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89(2,",rjeu) for r=

2 4

“4

16

a“

'
t
:
t

2.000E+00 4. 000E+00
1,741E+00  3.031E+00
1,515E+00 2, 297E+00
1,3196+00 1. 740E+00
1.148E+00 1.318E+00
1,000E+00 1.000E+00
8. 733-01 7,635E-01
7, 083E-01 5. 945E-01
6,875E-01 4. 049E-01
6,341E-01  4.2996-01
6,098E-01 4, 225E-01
6, 116E-O1 4,512E-01
6.332E-01 5. 021E-01
6,673E-01  5.634E-01
7.082E-01 6. 274E-01
7.S26E-01 6, 908E-01
7. 9ME-01 7, 527E-01
8.488E-01 8, 137E-01
8.960E-01 8, 745E-O1
9.A70E-O1 9. 363-01
1,000E+00 1. 000E+00

83(2,N,r,au) for r=

2 4

2.000E+00 4. 000E+00
1, 7A1E+00 3,.031E+00
1.516400 2, 297E+00
1,3196+00 1, 741E+00
1.148E+00 1, 3196400
1,000E+00 1, 000E+00
8.721E-01 7. 6116-01
7.0ATE-O1 5.875E-O1
6.803E-01 4, 720-01
6.2358-01 4, 121€-01
S.978E-01  4,.036E-01
6.011E-01 4. 355E-01
6, 256-01 4, 914E-01
6,626E-01 5S. S7IE-O1
7.086E-01 6. 2408-01
7.513E-01 6. 891E-01
7.962E-01 7. 5206-01
8.465E-01 8, 1338-01
8.959E-01 8, 7A4E-O1
9.469E-01 9. 363E-01
1,000E+00 1.000E+00

8, 0006400
5.2776+00
3, 4816400
2. 2956400
1.513640
1,.0006+00
6.6836-01
4.637E-01
3.531E-01
3. 1526-01
3, 5206-01
3.0N6E-01
4, 983€-01
5.3106-01
6.061E-01
6, 753E-01
7.AL9E-O1
8.063€-01
8, 678E-01
9. 940-01
1,.0006+00

2,.296E+00
1, 515E+00
1,000E+00
6, C47E-01
4, 5396-01
3. 3596-01
2. 9286-01
9. 094E-O1
3.065E-01
4 AXE-O1
5. 242E-01
6. 0166-01
6, 73bE-O1
7.411€-01
8, 059-01
8, 696E-OL
9, SEO
1, 000E+00

1, 600E+01
9, 188E+00
5.275E+00
3.027E+00
1, 737E+00
1,000E+00
5. G56E-01
3, 650E-01
2, 668E-01
2. 504E-O1
2. 880-01
3.568E-01
4, 3B4E-01
‘5. 210E-01
5. 990E-O1
6.714E-01
7.394€-O1
8, 04SE-O1
8, 687E-O1
9, SAE-O1
1,.000E+00

1, 600E+01
9, 189E+00
5.277E+00
3. 030E+00
1, T39E+00
1, 000E+00
5.808€-01
3.527E-01
2,467E-O1
2.B2E-01
2, 634E-01
3. 378E-O1
4. Z9E-O1
5.199E-01
‘5. 9SAE-O1
6. 697E-01
7, 3B4E~O1
8, 042E-01
8. 686E-O1
9. ME-O1
1..000E+00

2. 138E-O1
2. 6636-01
3. 4906-01
4. S2E-OL
5.178E-01
5. 97-01
6. 705E-01
7. 3886-01
8. O41E-O1
8, 6846-01
9, S3E-01
1, 000E+00

rd

3. 2006401
1, 600E+01
7, 99REHO0
3. 997E+00
1, 998E+00
1, 000E+00
5.079€-01
2, 764E-01
1, QB1E-01
1, 866E-01
2. 407E-O1
3, 2536-01
4. 196E-O1
S.107E-O1
5. 9S7E-01
6. 687E-O1
7, 380E-O1
9. 0386-01
8. 6836-01
9. 3326-01
1,.0006+00

37

6, 400E+01
2. 78SE+01
1,2126+01
5. 268€+00
2. 290E+00
1.000E+00
4512-01
2.9ME-OL
1, 730-01
1, 9248-01
2, S6E-O1
3. 400E-O1
4, 006-01
5, 1686-01
5. 9698-01
6. 702-01
7.SR6E-01
8.040E-01
8. OB4E-O1
9. 3328-01
1, 000E+00

cl

6, 400E+01
2. 786E01
1. 2128401
5. 274E+00
2. 294E+00
1, 000E+00
4, M44E-O1
2. 186E-O1
1,494E-01
1. O46E-01
2, 294E-01
3. 200E-01
417-01
5.097E-O1
5. 9328-01
6, 685E-O1
7. 378E-O1
8.007E-01
8,.682E-01
9, 332E-01
1,.000E+00

1, 2806+02
4, ASE+OL
1, 836E+01
6. FS0E+00
2.6296+00
1,.000E+00
3. 967E-O1
1912-01
1, 485-01,
1,.8046-01
2. 5038-01
3,378E-O1
4.272801
‘5. 165E-01
5. 967E-01
6, 7028-01
7. 3B6E-O1
8, 040E-O1
8, 6B4E-O1
9, 392E-01
1,.000€+00

128

1, 2806+02
4.50E+01
1, 837E+01
6, 959E+00
2, 638E+00
1, 000€+00
3.891E-01
1, 747E-01
1,240€~O1
1,S19€-O1
2. 2386-01
3. 1776-01
4, 1048-01
5.094E-O1
S.931E-O1
6. tAME-O1
7.301
8.037E-01
8.662E-01
9, E-O1
1,0006+00

9. 1706100
3.019600
1.000E+00
3,493E-O1
1, S88E-01
1,3238-01
1, 7348-01
2.476E-O1
3, 3698-01
4, 2898-01
5. 164E-01
5. 967E-O1
6. 702E-01
7. 386E-O1
8.080E-O1
8.684E-01
9, 32E-01
1, 000E+00

r=

2. 560E+02
8. M4E+01
2.78501
9, 182E+00
9,026E+00
1,000€+00
3.A10€-01
1, 415E-01
1,072E-01
1, 478-01
2. 2108-01
9, 168E-O1
4. 161E-01
S.070E-01
5.930E-01
6, 6B4E-01
7, 37-01
8.037E-01
8.662E-01
9. 332E-01
1,000E+00

1,470E+02
4. 2196401
1,210€+01
3.467E+00
1,000E+00
3.081E-O1
1. SE-01
1,216E-01
1,695E-01
2, 463E-01
3.365E-01
4, 2886-01
5, 164-01
5. 967E-O1
6. 7026-01
7. 386E-O1
8,040E-O1
8, 684E-01
9, 332E-01
1,000E+00

1,470E+02
4.216401
1,212E+01
3. 47SEHO0
1,.000€+00
2, 9926-01
1. 164E-O1
9.614E-02
1.405E-01
2.196E-O1
3.163E-01
4, 160E-O1
3.092E-01
5. 930E-01
6, 604E-O1
7. 378E-01
8.037E-01
8.682E-01
9. 3328-01
1,000E+00

1, 157E-01
1, 146-01
1,672-01
2.4SE-01
3.363E-01
4. 2B7E-O1
‘5. 1648-01
5.967E-O1
6. 702-01
7. SO6E-O1
8.040E-01
8. 6O4E-01
9, 332E-01
1, 000E+00

1024

1.024€+03
2.5002
6. 398E+01
1, S99E+01
3. 992E+00
1,.000E+00
2. 6276-01
9. 729E-02
8, 884E-02
1.381E-01
2. 1896-01
3.162E-01
4. 1598-01
5.092E-01
5.930E-01
6. 084E-01
7.378E-01
037-01
8.682E-01
9, 332-01
1.0006+00


