dentfcaton method therefore uses p = -round (δ) d, where round (δ) s δ rounded to the nearest nteger and d s the number of tmes that the data s dfferenced to brng δ down to <.5. If z s a -average of frequency data y(t), then = p; f z s a -sample of phase data x(t), then = p +, where s the usual power law exponent f, thereby determnng the nose type at that averagng tme. The propertes of ths power law nose dentfcaton method are summarzed n Table. It has excellent dscrmnaton for all common power law noses for both phase and frequency data, ncludng dffcult cases wth mxed noses. 6 ose ID Algorthm The basc lag autocorrelaton power law nose dentfcaton algorthm s qute smple. The nputs are a vector z,, z of phase or frequency data, the mnmum order of dfferencng dmn (default = ), and the maxmum order of dfferencng dmax. The output s p, an estmate of the of the domnant power law nose type, and (optonally) the value of d. Done = False, d = Whle ot Done z = z r = = = r δ = + r ( z z)( z z) = + ( z z) If d >= dmn And (δ <.5 Or d >= dmax) p= ( δ + d) Done = True Else z = z z,..., z = z z = d = d + End If End Whle ote: May round p to nearest nteger The nput data should be for the partcular averagng tme,, of nterest, and t may therefore be necessary to decmate the phase data or average the frequency by the approprate averagng factor before applyng the nose dentfcaton algorthm. The dmax parameter should be set to or 3 for an Allan or Hadamard ( or 3-sample) varance analyss respectvely. The alpha result s equal to p+ or p for phase or frequency data respectvely, and may be rounded to an nteger (although the fractonal part s useful for estmated mxed noses). The algorthm s fast, requrng only the calculaton of one autocorrelaton value and st dfferences for several tmes. It s ndependent of any partcular varance. 7 Results The lag autocorrelaton method yelds good results, consstently dentfyng pure power nose for = to 4 for sample szes of about 3 or more, and generally dentfyng the domnant type of mxed noses when t s at least % larger than the others. For a mxture of adjacent noses, the fractonal result provdes an ndcaton of ther rato. For those reasons, and because t can handle all averagng factors, the new lag autocorrelaton method has replaced the B /R(n) bas rato method n Verson.4a and hgher of the Stable3 program [5]. 8 Examples Examples of the lag autocorrelaton method for power law nose dentfcaton are shown n Fgures and for sets of 4 ponts of pure and mxed smulated nose, as generated by the Kasdn-Walter method [6], wth approxmately equal Allan varances (=) summed for the mxed noses. Fgure 4 shows a composte plot of overlappng Allan devaton and lag ACF nose type for a par of SAO VLGB hydrogen masers. The nose type vares from whte/flcer PM at short averagng tmes to more dvergent random wal/flcer wal FM at longer averagng tmes. Fgure 5 shows a composte plot of overlappng Allan devaton and Lag ACF nose type for a Symmetrcom Cs-III hgh performance laboratory cesum frequency standard vs. a Symmetrcom MHM hydrogen maser. As s typcal for such devces, t dsplays whte FM nose out to an averagng tme of several days before reachng a flcer floor. Fgure 6 shows a composte plot of overlappng Allan devaton and Lag ACF nose type for a Symmetrcom Model 83 mltarzed rubdum frequency standard, agan vs. a Symmetrcom MHM hydrogen maser. It dsplays whte FM nose before reachng a regon of flcer and random wal FM nose at an averagng tme of about 4 seconds. 9 Lmtatons Before analyss, the data should be preprocessed to remove outlers, dscontnutes, and determnstc components. Smulatons usng sets of whte FM nose of varous

