?_Ź˙˙˙˙%öœ“lyšÂ?EDFBrowseButtons()ZƒmainRWj mainđ˙j ?’OĐ!Ÿj Ż ŻV đ˙ OŻm##ĆO,—l? ŻŻŻś/&;)z4˙˙-˙˙˙˙|CONTEXTó~|CTXOMAPšv|FONTEg|SYSTEM|TOPICŰ|TTLBTREEÄv|bm0"‡|bm1 ‰|bm10Ý|bm11oă|bm12–é|bm13Üî|bm2ťˇ|bm36ş|bm4”ż|bm5ĂÂ|bm6É|bm7ŹÍ|bm8ęŇ|bm9ׯ‡Lˆˆˆˆˆˆˆˆřˆˆˆˆˆˆˆˆˆˆˆˆˆˆ€ˆˆˆˆˆ€ˆˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆpˆˆˆˆˆˆpˆˆˆˆˆˆ€ˆˆˆˆˆ€ˆˆ€ˆ€ˆˆˆˆˆˆ€ˆˆ€ˆˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆp˙˙˙đ˙đ˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙-#ˆˆˆˆksO˙˙žzżs˙˙˙˙*vżˆ ˆˆˆˆ€ˆˆˆˆpˆˆˆˆˆ€ˆˆˆˆ€ˆ€€ˆˆˆˆˆˆ€ˆˆ€ˆˆ€ˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆp˙˙˙đ˙đ˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙ˆˆˆˆˆˆˆˆˆˆˆˆˆřˆˆˆˆˆˆˆˆˆˆˆˆˆˆ€ˆˆˆˆˆ€ˆˆ€€ˆˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆˆpˆˆˆˆˆˆpˆˆˆˆˆˆ€ˆˆˆˆˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆ€ˆˆˆˆˆˆˆˆˆ€ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆp˙˙˙đ˙đ˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙jbab˙˙˙˙ 9˙˙˙˙E1Đ˙˙˙˙rEwContents2 w& €€€€‚˙Contents7EŽ) "€€‚H€ƒ€‚˙ˇContentsDwÍŰ „‰€PČ:‚H€€‚€ƒăŠUô%€ ‰‚€ƒăY’€ ‰‚€ƒă‚Ą™€ ‰‚€ƒă•tň€ ‰‚€ƒă6p.ý€ ‰‚€ƒă›€ ‰‚€ƒăë:ňŹ€‰‚€ƒă˝¨*Ȁ‰‚€ƒăi7˙l€‰€ƒ‚ƒăQ=ˇ—€ € ƒ‰€ ‚€ƒăŁvF΀ ‰‚€ƒăHyh€ ‰‚˙This help file for the EDF program contains the following topics:ˇContentsˇAboutˇIntroductionˇLicenseˇProgram OperationˇEDFˇConfidence IntervalsˇClassic EDF AlgorithmsˇTotal & Thęo1 EDF AlgorithmsˇCombined EDF AlgorithmsˇSupportˇAcknowledgmentsIŽ5 :€(€PČ:‚H€ƒă〠‰‚€‚˙ˇReferences6ÍL1×ÉL{!About/ {& €€€€‚˙About(LŁ% €€‚H€‚˙\'{˙5 :€N€0‘€‘€€ € ‚€ ‚€€‚˙EDF.HLPVersion 1.5 of 10/17/03-Ł,* $€€0 ąą(O€‚˙żk˙ëT x€Ř€€‚€€€‚€‚€€†"€€€€€‚€€‚˙Microsoft WindowsŽ Help File For EDF EDF Calculations for Frequency Stability Analysis-,* $€€0 ąą(O€‚˙áŻëů2 2€_€€‚€€‚‚‚‚‚‚‚‚˙Ó2003 Hamilton Technical Services195 Woodbury StreetS. Hamilton, MA 01982 USAPhone: 978-468-3703Fax: 978-626-3006E-Mail: bill@wriley.comWeb: http://www.wriley.com(!% €€‚H€‚˙= ů^19rĺ€^”śEIntroduction6!”& € €€€‚˙IntroductionŮ^­ @ N€ł€‚H€ƒ€‚€€€€€€€‚˙ˇIntroductionThe EDF program is a small Microsoft WindowsŽ application to calculate the number of equivalent chi-squared degrees of freedom for estimates of various statistical measures of frequency stability. It can find the edf for the Allan, modified Allan, Hadamard, Total, modified Total, Hadamard Total, and Thęo1 variances, as applicable, for either overlapping or non-overlapping (short or long stride) samples by means of classicial emperical formulae or a new combined (full or simplified) algorithm. Besides the choice of algorithmic method, variance type, and stride, the edf values depend on the power law noise type (white PM, a=2 through random run FM, a=-4), the number of (phase) data points, and the averaging factor. The EDF program is intended mainly as a convenient way to become familiar with and compare these methods, but it can also be used to validate code for implementing these algorithms, and to determine edf values for an actual analysis.