B3(2,K,r,mu) for r= t
m\ re 1 2 4 8 16 2 Cd 12 me 512 1024

1,000E+00 1,000E+00 1,000E+00 1.000E+00 1,000E+00 1.000E+00

1,000E+00 1.000E+00 1,000E+00 1,000E+00

1,000E+00 1,000E+00 1.000E+00 1. 000E+00 1,000E+00 1,000E+00 1.000E+00 1.000E#00 1.000E+00 1.000E+00

2 \!
A18 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E#00 1.000E+00 1.000E#00 1.000E+00 1.000E+00 1.000E+00 1.000E+00
bat 1,000E+00 1.000E+00 1.000E+00 1, 000E+00 +00 1,.000E+00 1,000E+00 1.000100 1.0006+00 1.000E+00 1.000E+00
“1.4 +00 1.000E100 1,000E+00 1.000E+00 1,000E100 1.000E+00 1.000E+00
+00
+00

1,000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00

1
1
L
1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.
1
1,000E+00 1,000E+00 1. 000E+00 1,000E+00 1
1

:

1

1

:

:

:

+ — 1,000E#00 1, 000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00
16 + — 1,000E#00 1,000E+00 1,000E+00 1.000E+00 1,000E+00 1.000E100 1,000E+00 1.000E+00 1.000E+00 1,000E+00 1,000E+00

+ 1,000E+00 1.000E+00 1..000E+00 1.000E+00 1.000E#00 1.000E#00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1. 000E+00

1 1,000E+00 1.000E+00 1.000E+00 1,000E+00 1,000E+00 1.000E#00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

1 1,000E+00 1, 000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

+ 1,000€#00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000100

1 1,000E+00 1. 000E+00 1.000E+00 1. 000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

1 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E#00 1.000E+00 1,000E#00 1.000E#00 1.000E+00 1, 000E+00

+ 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E#00 1.000E#00 1.000E+00 1,000E+00 1.000E+00 1,000E+00

+ 1,000E+00 1,000E+00 1,000E+00 1,000E+00 1.000E+00 1.000E#00 1.000EH00 1.000E+00 1.000E+00 1.000E+00 1.000E+00
1.2%  1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00
1,42 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E#00 1,000E#00 1.000E+00 1,000E+00 1,000E+00 1.000E+00
1.6! — 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000600 1,000E+00 1,000E+00
1.8 3 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1. 000E+00
2 8 — 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1. 000E+0)

83(2,4,rau) for f= 1,01

m\ Me of 2 a. 8 S12 1024

1.024E+03

1,000E+00 2,000E+00 4. 000E+00 8,000E+00 1.600E+01 3. 200E+01 6.400E+01 1, 2B0EH02 2.560602 5.120E+02

1,000E+00 9, 9386-01 9.920E-01 9.914E-01 9.919E-O1 9.912E-O1 9.912E-O1 9.912E-01 9.912E-01 9.912E-01 9. 9126-01
1,000E+00 9, 944E-O1 9.928E-01 9. 923-01 9.9Z2E-O1 9,922E-Ol 9,9Z2E-01 9.9Z2-O1 9.922E-01 9.922E-01 9.922E-01
1,000E+00 9, 950E-01 9.936E-01 9,932E-01 9.932E-01 9.931E-O1 9.931E-O1 9.931E-01 9.931E-01 9.931E-01 9.931E-01
1,000E+00 9, 9S6E-01 9. 945E-01 9.942E-O1 P.941E-O1 9,941E-O1 9,941E-O1 9.941E-01 9.941E-01 9.941E-01 9, 941E-01
1,000E+00 9, 9636-01 9.9S4E-01 9.952E-01 9.9SIE-O1 9. 9S1E-O1 9.9S1E-O1 9.9S1E-01 9.9S1E-01 9. 9SIE-O1 9. 951E-b1
1,000E+00 9.970E-OL 9,963E-01 9.961E-O1 9.961E-O1 9.961E-O1 9.9O1E-O1 9.961E-Ol 9.9S1E-O1 9.9O1E-01 9. 961E-01
1.42 1,000E+00 9.978E-01 9.972E-01 9.971E-01 9.970E-01 9.970E~01 9. 970E-O1 9.970E-01 9.970E-01 9. 970E-01 9.970E-01
1663 1,000E+00 9, 965E-01 9, 9B1E-01 9.981E-01 9.980E-O1 9.9B0E-01 9. 980E-01 9.900E-01 9,90E-01 9.980E-01 9, 980E-01
1,84 — 1,000E+00 9.992E-01 9.991E-01 9.990E-O1 9.990E-01 9.990E~01 9. 990E-O1 9. 990E-01 9.990E-01 9. 9ME-O1 9.990E-01
2 § — 1,000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.0006+00 1,000E+00 1.,000E+00 1,000EH00 1,000E+00 1. 000E+00

Dot
71.8 1,000E+00 1.300E+00 1. 771E+00 2.545E+00 3,.852E+00 6.095E+00 9. 968E+00 1,669E+01 2.834E+01 4.867EH)1 8, 401E+01
“1.6 +  1,000E+00 1. 105E+00 1.234E+00 1.407E+00 1,450E+00 2.0046+00 2.532E+00 3.324E+00 4.518E+00 6.324E+00 9, 057E+00
“14 1 1,000E+00 1,037E+00 1.073E+00 1.110E+00 1, 155E+00 1. 2096100 1.278E+00 1.367E+00 1.446100 1.437E+00 1. 8386-00
“1.2 5 1,000E+00 1, 011E+00 1.019E+00 1,026E+00 1.032E+00 1.039E+00 1.046E+00 1.054E+00 1.063E+00 1.079E+00 1.084E+00
1 §  1,000E#00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000600 1,000E+00 1.000E+00
~«B8 tf  1,000E+00 9, 9S6E-01 9. 933E-01 9. 921E-01 9.913E-01 9,908E-01 9. 905E-01 9.902E-01 9.899E-01 9.B97E-O1 9.876E-01
7.6 § —1,000E#00 9,.938E-01 9.910E-01 9,898E-01 9,872E-01 9.887E-01 9.887E-01 9.884E-01 9.885E-01 9,884E-01 9,8B4E-01
~4 1} 1.000600 9,.932E-01 9, 905-01 9,875E-01 9.891E-01 9. 8896-01 9.688E-01 9,888E-01 9,887E-01 9.887E-01 9. 887E-01
~s2 1 1,000E+00 9,931E-01 9, 907E-01 9,899E-01 9.896E-O1 9,895E-01 9,895E-01 9. 894E-01 9. 894-01 9.894E-01 9. 894E-01
0 —$ — 1,000E+00 9. 9348-01 9.9139E-01 PAOSE-O1 9.904E-01 9.903E-01 9. 903E-01 9.903E-01 9.903E-01 9.903E-01 9.903E-01

:

'

t

1

'

33


random-walk FM noise («4 = +1, @ = -2). The asymptotic limit is approached as
the product of number of samples averaged and the initial data sample time,
tT), becomes larger than the dead time (M > r). The variances so obtained

differ only by a constant, which can be calculated as given in this paper.

A knowledge of the appropriate power-law spectral model is required to

translate a distributed dead-time variance to the corresponding value of the
Allan variance. In principle, the power-law spectral model can be estimated
from the r* dependence, using the variance analysis on the data as outlined

above.

9. References

[1] J.A. Barnes, "Tables of Bias Functions, B, and B,, for Variances Based on
Finite Samples of Processes with Power Law Spectral Densities," NBS Tech.
Note 375 (1969).

{2] J.A. Barnes, "Atomic Timekeeping and the Statistics of Precision
Signal Generation," IEEE Proc. 54, No. 2, pp. 207-220, Feb. 1966.

{3] D.W. Allan, "Statistics of Atomic Frequency Standards, "IEEE Proc.

54, No. 2, pp. 221-230, Feb. 1966.

[4] RFC Vessot, L. Mueller, and J. Vanier, "The Specification of
Oscillator Characteristics from Measurements Made in the Frequency
Domain," IEEE Proc, 54, No. 2, pp. 199-207, Feb. 1966.

[5] D.W. Allan and J.A. Barnes, "A Modified “Allan Variance" With
Increased Oscillator Characterization Ability," Proc. 35th Annual
Frequency Control Symposium, USAERADCOM, Ft. Monmouth, NJ, May 1981,
pp. 470-475.

[6] D.W. Allan, "Time and Frequency (Time-Domain) Characterization,
Estimation, and Prediction of Precision Clocks and Oscillators," IEEE
Transactions on UFFC, November 1987.

[7] P. Lesage and T. Ayi, "Characterization of Frequency Stability: Analysis
of the Modified Allan Variance and Properties of its Estimate," IEEE
Trans. Instrum. Meas., IM-33, no. 4, pp. 332-336, Dec. 1984.