3 sample szes have shown that acceptable results can be obtaned from the lag autocorrelaton nose dentfcaton method for 3, where s the number of data ponts, as shown n the Table. The table shows the percentage of estmates that dffer from the expected value of by more than a half-nteger nose type. Ths percentage ncludes whatever error s due to the smulated nose data tself. % Range to to to to to to +. Concluson Ths paper has descrbed a method for power law nose dentfcaton based on the lag autocorrelaton functon. It s a fast and effectve way to support the settng of confdence ntervals and to apply bas correctons durng a frequency stablty analyss. References. D. Howe, R. Beard, C. Greenhall, F. Vernotte and B. Rley, A Total Estmator of the Hadamard Functon Used by GPS Operatons, Proc. 3 nd Precse Tme and Tme Interval (PTTI) Systems and Applcatons Meetng, ovember, pp J.A. Barnes, Tables of Bas Functons, B and B, for Varances Based on Fnte Samples of Processes wth Power Law Spectral Denstes, BS Techncal ote 375, Jan Table : Percent Incorrect Power Law ose Estmates vs. Sample Sze It would be nterestng to devse an mproved lag 3. G. Box and G. Jenns, Tme Seres Analyss, Forecastng and Control, Chapter, Holden-Day, Oaland, Calforna, 976, ISB autocorrelaton nose dentfcaton method (perhaps based on reflected data le the Total varance) that could wor down to smaller sample szes. The algorthm tends to produce jumps n the estmated alpha for mxed noses when the dfferencng factor, d, changes (although the 4. P. Brocwell and R. Davs, Tme Seres: Theory and Methods, nd Edton, Eq. (3..9), Sprnger-Verlag, ew Yor, Stable3, Software for Frequency Stablty Analyss, Hamlton Techncal Servces, 95 Woodbury Street, alpha value when rounded to an nteger s stll consstent). South Hamlton, MA 98 USA, As shown n Fgures 3, ths can be avoded by usng the same d for the entre range of averagng tmes, at the expense of hgher varablty when a lower d would have been suffcent. Further research s encouraged to develop 6..J. Kasdn and T. Walter, Dscrete Smulaton of Power Law ose, Proc. 99 IEEE Frequency Control Symposum, pp , May 99. even better ways to dentfy power law nose. ose Type Phase Data* x(t) Lag Autocorrelaton, r d= ACF of d= d= d= Phase Data x(t) y(t) x(t) y(t) x(t) y(t) W PM -/ -/ -/3 -/3-3/4 F PM.7 -/3 -/3-3/5-3/5-5/7 W FM -/ -/ -/3 F FM /3 -/3-3/5 RW FM - -/ * The dfferencng operaton changes the appearance of the phase data to that shown rows hgher. Shaded values are those used for nose ID for the partcular nose and data type Table : Lag Autocorrelaton for Varous Power Law oses and Dfferences 3

4 Lag Autocorrelaton Estmated Pure Integer Power Law ose Type (=4) Alpha # W PM, = + Max = +.47 Avg = Mn = F PM, = + Max = Avg = Mn = W FM, = Max = +.4 Avg = -.4 Mn = -.9 F FM, = - Max = Avg = -.45 Mn = -.95 RW FM, = - Ma x= -.86 Avg = Mn = -.38 Fgure : Examples of Lag Autocorrelaton ose ID for Pure Integer Smulated Power Law oses Lag Autocorrelaton Estmated Mxed Integer Power Law ose Type (=4) Alpha # W PM & F PM, = +.5 Max = +.74 Avg = (.359) Mn = +.7 F PM & W FM, = +.5 Max = Avg = +.38 (.373) Mn = +.57 W FM & F FM, = -.5 Max = -.38 Avg = (-.656) Mn = -. F FM & RW FM, = -.5 Max = Avg = -.64 (-.6) Mn = -.88 ote: Theoretcal values shown n ( ) Fgure : Examples of Lag Autocorrelaton ose ID for 5% Mxture of Adjacent Integer Power Law oses dmn= dmn= dmn= Fgure 3: ose Analyss of the Rubdum Frequency Standard of Fgure 6 for dmn =, and 4

5 Overlappng Allan Devaton, σ y SAO VLGB H-Maser S/ SAO6 vs SAO8 ose Type - WPM Lnear Frequency Drft Removed Stablty ose Type FPM WFM RWFM Lag ACF ose ID Extends to # Analyss Ponts FWFM Averagng Tme,, Seconds Fgure 4: Frequency Stablty and ose Analyss of Two Hydrogen Masers Overlappng Allan Devaton, σ y Symmetrcom Cs-III Hgh Performance Cs Freq Std S/ ose Type WFM RWFM ose Type Stablty Lag ACF ose ID Extends to # Analyss Ponts Averagng Tme,, Seconds Fgure 5: Frequency Stablty and ose Analyss of a Cesum Frequency Standard Symmetrcom 83 Mltarzed Rb Freq Std S/ 9583 Overlappng Allan Devaton, σ y WFM - -3 ose Type Stablty Lnear Frequency Drft Removed -. RWFM Lag ACF ose ID Extends to # Analyss Ponts ose Type Averagng Tme,, Seconds Fgure 6: Frequency Stablty and ose Analyss of a Rubdum Frequency Standard 5