+”Ř ( €€(‚H€‚˙°|­ ˆ 4 8€ř€‚H€ƒ€‚€€€‚‚˙ˇNoise TypesThe allowable noise types and a for each of the supported variance types are shown in the table below:+hŘ ł Ă#VƒĐQKKKLKKK€€˙€€€‚˙&€€ ‚H€€‚˙€,€ ‚H˙€.€‚˙€<€ ‚H‚˙€B€ ‚H˙€D€‚˙€R€ ‚H‚˙€X€ ‚H˙€Z€‚˙€h€ ‚H‚˙€n€ ‚H˙€p€‚˙€~€ ‚H‚˙€†€ ‚H˙€ˆ€‚˙€–€ ‚H‚˙€ž€ ‚H˙€ €‚˙€Ž€ ‚H‚˙€ś€ ‚H˙€¸€‚˙ €Ć€ ‚H€‚˙˙˙VarianceType/aWH PM2FL PM1WH FM0FL FM-1RW FM-2FW FM-3RR FM-4')ˆ Úţ#́RQKKJKKKK€€ —‚H˙€€€‚˙€€˙€€€€‚˙€€€€‚˙€(€€€‚˙€2€€€‚˙€<€€€‚˙€F€˙€H€‚H‚˙€L€‚H˙€N€‚˙˙˙Allanˇˇˇˇˇ+ł @í#ށVQKKJKKKK€€€‚˙€€˙€€€€‚˙€"€€€‚˙€,€€€‚˙€6€€€‚˙€@€€€‚˙€J€˙€L€‚H‚˙€P€‚H˙€R€‚˙˙˙ModifiedˇˇˇˇˇÚ @! /ÚAÝ#Ё^QKKJKKKK€€€‚˙€€˙€€€€‚˙€"€€€‚˙€,€€€‚˙€6€€€‚˙€@€€€‚˙€J€€€‚˙€T€€€‚˙˙˙Hadamardˇˇˇˇˇˇˇ") @:Bů#RQKKJKKKK€€˙€€€‚˙€€˙€€€€‚˙€€€€‚˙€(€€€‚˙€2€€€‚˙€<€€€‚˙€F€˙€H€‚H‚˙€L€‚H˙€N€‚˙˙˙Totalˇˇˇˇˇ,ASCí#ށXQKKJKKKK€€€‚˙€€˙€€€€‚˙€$€€€‚˙€.€€€‚˙€8€€€‚˙€B€€€‚˙€L€˙€N€‚H‚˙€R€‚H˙€T€‚˙˙˙Mod Totalˇˇˇˇˇ 0:B`DÝ#Ё`QKKJKKKK€€€‚˙€€˙€€€€‚˙€$€€€‚˙€.€€€‚˙€8€€€‚˙€B€€€‚˙€L€€€‚˙€V€€€‚˙˙˙Had Totalˇˇˇˇˇˇˇ&*SC†Eü#ȁTQKKJKKKK€€˙€€€€‚˙€€˙€€€€‚˙€ €€€‚˙€*€€€‚˙€4€€€‚˙€>€€€‚˙€H€˙€J€‚H‚˙€N€‚H˙€P€‚˙˙˙Thęo1ˇˇˇˇˇ0`DśE) "€€€‚‚‚€‚˙8†EîE1BÉŹ‚îEF0HLicense1 śEF& €€€€‚˙License6 îEUF) "€€‚H€ƒ€‚˙ˇLicenseŰŻF0H, &€_€‚€‚‚‚€‚˙A license is hereby granted for the use of this program. EDF may be freely installed, used, and distributed to others as a service by its author to the time and frequency community. No third-party licensed code is used.This program has been extensively tested, but it is never possible to declare a program completely bug-free. No warranty is made, nor is any liability assumed, in connection with the use of the program.BUFrH1œ ĺ€-rH­H$„Program Operation;0H­H& €*€€€‚˙Program Operation¨grHUIA R€Đ€‚H€ƒ€‚€€‚‚€†"€€‚˙ˇEDF ProgramThe EDF program has a single dialog box as shown in the following screen shot:+­H€I( €€(‚H€‚˙LUI˙I3 6€˜€PČ:‚H€€ƒ€‚€‚‚˙ˇControlsThe EDFprogram dialog box contains the following controls:^%€I]J9 B€J€Pт3}„UŐ€€ƒ€ƒƒƒ€‚˙Dialog BoxALTControlControlW!˙I´J6 <€B€Pт3}O„UŐ€ƒƒƒƒ€‚˙ControlKeyTypeDescriptionş]JÄLV z€u€Pт3}„UŐ€ƒƒƒ‚ƒƒƒ‚ƒƒƒƒ‚ƒƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚˙Variance TypeVComboChoose the variance typeNoise TypeNComboChoose noise type# Data PointsDEditEnter # phase data pointsAveraging FactorAEditEnter averaging factorTypeGroupEDF algorithm typeOriginalLRadiobuttonOriginal classic algorithmsFullURadiobuttonFull combined algorithmSimpleIRadiobuttonSimplified combined algorithmStrideGroupAnalysis stride typeLongGRadiobuttonLong stride (non-overlapping)V ´JNI `€€Pт3}„UŐ€ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒ‚ƒƒƒƒ‚˙ShortTRadiobuttonShort stride (overlapping)CalcCPushbuttonCalculate edfCloseSPushbuttonClose programCopyOPushbuttonCopy results to clipboardViewWPushbuttonView clipboard contentsHelpHPushbuttonInvoke this help fileEDFTextDisplay edf value-ÄLGN* $€€PČ:(‚H€‚˙>N…N. ,€ €PČ:‚H€€ƒ€‚˙ˇOperationfţGN÷€h ž€ý€‚€€€€€€€€€€€€€€€€€€€€€€€‚˙Select the desired Variance type , Noise type, algorithm Type, and Stride. Enter the # of phase Data Points and Averaging Factor. Press Calc to calculate the corresponding edf value. Then, optionally, press Copy to copy the results to the Windows clipboar…N÷€0Hd, and View to display the clipboard contents. The clipboard viewer will remain open to display subsequent results that are copied to the clipboard. Invoke this Help file as desired, and end the program with the Close button.+…N"( €€(‚€‚˙; ÷€]. ,€€PČ:‚H€€ƒ€‚˙ˇLimitsÝ´":ƒ) €i€‚€€‚˙The edf calculations are subject to a number of restrictions and limits, most of which are automatically indicated by the program controls. For example, fewer noise types are associated with the non-Hadamard variance types, and a certain minimum number of analysis points are required depending on the varianvce type, # of data points and averaging factor. For programming reasons, the # of data points is limited to 999,999,999.-]gƒ* $€€PČ:(‚H€‚˙; :ƒ˘ƒ. ,€€PČ:‚H€€ƒ€‚˙ˇErrors‚Ygƒ$„) "€˛€‚€‚€‚˙Errors in the edf calculations are indicated by an error message in the results box.4˘ƒX„1gŹ‚ĺX„…„ťˆEDF-$„…„& €€€€‚˙EDFg0X„ě†7 <€a€‚H€ƒ€‚€€€€‚˙ˇEDFThe equivalent # of chi-squared degrees of freedom (EDF) for a second-moment statistic such as the Allan variance is the basis for determining the confidence level of the particular variance estimate. The EDF depends on the statistic type (e.g. overlapping Allan variance, modified Allan variance, etc.), the # of data points (generally expressed as N, the # of phase data points), and the power law noise type (white FM noise, etc., generally expressed as a, the exponent of Sy(f), the spectral density of the fractional frequency fluctuations).Š~…„•ˆ+ $€ý€‚H€‚€‚‚˙The EDF can be estimated either by an empirical formula or by a theoretical expression (often based on an autocorrelation). These calculations are performed for each of the statistics as described in the Help topics below. They are divided into two categories, original (classic) EDF algorithms and the new combined EDF algorithm. The latter has two forms, full and simple.