[8] N.D. Faulkner and E,V.I. Mestre, "Time-Domain Analysis of Frequency

10


BIUNeju) for p= yt

wl me & 8 16 2 “ 128 Be 512 1024

"2 1,000E+00 1.006100 1. 000E+00 1.000E+00 1.000600 1.000E#00 1.0006+00 1.000E+00 1. 000E+00
“1.8 1.0936+00 1, 226E+00 1. 498E+00 1. SH7EH00 LL S79E#001,SB6E#001,585EH00 1,585E100 1, S8HE+00
“he 1,208E+00 1.522600 2.021E100 2. 3296100 2. 449E100 2, 490E#00 2.502E100 2. S0SE+00- 2, SOSEH00
“ld 1.337600 1. 907E+00 2, 832E#00 3.477E100 3.772800 3.892800 3. 9386400 3, FSHEH00 3.940610
-h2 1.A9LE+O0 2, 396EH00 «3. 931EH00 5. 125E#00 5. 751EH00 6, 046E+00 6, 177E400 6. 235E+00 6. 2606400

'
1
’
1
+ 1.667600 3.000600 5.375E00 7.429600 8, 6S9E1009.312E100-9.652E100 9. 825E100 9, 9128400
“Bt 1,860E+00 3,7206+00 7. 2046100 1,055E101 1.281E01 1.A19E01 §1,502E+01 1, 550E101 1. S7TE*O1
1 2,065E+00  4,540E100 9.4236+00 1. 4G1E+01 = 1.859610] 2.137601 2,326E+01 2.452601 2. SOE
1 2.2736+00  S.430E#00 1, 1996401 L.97IEOL 2.642801 3. 177EHO1 = -3.592E01 3. 9106401 4, ISGE+01
+ 2.472600 6.344600 1.4B1E#01 2, 5@5E+01 3.679601 4, GH4EH01 S.SAZEH01 96. 3146401 6, 989EH01
t  2.659E+00 7. 2366400 1. 779EH01 9, 900E#01 5.003601 6. 771EO1 «8, S64E01 —1.087EH02 1. 219602
+ 2,810E+00 8, 065E+00 2,080E+01 A. 1IIEHO! =. 6P0EH01 9. TS0EH01 «1. SS1E#02 1, 7416102 2.214602
+ 2,9N0E+00  8,B04E+00 2.376401 S,O1SE+01 8.G16E401 1, 397E#02 2.084602 2.998602 4, 19HE402
+ 3,0436+00  9,MMSEFOD 2.663801 6. 014EH01 1, 14961021. 999EH02 3, 295EH02 5. 265EH02 8. 2546102
1 3.123600 9. 987EHO0 2, 998E+01 7. 118E01 1, 487E#02 2.842602 5.269802 9. ASHEN 1.67803.
1 § 3, 1B4E+00  L.0485E+01 3. 2046401 8. 47EH01 1. 918EH02 4, 1146902 8.523602 1. 735EH03 3, SOOEH03
+ 3,290E+00  1,084E+01 3.463E+01 9. 728E+01 2, 47061025. 94EHO2 «1. 37HEHOS «3, 23261037, ASIEHOS.
1 3.265EH00 1. 118EHO1 «3, 721E+01 1. 130E02 3, 18SEHO2 8, 634EH02 2, 304E103 4, 10361031, 613E404
t 3.293400 L.147E+01 «3, 982E+01 1. O9EHO2 4 LISEHO2 1. 2636403 3.B49EH0S «1. 1GHE104 9, SSSEHO4
&  Q.S15E400 1.174801 4.252601 1.SI7E+02 5. SQ0E+02 1. 858E1036.447E+03 «2. 250EH04 7. R0EH04
t 3.333400 1, 200E+01 4.S33EH01 1. 7H0E02 «6, 9336402 2, 752EH03—1.097EHO4 4.378604 1.749105

BIINryu) for r= 3

Bs

1024

1,000E+00 1, 000E+00 1.000E+00 1, 000E#00 1.00000 1, 000E+00 1,000E100 1, 000E+00
1,226E+00 1, 25961001, 265E+00 1, 265E+00 1. 263EH00—1.262E400 1. 261E100 1, 261E+00
1.479600 1.564E+00 1.589E100 1. 59ME+001.594E1001,593E400 1. SPIEHO0 1, SPLEHOO
1,759E+00 1.925E+00 1. 98BE+00 2.00900 2. 014E+00  2,015E#00 2.015E+00 2. 014£+00
2,068E+H0 2, 351E+00 2, 479EH00 -2,SHHEH00 2, SSKEH00 2.585500 2.569800 2, 570EH00
2,405E+00  2.850E+00 3. 087E100 9.209600 3. 271E#00 3. 02EH00 3. 318E+00 —3.326E+00
2, 769E+00  3.435E+00 3. BATEH00 4, 09736400 4,.2398#00 4.325E+00 4.376610 4. 0SEH00
1,943E+00 3. 160EH00 4, 122E+00 4, BOMEHO0 5. 275E#00 5. 59SEH00 5.B12EH00 §5,.958E100 6. 056EH00
2.076E+00 3. 577E+00 4, 927EH0 6. 024EHO0 6. BBPEHO0 7, 564EH00 9 8.086E+00 8. 4B7E100 8. 795E100
2.2036+00 4,021E+00-5.877E#00 7, 597E#00 9. 147EH0O «1.053E401 1.175E+01 «1, 282E401 1. S7TE*0L
2.325E+00 44926400 7.000600 9. SSIEHOO 1, 2986401 LLSISEHO] «1. 796E+01 2.079801 2,.3469E+01
2.440E+00 49946400 8.338400 1. 23KE+01 «1. 710EH01 «2.20001 2. B97EH01 = 3,.632E+01 4, 479401
2, 550E+00  5.590E+00 9.940E+00 1,S99EH01 2, 411EH01 2, APGEHOL «4, 926601 6, 8236401 9, SI0E+01
2,654E+00 6, 106E+00 1. 187E#01 2, 089E+01 3.A72E+01 9 S.SBIEHOL §= 8. 786EH01 1.365E102 2. 103802
2.754400 6.729600 1.422E401 2, 75GEV01 S. 07801 «=, 1B7EHO1 §=—1.632E02 2.874802 5, 037EH02
2,852E+00 7.406E+00 1, 7086101 3,679E+01 7, 6186401 1.552602 3.132E+02 6, 29284021. 261E#03
2,9N7E+00  B.LATEHOO © 2, 059E#01 4, 42EH01 1. ISTE#02 «2.678602 170EH02 1. A19EH03 (3, 261E 403.
3.042600 89636400 2.493601 6. J09EH01 «1. 7BIEHO2 «4, 709EH02 1. 241E+03 3.272603 8, S31EHO3
3.137E+00 9. 868E+00 3, 030E+01 9, 18SEH01 2. 774E+02 8, SB81EH2 2,555EH03 7.679603 2. 324E+04
3,234E+00 1, 087E+01 3,699EH01 1. 267E402 4. 366EH02 L.SIIEHOS «5. 2456103 1.823604 6, 43EH04
3.33%+00  1.200E+01 4. S33EH01 1. 700EH02 6, 933E402 2, 7SZEHOS 1. 097EH04 4, 3786404 1. 7498405

