Handbook of Frequency Stability Analysis

where the 0 z i # (m) terms are the phase averages from the triply extended subsequence, and the prefix 0 denotes that thelinear trend has been removed. At the largest possible averaging factor, m = N/3, the outer summation consists ofonly one term, but the inner summation has 6m terms, thus providing a sizable number of estimates for the variance.Reference for Modified Total VarianceD.A. Howe and F. Vernotte, “Generalization of the Total Variance Approach to the Modified Allan Variance,” Proc.31 st PTTI Meeting, pp. 267-276, Dec. 1999.5.2.13. Time Total VarianceThe time total variance, TTOT, is a similar measure of time stability, based on themodified total variance. It is defined as32 τ2σx( τ) = ⋅ Modσtotal( τ )(28)3The time total variance is ameasure of time stabilitybased on the modified totalvariance.5.2.14. Hadamard Total VarianceThe Hadamard total variance, HTOT, is a total version of the Hadamard variance.As such, it rejects linear frequency drift while offering improved confidence at largeaveraging factors.An HTOT calculation begins with an array of N fractional frequency data points, y iwith sampling period τ 0 that are to be analyzed at averaging time τ =m τ 0. HTOT iscomputed from a set of N − 3m + 1 subsequences of 3m points. First, a linear trend(frequency drift) is removed from the subsequence by averaging the first and lasthalves of the subsequence and dividing by half the interval. Then the drift-removedThe Hadamard total variancecombines the features of theHadamard and total variancesby rejecting linear frequencydrift, handling more divergentnoise types, and providingbetter confidence at largeaveraging factors.subsequence is extended at both ends by uninverted, even reflection. Next the Hadamard variance is computed forthese 9m points. Finally, these steps are repeated for each of the N − 3m + 1 subsequences, calculating HTOT as theiroverall average. These steps are shown in Figure11.26