&솻ˆ# €€€‚˙E•ˆ‰1B-D ‰>‰OConfidence Intervals>ťˆ>‰& €0€€€‚˙Confidence IntervalsC‰‰) "€4€‚H€ƒ€‚˙ˇConfidence Intervalsr#>‰óŒO l€I€””€‚€‡"€€‚€€€€€€€€‚˙Sample variances are distributed according to the expression:where c˛ is the Chi-square, s˛ is the sample variance, s˛ is the true variance, and edf is the equivalent number of degrees of freedom (not necessarily an integer). The edf is determined by the number of analysis points and the noise type. The statistical measures used for time-domain frequency stability analysis often establish single or double-sided confidence intervals with a selectable confidence factor, based on c˛ statistics. The general procedure is to choose a single or double-limited confidence factor, p, calculate the corresponding c˛ value, determine the edf from the variance type, noise type and number of analysis points, and thereby set the statistical limit(s) on the variance. For a double-sided limits:6‰)1 2€ €””€‡"€€‚˙&óŒO# €€€‚˙G)–1:ĺ&‚–ÖëĹClassic EDF Algorithms@OÖ& €4€€€‚˙Classic EDF AlgorithmsI–Ž, (€:€‚H€ƒ€€‚˙ˇClassic EDF Algorithms+֍JŽ( €€(‚H€‚˙m;ގ2 2€w€‚H€ƒ€‚€€‚‚˙ˇOverlapping Allan Variance EDFThe classic empirical formulae for the edf of the overlapping Allan variance, as given by D.A. Howe, D.W. Allan and J.A. Barnes in their paper "Properties of Signal Sources and Measurement Methods" at the 35th Annual Symposium on Frequency Control in May 1981 are as follows:×XJŽšŔ ΀¸€O‚H€ƒƒƒ€‚€ƒƒ€†"€€‚€ƒƒ€ˇšŔO†"€€‚€ƒƒ€†"€€‚ƒƒ€†"€€‚˙Noise AlphaEDF FormulaW PM 2F PM 1W FM 0FFM -1GˇáŔ6 <€$€‚H€ƒƒ€†"€€‚˙RW FM -2+šŔ Á( €€(‚H€‚˙P'áŔ\Á) "€N€‚H€ƒ€‚˙ˇOverlapping Hadamard Variance EDF”k ÁđÂ) €×€‚€ €‚˙The edf for the fully overlapping Hadamard variance (HVAR) can be found by an algorithm developed by C.A. Greenhall based on its generalized autocovariance function. The HVAR edf is found either as a summation (for small m cases with a small number of terms) or from a limiting form for large m where 1/edf = (1/p)(a0-a1/p) with the coefficients as follows: &\ÁĂ# €€€‚˙ƒđ™Ăm#Ş€,= = €€˙ €€„„€‚˙€€„„‚˙€$€„„‚˙˙˙Noise Typea0a1}ĂÄi#˘€(= = €€„„[˙€€„„€‚˙€€„„‚˙€€„„‚˙˙˙W FM0.780.50k™ĂÄX#€€&= = €€„„€‚˙€€„„‚˙€€„„‚˙˙˙F FM1.000.62lÄíÄX#€€(= = €€„„€‚˙€€„„‚˙€€„„‚˙˙˙RW FM1.030.61lÄYĹX#€€(= = €€„„€‚˙€€„„‚˙€€„„‚˙˙˙FW FM1.060.53líÄĹĹX#€€(= = €€„„€‚˙€€„„‚˙€€„„‚˙˙˙RR FM1.300.54&YĹëĹ# €€€‚˙MĹĹ8Ć1=D ž8Ć~Ì Total & Thęo1 EDF AlgorithmsF ëĹ~Ć& €@€€€‚˙Total & Thęo1 EDF AlgorithmsO#8ĆÍĆ, (€F€‚H€ƒ€€‚˙ˇTotal & Thęo1 EDF Algorithms+~ĆřĆ( €€(‚H€‚˙e"ÍĆ]ÉC T€E€‚H€ƒ€‚€€€ €€€€€‚˙ˇTotal Variance EDFThe EDF for the Total variance is determined by the method described in the paper "Total Variance Explained" by D.A. Howe presented at the Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium in April 1999. It applies to -2 Ł a Ł 0. The edf for theTotal variance is given by the formula b(T/t) -c, where T is the length of the data record, t is the averaging