20

1.000E+00
1. 583E+00
2, 504E+00
3.962E+00
6. 278E+00
1,0006+01
1,616E+01
2. 702E+01
4.92281
1, 136E+02

2.013E+00
2.571E+00
3.333610
4, 447E+00
6, 252E+00
9.TTCEHOO
2,017E+01


BO(2,Kyrjqu) for rs 256

Mm \ 1 2 123 Be ‘312 1024

1,000E+00 2.000E+00 4. 000E+00 1, 2R06+02 2.S60E+02 5.120E+02 1.024E+03

3 !
“1.8 $ 1,000600 1.7H1E#00 3,031E+00 4OS0EHO1 8, 445E+01 1.470E102 25606402
“1e6 + 1.000600 1,516E+00 2.297840 1.838601 2.784EH01 4.222401 6.399401
“1.4 1 1,000E+00 1.319€+00 1. 7416400 2.297600 3.0G1E+00 3.996400 5,Z77E100 6.962800 9. 187E+00 1.2126+01 1. 599EH01
“1.2 | 10006400 1,1496#00 1.319E#00 1,S1SE+O0 1,7HOEHOO 1.996100 22966400 2.637E+00 3.029600 3,479E+00 3.966400
=I 1,000E+00 1,0006+00 1.0006+00 1.000600 1.0006+00 1.000E+00 1.0006+00 1.0006+00 1.0006+00 1.000+00 1..000€+00
~.B + 1.000400 8,714E-O1 7,S97E-Ol 6,426E~O1 5. 7B1E-O 5.O4SE-O1 4.405E-01 3.048E-01 3.363E-01 2,941E-01 2,573E-01
-.b 1 10006400 7.8236-01 5,8306-01 4,475E-01 3,4506-01 2.67HE-01 2,086E-01 1.640601 1.303E-01 1.047E-01 8.533E-02
“41 1.000600 6,750E-01 4, 6246-01 3.235E-01 2,921E-O1 1. 7206-01 1,323E-01 1.062601 8.879E-02 7. 7636-02 7.014€-02
~.2 t  1,0006+00 6, 152E-O1 3,981E-O1 2. 7896-01 2.054E-01 1.655E-01 1,427E-01 1.2966-01 1,Z21E-01 1,178E-01 1, 1836-01
0 1.00000 5.0B2E-01 3,688E-01 2,9136-01 2.496E-~01 2.201E-01 2,0B4E-01 2.026E-01 1.997E-OL 1.982E-01 1.975E-01
«2% 1,000E+00 5.929E-01 4, 2336-01 9,S25E-01 3. 2266-01 3. 099-01 3.085E-01 3,022E-01 3.011E-O1 3.007E-01 3. 008E-01
<4 + 1.000400 6, 2028-01 4,G97E-O1 4,347E-01 4.170E-01 4.105E-O1 4.0B1E-01 4.0728-01 4.059E-01 4.058E-01 4. 067E-01
+6 t 1.000600 6.596E-01 5.SQ0E-01 5.198E-01 S.094E-O1 S.OSIE-O1 S.CSIE-O1 5.OS7E-01 5.0ASE-O1 S.OME-O1 S.O46E-01
+8 + 1.000600 7,042E-01 6, 2216-01 5. 995E-01 5.93GE-O1 S.91E-O1 S.9IIE-O1 5.910E-O1 S5.910E-O1 5.9106-01 5,910€-01
1 1.000600 7507-01 6.080E-01 6,727E-01 4,488E-01 6.679E-O1 6.670E-01 6.476E-01 6,675E-01 4.675E-O1 6.4756-01
1.21 1.000600 7,9B1E-O1 7.S16E-O1 7,408E-O1 7,262E-O1 7.376E-O1 7.37SE-01 7.374E-01 7.374E-01 7.374E-01 7.374€-01
1.41 1.000600 8.A64E-01 8,132E-01 8,058E-01 8,041E-01 8.034E-01 8.03KE-01 8,03SE-01 8,035E-01 8.035E-01 8.035E-01
1.6 1,0006+00 8,958E-01 8,743E-O1 0,694E-O1 8.685E-01 8.682E-01 8.682E-01 8.682-01 8.682E-01 8,482E-01 8,482E-01
1.8 1,0006+00 9,4696-01 9.362E-01 9.339E-01 9,333E-01 9.9328-01 9.332E-01 9.332E-01 9.5326-01 9.3328-01 9. 302E-01
2 t  1,0006+00 1,000E+00 1.0006+00 1.000610 1,0006+00 1.000E+00 1.00000 1.0006#00 1.000400 1.0006+00 1, 0006+00
53(2,yr,eu) for = 512
MN het 2 4 8 16 2 “ 128 4 512 1024

1.000E+00 2,.000E+00 4..000E+00 1,Q00E+01 3,200E+01 6.400E+01 1. 280E+02 2.560E+02 5,120E+02 1.024603
1,000E+00 1. 741E+00 3. 031E+00 9, 190E+00 1.G00E+01 2,786E+01 4. 850E+01 8.445E+01 1.470E+02 2.560402
1,000E+00 1,516E+00 2. 297E+00 5. 2786+00 8.000E+00 1.213E+01 1.838601 2, 784E+01 4.222E+01 6, A00EH01

1,000E+00 1. 3196+00 1, 7416400 2.297E+00 3. 031E+00 4,000E+00 5.277E+00 6. 964E+00 9, 1S9E+00 1.212E+01 1, Q00E+01
1,000E+00 1, 149E+00 1, 319E*00 1,516E+00 1. 741E+00 2.000E+00 2.297E+00 2.638+00 3,030E+00 3.481E100 3. 9986100
1,000E+00 1.000E+00 1,000E+00 1.000E#00 1,000E+00 1.000E+00 1,000E#00 1.000E+00 1.000E#00 1.000E+00 1,000E+00
1,000E+00 8. 711E-O1 7.S89E-O1 6.614E-O1 +5, 765E-01 5.026E-01 4.383E-01 3.823E-01 3.336E-01 2.911E-01 2,5428-01
1,000E+00 7, 606-01 5,G00E-01 §,4336-01 3,398E-01 2.614E-01 2.020E~01 1.570€-01 1, 2296-01 9.709E-02 7. 7S51E-02
1,000E+00 6.712E-O1 4,557E-O1 3.1436-01 2.213E-01 1,601E-01 1,19BE-Ol 9.316E-02 7.S61E-02 6.403E-02 5, 439E-02
1,000E+00 6.085E-01 3.870E-01 2,619E-O1 1.897E~O1 1.487E-O1 1.Z59E-01 1.118E-01 1,041E-01 9,970E-02 9. 716E~02
1,000E+00 5, B03E-01 3, 761E-01 2.7646-01 2.2746-01 2.0G1E-O1 1.911E-01 1.851E-01 1.821E-O1 1.806E-01 1. 7996-01
1.000E+00 5.869E-01 4.136E-O1 3.419E-O1 3,108E-01 2.978E-O1 2.9228-01 2.898E-O1 2.887E-01 2.883€-01 2. 9B1E-01
1,000E+00 6, 1626-01 4, 7B1E-OL 4. 2846-01 4, 1048-01 4.039E-01 4.014E-01 4,005E~O1 4,002E-01 4.001E-01 4. 000E-01
1,000E+00 6.576E-01 5.S04E-01 5.169E-01 S.Ob4E-O1 S.OS1E-O1 S.O21E-O1 S.O1GE-01 S,016E-O1 S.O16E-O1 5.016E-01
1.0006+00 7,083E-01 6, 211E-O1 S.9@4E-O1 5.92Z1E-01 5. 904E-O1 5.897E-01 S.G78E-Ol S.898E-01 5.898E-01 5, 898E~01
1.000E+00 7, S0GE-01 6.879E-01 6.723E-O1 4.O84E-O1 6.674E-01 6.472E-O1 6.671E-01 6.671E-01 6.671E-01 4,671E-01
1.2.8 1,0006+00 7, 9006-01 7,S15E-O1 7.406E-O1 7.9B1E-01 7.375E-O1 7.379E-01 7.373E-01 7.373E-01 7.37%-01 7.37%-01
1,4  1,000E+00 8,463E-01 8,132E-01 8,057E-O1 8.0M0E-O1 8.0Q6E-01 §,030E-01 8.035E-01 8.035E-01 B.0G5E-01 8.0G5E-01
1% 5 1,000E+00 8, 930E-01 8. 7436-01 G.O08E-01 6.682E-01 8,682E-01 8.681E~91 8.681E-01 8.681E~O1 8.481E-01
1,61  1,000E+00 9,449E-01 9.3626-~O1 9.S09E-O1 9,353E-O1 9.SR2E-~OL 9.332E-01 9.392E~01 9.332E-O1 9,332E~01 9. 5326-01
2 1 1,000E+00 1,0006#00 1. 000E+00 1.000E+00 1.006100 1.000E+00 1.000E+00 1. 0006100 1.000E+00 1.000E+00 1.000E+00

38


VARIANCES BASED ON DATA WITH DEAD TIME BETWEEN THE MEASUREMENTS

James A. Barnes
Austron, Inc.
Boulder, Colorado 80301

and

David W. Allan
Time and Frequency Division
National Institute of Standards and Technology
Boulder, Colorado 80303

The accepted definition of frequency stability in the time domain is
the two-sample variance (or Allan variance). It is based on the
measurement of average frequencies over adjacent time intervals,
with no "dead time" between the intervals. The primary advantages
of the Allan variance are that (1) it is convergent for many
encountered noise models for which the conventional variance is
divergent; (2) it can distinguish between many important and
different spectral noise types; (3) the two-sample approach relates
to many practical implementations; for example, the rms change of an
oscillator’s frequency from one period to the next; and (4) Allan
variances can be easily estimated at integer multiples of the sample
interval.

