NIST Technical Note 1337: Characterization of Clocks and Oscillators

while the Hanning ""!Odo"" yIelds1.305 2 (f) )/.V b, 1==1:2.205 2 (/) )IN" j == 2:1.315 2 (/j )IN" j =: 3;1.155 2 (/) )/N" ] =: 4;1.095 2 (/))/N" J =: 5:1.065 2 (!i )/N" j =: 6;1.045 2 (/j )iN" j =: 7;5'2U))IN" 8 .$ j :s: 511 to within 3%.Except for the few lowest frequenci~. the results for the Hanningwindow agree with our experimental results and with standardstatistical theory; however. the factor of five in the variancefor the uniform window disagrees with our experiments and withstandard theory (although it has been verified by Monte Carlotechniques). The cause of this di!lcrepancy is under investigation,but we think it is due to the band-limited nature of theexperimental data.Third, for a random-IUD process, the variance computationsBJ'E not useful since the variance is domillated by the factthat the expected value of the sample variance for each block ofsampl~ increases with time. The agreement which we found betweenstandard statistical theory and our experimental resultson the liN. rate of decrease of variance is undoubtedly dueto the band-limited nature of the experimental data. We will. a.ttempt to verify these conciusions in the future using MonteCarlo techniques.-7]de(V)'0"e/OIVA.IJlIATHexample we have cho~n to take .vb = 1000 blocks of :be va,.,;ouspower-law noise types examined m IlLS above and compare thevalue of the spectral estimate with that obtained from .\"6 = 32blocks (see Figure 4). Since the variance of the 1000 block datais about 32 times smaller than that of the 32 block data. it canserve as an accurate estimate of the "true value. ~ Let 5 1000 (f) )represent this quantity at the loth cha.n.nel (frequency). By subtractingthe 1000 block data from the 32 block data at the l-thcha.n.nel, we then have one estimate of the error for the 32 blockdata; by repeating this procedure over N c different cha.n.nelsand N r clifi'erent repliCAtions, we can obtain accurate estimatesof the variance for the 32 block data. Let 5 32 i (f) ) represent thespectral estimate for the 32 block data at the j-th cha.n.nel andthe i-th repliCAtion. To compensate for the variation in the levelof the spectral estimates with cha.n.nel, it is aecessary to dividethe error at the j-th cha.n.ne1 by the "true value~ 5 1000 (/)), Themean square fractional error of the 32 block data for the noisetype under study is given byIt is assumed that all channels with biM - as indicated in Table1 - have been excluded in the summation CfIIer j. It is alsoimportant that the changes in the spectral density not exceedthe dynamic range of the digitizer because under this conditionthe quantization errors - in addition to causing biases in thespectral estimates as discussed earlier - can lead to situationswhere the variance does not improve as N 6 increases_ These Vll.!­ues can be sealed to any number of blocks N, if care is taken toavoid these quantization errors. Upper and lower approximate67% confidence limits for SUj) - the true spectral density atchannel j - using Hanning, uniform and the proprietary 'ilattenedpeak" windows for N 6 approximately independent blocksare given bys;~;~. L~=""H""Z--------------;-S"'TC=-., -1'""OO::::-:OOO=-:Ha-:-lFigure 4. Comparison of the spectral estimate of /-4 powerlawnoise with 1000 samples with that obtained with 32 samples.The text expwIUl how these two curves are used to obtainthe fractional RMS confidence of the spectral estimate for 32samples.IV.B.Experimental DeterminationThe following procedure can be used to experimentally determinethe VlUiance of the spectral estimates of virtually anytype of noise spectrum with any type of window for a particularinstrument. Since the spectral density of interest is in generalnonwhite, we must determine both the "true value" and a wayto nonnalize the fractional error of the estimate lila a functionof the number of SIlmples. This can be done by making IJlIelof the above th~retical ane.lysis that shows that the varianceshould decrease as the square root of the number al samplesIsince they are approximately statistically indepf!Ildf!Ilt (in fut,lexa.ctly 90 in the ca.ses of white and random-waJ.k noise). As anwhere S(Ii) is the spectral estimate given by Equation (1) andV(a,N,) is the fractionalll!lriance given in Table 2 for fQ andQ = 0, ·2, -3 and -4 (these results were obtained by averagingover NrNc = 1200 channels). The variances obtained are veryclose to thoee obtained from standard statistical analysis forwhite noise, i.e.,Table 2.power lawConfidence Interval! for ITT Spectral Estimateswindownoise type uniform Hanning Battened peakr 1.02/,fN; 0.98/,fN; O.9S/v'N;/-2 l.02/..;'N; 1.04/,fN; 1.04/.;:v;r 3 unusable 1.04/ ,J'lV; 1.04/y'N;/-4 unusable 1.041,J'lV; l.04/v'NbV. CondlUioDJIWe have introduced experimental techniques to evaluatethe stati"tieal properties of FFT spectral estimates for commonnoise types found in oecillators, ampli£.ers. mixers and similarTN-252