time, and b & c are coefficients that depend on the noise type as shown in the following table:(řƅÉ% €€‚H€‚˙ ]ÉĘp#°€@ëL K €€˙&€€„„€€ ‚˙€4€„„‚˙€:€„„‚˙˙˙ Power Law Noise Typebc‚…É—Ęi#˘€2ëL K €€„„~˙€€„„€ ‚˙€€„„‚˙€&€„„‚˙˙˙ White FM1.500.00rĘ ËX#€€4ëL K €€„„€ ‚˙€€„„‚˙€(€„„‚˙˙˙ Flicker FM1.170.22z—ĘƒË[#†€>ëL K €€„„€ ‚˙€$€„„‚˙€0€„„€‚˙˙˙ Random Walk FM0.930.36+ ËŽË( €€(‚H€‚˙`)ƒËÎ7 <€S€‚H€ƒ€‚€€€ €‚˙ˇModified Total Variance EDFThe EDF for the modified Total variance is determined by the method described by D.A. Howe and F. Vernotte in their paper "Generalization of the Total Variance Approach to the Modified Allan Variance" presented at the 31th Precise Time and Time Interval Meeting in December 1999, and private communications with D.A Howe in March and May 2000. It applies to -2 Ł a Ł 2, and uses the same formula as the Total variance with different coefficients (subject to certain restrictions) as shown in the following table:(ŽË6Î% €€‚H€‚˙‹ÎÁÎm#Ş€<äK I €€˙ €€„„€‚˙€0€„„‚˙€6€„„‚˙˙˙Power Law Noise Typebc6ÎBĎi#˘€0äK I €€„„Ž˙€€„„€‚˙€€„„‚˙€$€„„‚˙˙˙White PM1.902.10qÁÎłĎX#€€2äK I €€„„€‚˙€€„„‚˙€&€„„‚˙˙˙Flicker PM1.201.40oBĎ.X#€€.äK I €€„„€‚˙€€„„‚˙€"łĎ.ëŀ„„‚˙˙˙White FM1.101.20qłĎŸX#€€2äK I €€„„€‚˙€€„„‚˙€&€„„‚˙˙˙Flicker FM0.850.50u.X#€€:äK I €€„„€‚˙€"€„„‚˙€.€„„‚˙˙˙Random Walk FM0.750.31+Ÿ?( €€(‚H€‚˙ž@Ý^ Š€€‚H€ƒ€‚€€€ €€€€€€€€€€€€€€‚˙ˇHadamard Total Variance EDFThe EDF for the Hadamard Total variance is determined by the method described by D. Howe, R. Beard, C. Greenhall, F. Vernotte and B. Riley their paper "A Total Estimator of the Hadamard Function Used for GPS Operations" presented at the 32nd Precise Time and Time Interval Meeting in November 2000. It applies to -4 Ł a Ł 0, and uses the formula (T/t)/(b0+b1t/T), where T is the length of the data record, t is the averaging time, and b0 & b1 are coefficients that depend on the noise type as shown in the following table:(?% €€‚H€‚˙$ݢy#€HäK I €€˙ €€„„€‚˙&€0€„„€€‚˙&€<€„„€€‚˙˙˙Power Law Noise Typeb0b1ƒ%i#˘€4äK I €€„„Ž˙€€„„€‚˙€€„„‚˙€&€„„‚˙˙˙White FM0.5591.004s˘˜X#€€6äK I €€„„€‚˙€€„„‚˙€(€„„‚˙˙˙Flicker FM0.8681.140w%X#€€>äK I €€„„€‚˙€"€„„‚˙€0€„„‚˙˙˙Random Walk FM0.9381.696Ś'˜ľ#΀NäK I €€„„˙ €€‚H€€‚˙€(€‚H˙€*€€ €‚˙€<€€€‚˙˙˙Flicker Walk FM0.9742.554„#9a#’€FäK I €€€€‚˙€"€€ €‚˙€4€€ €‚˙˙˙Random Run FM1.2763.149(ľa% €€‚H€‚˙+9Œ( €€(‚H€‚˙Ůaœ 7 <€ł€‚H€ƒ€‚€€€ €‚˙ˇThęo1 EDFThe EDF for the Thęo1 (Theoretical Variance #1) statistic is determined by the method described in the paper "Estimation of Very Long-Term Frequency Stability Using a Special-Purpose Statistic" by D.A. Howe and T.K. Peppler presented at the 2003 IEEE International Frequency Control Symposium in May 2003. It applies to -2 Ł a Ł 2, and the edf for the Thęo1 variance