In 1974 a table of bias functions which related variance estimates
with various configurations of number of samples and dead time to
the Allan variance was published [1]. The tables were based on
noises with pure power-law spectral densities.

Often situations occur that unavoidably have dead time between
measurements, but still the conventional variances are not
convergent. Some of these applications are outside of the time-and-
frequency field. Also, the dead times are often distributed
throughout a given average, and this distributed dead time is not
treated in the 1974 tables.

This paper reviews the bias functions B,(N,r,p#), and B,(r,y) and
introduces a new bias function, B,(2,M,r,#), to handle the commonly
occurring cases of the effect of distributed dead time on the
“computed variances: Some convenient and easy-to-interpret
asymptotic limits are reported. A set of tables for the bias
functions are included at the end of this paper.

Key words: Allan variance; bias functions; data sampling and dead
time; dead time between the measurement; definition of frequency
stability; distributed dead time; two-sample variance


10(2,,rmu) for r= 01
wi me 1 2 4 s
“21 1,0006+00 2.000600 4.000600 8. 0006+00
“1.81 1,0006+00 2.214600 $,042E+00 1, 829E+01
“16 1. 1,0006+00 2.4196+00 7.398600 2, 6988401
“1.4 1 1.000600 2.617600 8. 745E+00 3, 2956+01
“1.2 1 1,0006+00 2.811600 9,937E*00 3, 0436001
“11 1.00000 3,0006+00 1.100601 4, 3006+01
=.B 1 1.000600 3, 180600 1. 1916401 4, 4S1E+01
“16 1 1,000E#00 3.557800 1, 258601 4,0806+01
<4 1 1.000600 3.444600 1.285601 4.012601
=.2 $  1,000E+00 3.449600 1, 282601 4. 4006401
0 1 1.000600 3.368600 1. 1S1EMO 3, 8646+01
12+ 150006+00 3.16600 9.992600 3, 0906401
+4 1.000600 2.080600 8, 2516400 2.314601
+b 1 1.000600 2.574600 6.579E400 1, 6S1E+01
+8 1 1.00000 22706400 5. 1236400 1. 1416401
1 + 1.000600 1.989600 3.940600 7.728600
1.2.1 1.00000 1.737600 9.008600 5, 177E+00
1.41 1,0006#00 1.514640 2.289600 3.466400
1.61 1.000600 1.3196+00 1.738600 2. 2856+00
1.8 | 1.000600 1.1496+00 1,3196+00 1,S16+00
2} 1.000600 1.0006+00 1.0006+00 1,.0006+00
H 83(2,Mr,au) for r= 03

1,000€+00
1,000E+00
1.000€+00
1, 000E+00
1.000E+00
1, 000E+00
1.000€+00
1,.000E+00
1,000E+00
1.000E+00
1,000€+00
1.000€+00
1, 000€+00
1,000E+00
1,000E+00
1, 000E+00
1,000E+00
1.000E+00
1, 000€+00
1,000E+00
1.000E+00

2

2,.000€+00
2, 216+00
2. 422E+00
2, 673600
2, 816E+00
3. 000E#00
3.1G4E+00
3..290E+00
3, 356E+00
3.339100
3. 2298100
3.036E+00
2. 787E+00
2. S10E+00
2. 2328+00
1, 967E+00
1, 725E+00
1. 508E+00
1,316E+00
1, 148E+00
1, 000E+00

7. 4AIEHO0
8. 817400
1. 000E+01
1, 1006+01
1. 17SE+01
1, 2076401
1.213801
1, 1S7E+OL
1,0526+01
9. 146E+00
7. 6M0E+00
6, 186E+00
4, 897E+00
3.818E+00
2, 946E+00
2. Z59E#00
1.725600
1, 3146+00
1..000E+00

9. 378E+01
3, 910E+01
4, 00E+01
4, S23E+01
4, SA3E+01
4, 3316401
3.896E+01
3. 298E+01
2. 635E+01
2,0026+01
1, 461E+01
1,068E+01
7.179800
4, 907E+00
3.322E+00
2. 2348100
1,497E+00
1,000€+00

1, 600E+01
7, 138E+01
11198402
1 ALIE+02
1, O7EW02
1. 7106#02
1, 7196402
1, 6326402
1, AS3E+02
1, 2146002
9. M6ZE+01
6, 912E+01
4, THEOL
3. 153E+01
2.013601
1, 254401
7, 688E+00
4. 6SBE+00
2. 801E+00
1,.676E+00
1,.000€+00

3. 2006+01
2. S36E+02
4. 1516902
5, 3308+02
6. 287E*02
6. 8306402
7.063E+02
6. 877E+02
6.213802
S.131E+02
3. 826E+02
2.579802
1,5958+02
9. Z28E+01
‘S.070E+01
2. 731E+01
1, 435E+01
7. 440E100
3. 8286+00
1. 960E+00
1, 000€+00

31

6, 400E+01
1.173803
1, 688E+03
2, WE03
2. SES
2.67088
2. SR0E+03
2, WES
1. 957EHS
1, 486E+03
1.015803
6. 2336402
9, 5216402
1,851E+02
9. 200801
4, 0RE+01
2.142601
1, 008E+01
4, 78E+00
2, 166E+00
1, 000€+00

4
6, 400E+01
4, 880E102
7. 9SE02
8, 2856902
8. GEO?
8, 2216402
7, 4SSE+02
6, 378E+02
5, 1266402
3, 851E+02
2.702602
1, 76E+02
1, 1046+02
6.564401
3.772801
2, L16E+O1
1, 1686401
6.STIE+00
3, 450E+00
1, 860E+00
1, 000E+00

12

1, 2008+02
9.308603
5.073
5.96403
6. 2536103
6,.094E903
5. S70EH0S
4. 77H
3, T74E+03
2.712603
1, 783003
1.021603
5.442602
2, TO7EH02
1, 285E+02
5, 908E+01
2, BOOE+OL
1, 1626901
5.21 1£+00
2, 286E+00
1,0006+00

9. Ob7EHO2
8. OSE?
7, 247E02
5. 743102
4.262602
2. 9STEW2
1.9246+02
1, 184E+02
6, 975EH01
3. 973E+01
2, 2106901
1, 2098401
6,538E+00
3.511E+00
1,876E+00
1, 000E+00

2.360E+02
4. 836E+03
7, 227E48
0.2698+03
8, M9EH03
8,.047E+03
7, 208603
6, 085E+03
4, 6B6E+03
3, 904603
2.007EHS
1, 2008903
6, 2006402
3. 071E+02
14316402
6, 4O4E+01
2. 89EO1
1,.248E+01
5. 405E+00
2.328600
1,.0006+00

2, 560E+02
8, 075E+02
1, 065E+03
1, 1478403
1, 125E+03
1.039E+03
9, 122E#02
7, O0E+02
5. 974E#02
4, WOSE+02
3.040E+02
1, 970E#02
1, 209+02
7, 095E+01
4. 0Q0E+01
2.236601
1, Z20E+01
6, 583E+00
3.527E+00
1, 8B1E+00
1.000€+00

‘S12

‘5. 1206+02
6,0806+03
8. 7465903
9. 700E+03
9.671EHOS
9.02403
7. SOEHOS
6.577E403
S,ONEH03
3.521603.
2.216603
1. 2996403
6, SA9E102
3.183602
1,476E+02
6.6296+01
2.91 7E+01
1, 267E+01
5. 4596+00
2. 3M0E+00
1, 000E+00

1.024640
7. ASSEHO3
9: 0BZEHS
1,083E+04
1.036E+04
9. 312E+03
9. 200E+03
6, 791E+03
3.171403
2.503
2, HEH
1.277E+03
6, 627E+02
3,.215E+02
1, 4B8E+02
6,6736+01
2.933E+01
1,.2726+01
5, 473E+00
2, W3E+00
1,.0006+00

1.0246+03
1, 143E+03
1, 274€+03
1, 2836+03
1, 2126403
1,0936+03
9, ASIE+02
7.794802
6, 086E+02
4 M8EHO2
3, 074E+02
1, 987E+02
1.217E+02
7.136E+01
4. O49E+01
2, 244E+01
1, 2248+01
6.598E+00
3,.532E+00
1,882E+00
1,000E+00.