is determined by the following approximation formulae for each power low noise type:(ŒÄ % €€‚€‚˙×Xœ ›  ΀¸€O‚H€ƒƒƒ€‚€ƒƒ€†"€ €‚€ƒƒ€†"€ €‚€ƒƒ€†"€ €‚ƒƒ€†"€ €‚˙Noise AlphaEDF FormulaW PM 2F PM 1W FM 0FFM -1GÄ â 6 <€$€‚H€ƒƒ€†"€ €‚˙RW FM -2+›  ( €€(O‚H€‚˙(â 5 % €€‚€‚˙1 f * $€€‚H€‚€‚‚˙ &5 Œ # €€€‚˙Gf Ó 1§&‚ą Ó  ‚CCombined EDF Algorithm@Œ  & €4€€€‚˙Combined EDF AlgorithmÚŠÓ í 1 0€S€‚H€ƒ€‚€‚‚‚‚˙ˇCombined EDF AlgorithmThe combined EDF algorithm uses an analytical method rather than empirical formulae to find the edf for modified and unmodified Allan and Hadamard variances for overlapping and non-overlapping samples for all applicable noise types. It is based on modeling the phase as the first difference of a continuous-time pure power-law process.The inputs to the combined edf algorithm are as follows:™# †v#ź€F:îő€€˙"€€€€‚˙"€€€€‚˙"€0€€€‚˙˙˙TermDescriptionRemarks‘'í j#¤€N:îő€€€˙€€€‚˙€ €‚˙"€0€€€€‚˙˙˙aFM noise exponent-4 Ł a Ł 2ŞV†ÁT#x€Ź:îő€€€‚˙€€‚˙€>€‚‚‚˙˙˙dOrder of phase difference1=1st difference2=Allan variance3=Hadamard variance‚!O@a#’€B:îő€€€‚˙€€ÁO@Œ ‚˙.€,€€€€€€‚˙˙˙mAveraging factort/t0‘>Áŕ@S#v€|:îő€€€‚˙€€‚˙€&€‚‚˙˙˙FFilter factor1=modified variancem=unmodified variance–CO@vAS#v€†:îő€€€‚˙€€‚˙€&€‚‚˙˙˙SStride factor1=long (non-overlapping)m=short (overlapping)‰.ŕ@˙A[#†€\:îő€€€‚˙€€‚˙"€2€€€€‚˙˙˙N# phase data pointsSample period=t0)vA(B& €€(€‚˙<˙AdB, (€ €‚H€ƒ€€‚˙ˇReferenceď(B‚C/ ,€ß€‚€ €€ €‚˙C.A. Greenhall and W.J. Riley, "Uncertainty of Stability Variances Based on Finite Differences", Proceedings of the 35nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , December 2003 (to be published).8dBşC1˙ž•ƒ şCëCšFSupport1 ‚CëC& €€€€‚˙Support6 şC!D) "€€‚H€ƒ€‚˙ˇSupportV2ëCwD$ €d€€‚‚˙Support for the EDF program is available from:­~!D$E/ .€ü€PA‚Ă}€ƒƒƒ‚ƒ‚ƒƒ‚˙W. J. RileyTelephone (non-business hours):Hamilton Technical Services978-468-3703195 Woodbury StreetFax (any time):O*wDsE% €T€€ƒƒ‚˙S. Hamilton, MA 01982 USA978-626-3006J$E˝E, (€<€PA‚Ă}€ƒƒƒƒ‚‚˙E-Mail:bill@wriley.comüÔsEšF( €Š€€‚€‚˙Hamilton Technical Services develops software for frequency stability analysis, including the popular Stable programs. More information about those products can be found on the web at http://www.wriley.com.?˝EřF1ąX„ řF0GHAcknowledgment8šF0G& €$€€€‚˙AcknowledgmentăąřFH2 2€c€‚H€ƒ€‚€‚€‚˙ˇAcknowledgmentThis program is heavily based on the work of C.A Greenhall, whose contributions to the field of frequency stability analysis are gratefully acknowledged.; 0GNH1Ö •ƒ˙˙˙˙ NH‚HہReferences4H‚H& €€€€‚˙References=NHżH, (€"€‚H€ƒ€€ ‚˙ˇReferencesZ5‚HI% €j€””€ ‚˙The following references apply to the EDF program:6őżHOKA P€ë€PČ:‚H€ ƒ€€ €€ ‚ƒ‚ƒ€€ ‚ƒ‚˙1."IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology - Random Instabilities", IEEE Std 1139-1999 , July 1999.2.C.A. Greenhall, "Estimating the Modified Allan Variance", Proc. IEEE 1995 Freq. Contrl. Symp., pp. 346-353, May 1995.3.D.A. Howe & C.A. Greenhall, "Total Variance: A Progress Report on a New Frequency Stability Characterization", Proc. 1997 PTTI Meeting, pp. 39-48, December 1997.4.C.A. Greenhall, private communication, May 1999.DIĐM= H€‰€PČ:‚H€ ƒ‚ƒ€€ ‚ƒ‚ƒ‚ƒ€€ ‚˙5.D.A. Howe, private communication, March 2000.6.D.A. Howe, "Total Variance Explained", Proc. 1999 Joint Meeting of the European Freq. and Time Forum and the IEEE Freq. Contrl. Symp ., pp. 1093-1099, April 1999.7.D.A. Howe, private communication, March 2000.8.C.A. Greenhall, "Recipes for Degrees of Freedom of Frequency Stability Estimators", IEEE Trans. Instrum. Meas., Vol. 40, No. 6, pp. 994-999, December 1991.9.D.A. Howe, "Methods of Improving the Estimation of Long-Term Frequency Variance", Proc. 11th European Freq. and Time Forum , pp. 91-99, March 1997.¸wOK”€A P€ď€PČ:‚H€ ƒ€€ ‚ƒ‚ƒ€€ ‚ƒ€€ ‚˙10.J.A. Barnes and D.W. Allan, Variances Based on Data with Dead Time Between the Measurements", NIST Technical Note 1318 , 1990.11.C.A Greenhall, private communication, May 2000.12.D. Howe, R. Beard, C. Greenhall, F. Vernotte and W. Riley, "A Total Estimator of the Hadamard Function Used for GPS Operations", Proceedings of the 32nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , pp. 255-268, November 2000.13.C.A. Greenhall, "The Generalized AutocĐM”€Hovariance: A Tool for Clock Noise Statistics", TMO Progress Report 42-137 , Jet Propulsion Laboratory, Pasadena, CA 91109, May 1999.!óĐMľ. *€ç€PČ:‚H€ €€ ‚˙14. C.A. Greenhall and W.J. Riley, "Uncertainty of Stability Variances Based on Finite Differences", Proceedings of the 35nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , December 2003 (to be published).&”€ہ# €€€‚˙1ľ ‚1U˙˙˙˙˙˙˙˙ ˙˙˙˙ ‚0‚$ہ0‚" €€€˙1 ‚˙˙˙˙1˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙tkľ,HelvSystem8514oemFixedsysTerminalRomanScriptModernCourierMS SerifMS Sans SerifSmall FontsMarlettArialArial CEArial CYRArial GreekArial TURCourier NewCourier New CECourier New CYRCourier New GreekCourier New TURLucida ConsoleLucida Sans UnicodeTimes New RomanTimes New Roman CETimes New Roman CYRTimes New Roman GreTimes New Roman TURWingdingsSymbolVerdanaArial BlackComic Sans MSImpactGeorgiaPalatino LinotypeTahomaTrebuchet MSWebdingsMicrosoft Sans SeriMap SymbolsArial NarrowArial Unicode MSBatang@BatangBook AntiquaBookman Old StyleCenturyCentury GothicGaramondMS Mincho@MS MinchoMS OutlookPMingLiU@PMingLiUSimSun@SimSunAgency FBAlgerianArial Rounded MT BoBaskerville Old FacBauhaus 93Bell MTBerlin Sans 