Appendix

With reference to figure 1, the frequency sampling window has an equivalent
phase sampling window. The intent is to evaluate the variance, S(M), of the
sampled phase function in terms of the phase autocorrelation function, R(r).
The process here is to correctly account for terms and cross-terms coming from
squaring and averaging the samples for each M. The B,(2,M,r,u) function can

then be obtained from the relation,

By (2,M,x,p) = —S@O__
S(1) «Mat?

for appropriate M, r, and ». The denominator is just the two-sample variance
with dead time for MT and Mr (in accordance with the definition of
B3(2,M,r,n)). The factors common to the numerator and denominator are ignored
in the following.
For M = 1, the variance S(1) is just
;
S(1) = 4-R(O) - 4-R(r) - 4-R(T) + 2-R(T+r) + 2R(T-1),
where use has been made of the definition of the autocorrelation function,
R(T) = E[¢(t) - d(t+T)].
It is convenient to define a function G(T) as
G(T) = 2-R(T) - R(T+r) - R(T-r).
Similarly, S(2) can now be written in the form,

S(2) = 8-R(O) - 8-R(r) + 2-G(T) - 4:G(2T) - 2-G(3T).

Following this procedure, we can verify that the general S(M) is just

17


fu \ Ne

=
al
oh
“14
1.2
-l
78
ob
“A
2
0
2

Roan

:
t
:
a
t
'
of
:
’
:
'
’
t

BUN rm) for re 4

‘ 8 16 2 “ 2 TM 312 10
1,0006+00 1.0006+00  1,0006+00 1.0006+00 1.008500 L.80E*00 1.00000 1.000600 1.00000 1. 0006+00
9,97ME-01 9,9S7E-01 9.944E-D1 9, 9ME-01 9.93401 99GSE-01 ALVGAE“O1 9.9G3E-01 9.9G3E-01 9, 9336-01
9,9SZE-01  9.918E-01 9.89601 9.8BME-01 9.876501 9873E-01 P/BTOE-O1 9. B69E-01 9.889E-01 9. 968-01
9,941E-O1 9.8F8E-01 9.969E-01 9.851E-01 9,S40EO1 F.ROREC1 FROGIE“O1 9.827E-~01 9.629E-01 9, 8206-01
9,95%-O1 9.916E-01 9.8906-01 9.873E-01 9,.862E-01 9.8K! “P-SIE-O1 9.8SIE-01 9.850E-01 9. 8498-01
1.0006+00 1.006400 1.000E+00 1. 000E+00 1.008640) “1.0ROGE€D0 100000 1.000E+00 1.000400 1. 000E+00
1.010600 1,0196+00 1.02600 1.031E#00 1,035E+00 1.037E+00 1, 039E+00 1.080600 1,041E+00 1.0426+00
1,027E+00 1.0896+00  1.075E+00 1.070600 1.10700 -L.117E¥00 11246400 1.129E#00 11926100 1.140E#00
1,054E+00 1. 1096400 1. 160E+00  1,205E#00 1, 2436100 12746100 172996100 1.318600 1,2306+00 1.982640
1.0926+00 1.194600 1.298100 1.397E100 1.494E#00 155806100 458EH00 1.727E400 1.760600 2, 2046400
11446100 1.31864001.514E+00 1.726610 1,94E#00 207770 24146700 2.451E100 2.890640

1,2146900 1.495E+00 1.848E+00 2.27SE+00 2.783 2.STFECO 4.087600 4.865100 5. 7846400

1,9036+00 1, 7426+00 2.356600 9, 196E#00 4.326600 “B/BHEFOO “7,897EM@D 1.046401 1. 399E+01

1.415E+00 2,0826+00 3.129100 4,748E+00 7. 2296400 1. 10E*O1 16768401 2.550601 3.875E+01

1,555E+00 2.548100 4,2096+00 7.385E+00 1.278101 2.219401 S.GM0EHI 6.718601 1. 170€402
1,727E+00 3, 1826400 6.071E+00 1, 191EH01 2.556401 ACMEIEKDL 9.SQGEFD1 | 1.865E+02 3.724602

1,9386+00  4,045E+00 8.826640 1.974EH01 | 4.4725101 1.020802 2.395602 S.SSEHI2 1.298103
21996400 5.296400 1.300601 3, 9406401 B.684EHOI 2-77SEKO2 SABIE 1.575403 4, 1546403
2.504100 6.B19E100 1.952401 5.74501 1, 7ISECOZ "S1RKOE02 1SSERTS 4, 714EH031.4286904
2.879600  9.006E+00 2.960401 1.000102 3.431E#02 ULHBAEHOS A1126+03 14296404 4.972608

3.533640) 1.200601 4.5306401 1.760642 6,983E802 27ISZEH03 ALUOWTEKOH 4.378104 1, 7496405

BL(N,rjau) for r= 8

4 8 16 2 “ 128 B77 512 1026 IN
1,0006+00 1,0006+00 1.0006+00 1,0006+00 1,0008800 11.000E+00 10006400 1,0006+00 1.000600 1. 0006+00
9.9936-01 9,988E-01 9.985E-01 9.983E-01 9.982E-01 9, 9B1E-01 2.ABHEWOI 9, 981E-01 9. 981E-01 9. 981E-01
9.984E-01 9.9736-01 9.96601 9.962E-01 9.960E-01 9. 958E-01 9. TSGE-01 9.957E-01 9.957E-01 9, 957E-01
9.978E-01 9.961E-01 9. 950E-01 9. 944E-01 9, 940E-01 RMBTE-01 RAGKELO] 9.936E-01 9.935E-01 9, 9S5E-01
9.9906-01 9.9636-01 9.952E-01 9.945E-O1 9,980E-01 “9.938E-01 P.PRKE-O1 9.9G5E-01 9.995E-01 9, 9346-01
1.0006+00 1, 0006+00  1.000E#00 1. 000E#00 1.000E+00 \1.000E+00 1. 080E+00 1. 000E+00 1.006400 1. 0006+00
1,006E+00 10116400 1.014E+00 1.017E+00 1.020690 1VOZIEW0O 4022800 1,0226+00 1.023400 1.023600
1.017E+00  1.0336+00 1.047E+00 1.058E+00 1,067E+00 11.1073E+00 1 /078E¥00 1,081E+00 1,0836+00 1, 0886+00
1,0376+00 1,075E+00  1.110E#00 1,142E#00 1, 1686+00 11, 189E+00 1, 207E#00 1, 2206100 1, 231E#00 1, 264E+00
1,0696+00 1,145E+00 1.22300 1.296100 L,371EH00 1.A3SEH00 L.AMELOO 1.S44EH00 1,9926+00 1. 905E+00
1.116E+00 1, 255E+00 1, 413E+00 1,58HEH001,7636+00 L9H9EH0O 2. 197EH00 2.328100 2, 5206100

1.181E+00 1,4206+00 1.720400 2,083E+00 2.SI4EHID 3,0196+00 3,6056400 4. 2846100 5, OO4EHO0

1, 2686400 1,657E+00 2.202100 2,9466H00 3.945100 5.206100 7.062100 9.416100 1, 2526+

1.380600 1,9916+00 2. 950E+00 4.4336+00 6, 706600 1.OI7EW01 1.SM4EH01 2.945EH01 3. 5596401

1,S21E+00  2.453E#00 4. 101E+00 6.996E00 1, 206E+01 2.090101 3.6001 6.315401 1.099602
1.6966#00 3.087E+00 5.870100 1.1436+01 2.Z57EH01 AABIEHOL B.Q5EO1 1.704102 3, 545E02
1,910E+00  3,953E+00 8.590100 L.DITEOL 4.946401 “B5Q7REH0] 2.268EH02 5. 195EH02 1.192403
2.170600 5.135E+00 1.279601 3.276501 8, SISEKOL 2.290402 “S.BA2EHO2 1.544E+03 4.0716+03
2,486E+00 6. 7SIEH00 1.900601 5.678601 1.69502 “5.079102 1,5406H03 4.658409 1. 411EHO

2.870600 8.963400 2.94SEHOL 9,9SIEH01 3.419402 1.17903 4.090609 1.A2IEHH 4, 9456404
3,3336+00 1.200601 4, 5996401 1. 7606402 6.933402 _2.7S2EH03 LLLOTEUO 4.378404 1. 749EH05

23,



BI(Wrjeu) for r= OL

uN we 4

1,000E+00
1.091E+00
1.199E+00
1.328600
1,482E+00
1,6676+00
1, 8B4E+00
2. 138E+00
2.407E+00
2. 677EW00
2. 9126400
3, 0896100
3. 2036+00
3.2686+00
3. 302E+00
3.319400
3, 327E+00
3. 330E+00
3,3326+00
3,.330E+00
3.333100

1.000600
1. 2106+00
1,487E+00
1,.856E+00
2.346E+00
3..0006+00
3..860E+00
4, S9EH00
6, 2796100
7.714€900
9. 07SEH00
1, 016601
1.095E+01
1, 1438001
1, 170E+01
1, 185E+01
1.192801
1 196E+O1
1, 198EHO1
1, L99EHO1
1, 2006101

BLN, au)

for r=

1%

1,.000€+00
1, 380E+00
1, 873E+00
2. 6BBE+00
3,.880E+00
5.667E+00
8. 303E+00
1, 208E+01
1, 7036401
2,.296E+01
2. 9096401
3, 448E+01
3. 8S7E+01
A, 1ZSEH01
4.303601
4. 403E+01
4. AO1EOL
4. 494E401
4, S14E+01
4. S26E+01
4. S33E+01

16

03

1, 000E+00
1.SASE+00
2.45100
3. 991E100
6.598E+00
1. 100E#01
1, 6286401
2.977E901
4. O55E+01
6, 604E+01
9, 282E01
1, 1628402
1, S4E+02
1.49EH02
1,590E+02
1,6536+02
1,696E+02
1.7206+02
1, 7386+02
1, 750E+02
1, 760E+02

o

1,000E+00
1.772E+00
3, Z39E+00
6. 054E+00
1.145401
2, 1676901
4. OMEHO1
7. 200EH1
1.247E+02
1,.9768#02
2, 857E+02
3.763802
4.572802
5.221602
3. 7086402
6, 064E+02
6. 330E+02
6. 5906102
6, 6886+02
6.B19E+02
6, 733EH2

128

1.000E+00
2.089E+00
4. 4A0E+00
9. 5S00E+00
2.026E+01
4. ZRE+01
8.691E+01
1,697E#02
9. 106E+02
3. 2206902
7. FEW?
1,097E903
1.390E+03
1.649E+03
1. 069E+03
2. 053E#03
2.215603
2. 59EH03
2,493E+03
26236403
2, TS2E#03

Bs

1.0006+00
2.349600
5.505E+00
1,.2826+01
2. 947EHOL
6.628601
1.43EHO2
2, 97H
5.61402
1, 0866403
1.67203
2.440603
3.287803
4.142809
4. INES
5.042603
6.717803
7. OME+03
8.697603
9. 73TEHO3
1, 097EH04

512

1,000E+00
2. 43REHOO
6.002E+00
1,456E401
3. 486E+01
8, 190E+01
1, 068E+02
4.076E#02
8, 357E+02
1, 5806409
2. 7196403
4.263603
6.172403
8, 4196403
1. 104E+04
1, 412E+04
1, 782E+04
2. Z2AEHO4
2. 793EHO4
3, 493EHO4
4, STREHOA

52

1024

1, 000€+00
2.494E+00
6, 198E+00
1,5326+01
3. THEW
9. 0646901
2. 137E+02
4. 851002
1, 0436#03
2. 084E+03
3.826E+03
6, 4596903
1.014E+04
15146404
2, 189E+O4
3. 109E+04
4 EEF
6, 169E+04
8.697104
1, ZB1E#05
1, TA9EHOS

1024

1.000E+00
2.512600
6,.309E+00
1.SBS5E+01
3, 980E+01
1,.000E+02
2.519802
6, 4236402
1, 7198403
5.581103

1, 000E+00
1,0F1E+00
1,199E+00
1.329800
1, 484E+00
1,667E+00
1,878E+00
2, 114E#00
2.362600
2, BOMEHO0
2,8196+00
2.99E100
3, 118E+00
3.203600
3.257600
3,.290E+00
3. 309E+00
3.320E+00
3,.327E+00
3.331E+00
3, 3336400

e

1.000E+00
1,2116+00
1.A91E+00
1,8696+00
2. 355E+00
3,.0006+00
3, 825E+00
4, B38E+00
6. 005E+00
7. 242E+00
8, A30E+00
9, A60E+00
1.027E+01
1. 087E+01
1, 1286401
1, 1SSEHO1
1, 1726401
1, 164E+01
1 191E+O1
1. 196EHOL
1.200E+01

1..000E+00
1, 366E+00
1, 910E#00
2.719800
3. 918E+00
5.657E+00
8, 141E+00
1, 147E401
1, 566E+01
2.047E#01
2, S46E+01
3,013€+01
3, 4126401
3.728E+01
3. 966E101
4. 1ME+O1
4. 267E+01
4. Q0E+01
4. 431E+01
4, 487E+01
4, 533601

1,000€+00
1, 573E#00
2.532610
41338100
6. 768E100
1, 100E+01
1, 7S3E+01
2. 706E+01
3.993401
5. SB1E+01
7, WRE+01
9, 127E+01
1,077E02
1, 220€+02
1.339802
1,437E+02
1,518€+02
1. 588E+02
1, 650E#02
1, 706EH02
1,760E+02

1. 000E+00
1,827E+00
3. SALE+00
6.080800
1. 09736401
1. 928E+01
9, 204E+01
‘5. MEH
8. 524E+01
1, 259602
1, 7486102
2, 2B6EH02
2, 835EH02
3.372602
3.887EH02
4. 382602
4.867602
5.352602
5. B51E+02
6.374602
6, 7336102

19

1,000E+00
1. 9506+00
3. 782E+00
7, 268E+00
1. 378E+01
2, 560E+01
4.626E+01
8. O49E+OL
1, 335E+02
2.094E+02
3,096EH02
4. 324EH02
5. 7S0EH2
7. 357E#02
9.1S2E+02
1, 117E+03
1, 348E+03
1.615603
1, 990E+03
2. 3046103
2, TS2EW03

1, 000E+00
1.95E#00
3. 9636+00
7,815E+00
1,5256+01
2.929801
5.499E+01
9. 997EHOL
1, 7M4EH02
2. 897EH02
4, S68E+02
6, B56E+02
9, 68E+02
1. 376E+03
1,879E+03
2, 533E+03
3.392E003
4, S3SEH03
6,.0686+03
8, 142E+03
1.097E+04

1,.000E+00
2.010E+00
4.030E+00,
8. 0ME+00
1, S95E+01
‘3, 127E+01
6, 029E+01
1.134802
2,.066E+02
3,619E+02
6, 0826102
9, S4EH02
1, S#1EH03
2, 964E+03
3, 580E103
5.37503
8, 128E+03
1, 228E+04
1. B6GE+04
2, B4IEHOA
4, S78E+04

1, 000E+00
2.015E+00
4, 053E+00
8. 135E+00
1,626E+01
3. 2296401
6, MEH!
1, 2258102
2.314602
4 S8EH02
7. 611E#02
1.328403
2.277EH3.
3. 8686103
6, SHES
LA 1LAE+O4
1. 9036404
3,.276E+04
5. OB4E+04
9, F3GE+04
1. 7498105