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ItalicRavieRockwellRockwell CondensedRockwell Extra BoldInformal RomanScript MT BoldShowcard GothicSnap ITCStencilTempus Sans ITCTw Cen MTTw Cen MT CondensedViner Hand ITCVivaldiVladimir ScriptWide LatinWingdings 2Wingdings 3Berlin Sans FB DemiVerdana RefFencesMT SymbolMT Extra˙Letter Gothic MTTimes New Roman MTOCR-AOCR B MTQuickTypeQuickType CondensedQuickType MonoQuickType PiOCR-A IIQuickType IIQuickType II CondenQuickType II MonoQuickType II PiSubScript VNISubScript VNI2SuperScript VNISuperScript VNI2 € ˙  ) € ˙ ˙ ˙  0 ˙0 ˙   € ) /&;)Lz˙˙ţ˙˙˙˙ContentsrAboutÉIntroductionĺ€LicenseŹ‚Program Operation-EDFĺConfidence IntervalsD Classic EDF Algorithms&‚Total & Thęo1 EDF AlgorithmsžCombined EDF AlgorithmąSupport•ƒAcknowledgmentX„Referencesöř/&;)L4˙˙  ˙˙˙˙Q=ˇ—ž‚Ą™Éë:ňŹĺ˝¨*ČD 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ˆřˆŽ€ˆ€ˆˆˆˆpˆpˆƒ€ˆˆ‘ˆˆ€ˆˆ€ˆ€ˆˆˆˆˆŠ€ˆˆ€ˆˆ€ˆˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆŽˆ€ˆˆ€ˆˆ€ˆ€€ˆpˆpˆ”ˆˆˆˆ€ˆˆ€ˆ€ˆˆˆˆˆ€ˆˆ€ˆˆ€ˆ€ˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆ‹ˆ€ˆˆ€ˆ€€ˆpˆpˆ‚€ˆˆˆ€ˆ€ˆˆˆˆˆ€ˆˆ€ˆˆ€ˆˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆ‹ˆ€ˆˆ€ˆˆˆpˆpˆ‚€ˆ˘ˆˆ€ˆˆ€ˆ€ˆˆˆˆˆˆ€ˆˆ€ˆˆ€ˆˆˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆ‹ˆ€ˆˆ€ˆˆˆpˆpˆ€ˆˆˆˆ€ˆ€€ˆˆ“ˆˆ€ˆˆ€ˆˆ€ˆ€ˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆ‹ˆ€ˆˆ€€ˆˆ€ˆpˆpˆ‰€ˆˆˆˆˆ€ˆˆ„ˆˆ€ ˆ€ˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆ‹€ˆ€ˆˆ€ˆpˆpˆ‰€ˆˆˆˆˆˆˆ„ˆˆ ˆ€ˆŒp˙˙˙đ˙đ˙˙#˙ ˆřˆ€ˆˆpˆpˆ…€ˆˆˆ ˆˆ€ˆŒp˙˙đ˙đ˙˙#˙ ˆřˆƒˆ€ˆpˆpBˆ‹p˙˙˙˙˙đ$˙ ˆřˆƒ€ˆpˆpBˆp.˙ ˆř,ˆpˆpBˆp.˙ ˆř,ˆpˆpBˆp.˙ ˆř,ˆpˆpBˆp.˙ ˆř,ˆpˆpBˆp. ˆř,ˆpˆpBˆ/w ˆ-˙đˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpBˆ0˙ ˆ.ˆpBˆx.ˆ ˆ÷,wpˆpBˆp'˙đ ˆř,ˆpˆpBˆp'˙řw‚p ˆř,ˆpˆpˆˆ ˆp'˙‚řřˆ‚p ˆř,ˆpˆpˆˆ„ˆˆ ˆp'˙‚řřˆ‚p ˆř,ˆpˆpˆ„€ˆˆ ˆp'˙‚řřˆ‚p ˆřˆˆpˆpˆ€ˆˆˆ€ˆ€ˆˆˆ€ˆŒ€ˆˆˆ€ˆˆp˙˙đ˙˙đ˙˙˙˙đ˙˙đ˙…đ˙đ˙đ˙‚řřˆ‚p ˆř,ˆpˆpˆ’€ˆˆ€€ˆˆ€ˆˆ€ˆˆˆ€ˆŠˆˆ€ˆˆ€ˆp˙˙đ˙˙đ˙˙˙˙đ˙˙đ˙…đ˙đ˙đ˙‚řřˆ‚p ˆřˆ€ˆˆˆˆˆ€ˆpˆpˆŠ€ˆˆ€ˆˆ€ˆ‚€ˆ€ˆˆˆˆ€ˆˆp˙˙˙˙˙˙đ˙˙đ˙…đ˙đ˙‚řřˆ‚p ˆřˆˆ€ˆ€ˆˆ€ˆ€ˆˆ€ˆpˆpˆ’€ˆˆ€ˆˆ€ˆ€ˆ€ˆ‹€ˆˆ€ˆ€ˆˆ€ˆˆp˙˙˙˙˙˙đ˙˙đ˙…đ˙đ˙‚řřˆ‚p ˆř ˆ€ˆ‹€ˆˆˆˆ€ˆˆpˆpˆ’€ˆ€ˆ€ˆˆ€ˆˆˆ€ˆˆˆŽ€ˆˆ€ˆ€ˆˆ€ˆˆ€ˆp˙˙˙˙˙˙đ˙˙đ˙…đđ˙đđ˙ˆřřˆ€ˆˆp ˆř ˆ€ˆ‰€ˆˆˆˆˆpˆpˆ’€ˆˆ€ˆˆ€ˆˆ€ˆˆˆŽ€ˆˆˆˆˆ€ˆˆ€ˆ‰p˙đ˙đđ˙đ˙‹˙˙đđđ˙đđ˙ˆřřˆˆp ˆř ˆ€ˆ†€ˆˆ€ˆƒˆ€ˆpˆpˆ‘€€ˆˆˆˆ€ˆ€ˆˆˆ€ˆˆˆˆˆ€ˆˆp˙đ˙đđ˙đ˙˙˙đ˙˙đ˙…đ˙˙˙ˆřř€ˆp ˆř ˆ€ˆŒ€ˆˆ€ˆ€ˆˆ€ˆpˆpˆ…€€ˆˆˆ€'ˆp˙˙˙˙˙˙˙đ˙˙đ˙…đ˙˙˙‚řřˆ‚p ˆř ˆ€ˆŒˆˆˆˆ€ˆpˆpˆ…€ˆˆˆ€'ˆp˙˙˙˙˙˙˙đ˙˙đ˙đ˙đ˙‚řřˆ‚p ˆř ˆ€ˆ‚ˆpˆpˆ€ˆˆ€ ˆ€%ˆp˙˙˙˙˙˙˙đ˙˙đđ˙đ˙‚řřˆ‚p ˆřˆ…ˆ€ˆˆpˆpBˆp'˙‚řřˆ‚p ˆřˆ…€ˆˆpˆpBˆp'˙‚řřˆ‚p ˆř,ˆpˆpBˆp'˙‚řřˆ‚p ˆř,ˆpˆpBˆp'˙ř˙‚p ˆř,ˆpˆpBˆp'˙řˆ‚€ ˆř,ˆpˆpBˆp. ˆř,ˆpˆpBˆ/w ˆ-˙đˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpˆ4ˆpBˆ0˙ ˆ.ˆpBˆx.ˆ ˆ.ˆpBˆp'˙đ ˆ,wˆpBˆp'˙řw‚p ˆ+ˆ‚‡ˆp&ˆˆˆp'˙‚řřˆ‚p ˆ‚ˆ*‚‡ˆpˆˆ„ˆˆˆp'˙‚řřˆ‚p 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