1, 000E+00
2.016E+00
4, 064E+00
8, 191E+00
1, 651E+01
3.336401
6,.770E+01
1, M03E+02
3.099EH02
8.S71E+02


B3(2,M,rjeu) for re AS

4

4,000E+00 8.000E+00 1.400401 3, 200E+01

1,000E+00 2.116E900 4, Z21E+00 7.125E+00 9,0136+00 9, 6836400
1,000E*00 1.888400 3.977600 5.2236+00 6. 5ME+00 6.745E+00
1,000E+00 1,6746+00 2,676E+00 3, 791E+00 4.435E+00 4, 653E+00
1,000E+00 1,477E#00 2, 108E+00 2, 7326100 3.074E+00 3, 187E+00
1,000E+00 1.2996+00 1, 647E+00 1, 960E+00 2.120E+00 2, 172400
1,000E+00 1,141E#00 1, 285E+00 1.401E+00 1.457E+00 1.475E+00
1,000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1,000E+00

1
1.8 $  1,000E+00 2.2296+00 6.118600 2,227E401 4.419601 5. 757E+01
“1.6 1  1,000E#00 2, 048E+00 7. B56E+00 3,138E*01 6,097E+01 7. 99KE+01
“1.4 $  1,000E+00 2.654600 9,270E+00 3.4266+01 69196401 9.832640!
“1,2 1  1,000E+00 2.042600 1,0G2E+01 3,859E+01 7.133E+01 8. 9116901
“tt 1,0006+00 3,000E+00 1. 1006901 3,Q59E+01 6. 906E+01 8, 4S3E+01
7B 1 10006400 3, 117E+00 1.127601 3.694401 6.365601 7, 6455401
~+b 8 1.000600 3, 180E+00 1.114601 3. 305E+01 S.617E901 6.632601
~4 + 1,000E#00 3,179E+00 1.051601 2.981E+01 4.760701 5.532E+01
712 L.000E#00 3. 1116100 9.757E+00 2,5286+01 3.684E+01 4, 447E+01
O —$ — 1,000E+00 2. 980E+00 8. 683E100 2.069901 3.059601 3, 455E+01
+2 1 $,000E*00 2, 7996900 7,502E+00 1.6426+01 2.556101 2. 603E+0!
+4 t 1.000610 2.584600 6, 317E#00 1.269601 1.735E+01 1, 910E+01
+6 1.000800 2.352400 5. 208E+00 9. S91E+00 1,.262E+01 1, 372E+01

t

1

:

!

83(2,Nrjeu) for r= 3

m\ Me od 2 4 8 16 2

4.000E+00 8.000E+00 1.600E+01

3. 200E+01

“1.8 + — 1,000E+00 2,388E+00 5.582E+00 9,024E+00 1,341E401 1. 980E+01
“1.6% 1,000E+00 2.6696+00 6, 503E+00 9.614600 1. Z37E401 1. 521E+01
v1.4 + 1,0006+00 2.857E+00 6. 937E+00 9. 770E+00 1, 171E+01 1,317E+01
“1.2 §  1,0006+00 2,964E+00 70196400 9,562E+00 1,1026+01 1, 168E+01

1,000E+00 3,000E+00 6. 833E+00 9,063E+00 1.021E+01 1,077E+01
1,000E+00 2.976600 6,477E#00 8,422E+00 9.300E+00 9, 686E+00
1,000E+00  2.902E+00 60086400 7,655E400 8,339E+00 8. 610E+00
1,000E+00 2. 7896+00 5, 478E+00 6, 844E+00 7.374E+00 7, S66E+00
1,000E+00 2.648E+00 4.923E+00 6.030E+00 6. 4436100 6, SB0E+00
1,000E+00 2.487E+00 4,3746+00 5. 2646400 5.573E+00 5.671E+00
1,000E+00 2.316E+00 3.850E+00 4,548E+00 4, 7BIE+00 4. G51E+00
1.000E+00 2. 140E+00 3,362E+00 3.900600 4,079E#00 4, 1236400
1,000E+00 1.967E+00 2.919E+00 3.324E+00 3.450600 3, 4B6E+00
2,522E+00 2.820E400 2.910400 2, 735E+00
1,000E+00 1,639E+00 2, 171E+00 2.383500 2.446E+00 2. 4436+00
1,000E+00 1.489E+00 1.G54E+00 2, 0096100 2,051£100 2, 062E+00
1,000E+00 1,350E+00 1.597E+00 1,690E+00 1.716E+00 1,723E+00
1.000E+00 1. 2236400 1.367E+00 1,420E+00 1.4346+00 1.438E+00
1,000E+00 1,106E+00 1. 1696+00 1,192E#00 1.198E+00 1,200E+00
1,000E#00 1,000E#00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

3
3
3
3

32

4. 7148400
3.218E+00
2, 186E+00
1, 480E+00
1.000E+00

1. 2426401
1, 108E+01
9, BS0E+00
8.713600
7.633E+00
4.6246+00
5. 701E+00
4. 871E+00
4, 137E+00
3.495E+00
2.942E+00
2.468E+00
2. 06SE+00
1,725E+00
1, 4396+00
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19798401
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3. Z26E+00
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1.280E+02,
4, GA9EHOL
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9, 918E+00
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7. BSSEH00
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5.709E+00
4.878E+00
4, 140E+00
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2. 983E+00
2. 4b9E+00
2. 0668400
1,726E+00
1. 440E+00
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2. S80E+02
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1, 1388402
1,080E+02
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B.S91E+01
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5. 1208+02
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1.099402
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9. PAGE+00
6.896E+00
4, TREHO0
3. Z28E+00
2, 19TE+00
1.482E+00
1, 000E+00

1, 190E#01
9, 952E+00
8. 766E+00
T.GO4E+00
6.6436+00
5. 7126400
4, 878E+00
4. 141E#00
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2.4696+00
2, 066E+00
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6..896E+00
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2.191E+00
1,482E+00
1, 000E+00

1024
1,024E+03
2.065602
5.023E+01
2.026401
1, S48E+01
1, 132E+01
9. FS3EH00
8, 766E+00
7, O4E+00,
6, 6436400
5. 7136400
4,879E+00
4.142E+00
3. 499E+00
2. 944E+00
2.469E+00
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17266900
1. 440E+00
1, 2006+00
1,0006+00


ort! 37 Syinet?
White PM. 27
Flicker PM ‘ 1+

White FM 3 ( > \ 1 2 3
Flicker FM @= "BOT 4

5 @=-yu-1
Mod. oy-(t) wtH #

Random ~ot
Walk FN
Flicker —3L
Walk FM 8

Figure 2. A plot of the relationship between the frequency-domain power-law
spectral-density exponent a and the time-domain two-sample Allan variance
exponent p (a = -p-1, -2 <p < 2 anda2i1 for p = -2). Also shown is the
similar relationship between a and the modified Allan variance with exponent
on r of wp’ (a = -p'-1, -4 <p! < 2). The pointing arrows indicate the mu-
alpha relationship (a vs. w or pw’) for which the particular variance applies.

14


the averaging time r,. If we average groups of the measurements of y(t), then
the dead times between the original measurements are distributed periodically

throughout the new average measurements (see figure 4). Define

MHi-1 _
a= a a Yas (5)

where y, are the raw or original measurements based on dead time T,-r).
Also define the two-sample variance with distributed dead time as

o7(2,M,T,7) = HEL -Yiem)?] (6)
with r = Mry and T = MT,.

We can now define B, as the ratio of the N-sample variance with distributed
dead time to the N-sample variance with dead time accumulated at the end as in

figure 1:

BW Morya) = SMT) )

Although B,(N,M,r,#) is defined for general N, the tables in the Appendix
confine treatment .to the case where N= 2. There is little value in extending
the tables to include general N. Though the variances on the right in eq (7)
depend explicitly on N, T and r, the ratio B,(N,M,r,u) depends on the ratio

xr = T/r, and on uw as developed later in this paper.

In words, B,(2,M,r,#) is the ratio of the expected two-sample variance with
periodically distributed dead time, as shown in figure 4, to the expected two-
sample variance with all the dead time grouped together as shown in figure 1.
Both the numerator and the denominator have the same total averaging time and
dead time, but they are apportioned differently. The product B,(r,p) *
B,(2,M,r,#) is the distributed dead-time variance over the Allan variance for

a particular T, 7, M and p.


6,667E-01
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29

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2.979E-01
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7
6,667E-01
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6. 251E-01
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5, 7896-01
‘5.635E-01
5. 4836-01
5.3301
5. 186E-O1
5.042E-01
4, 900E-O1


BL(N,r,mu) for r= 256

1, 889E+00
2. 156E+00
2, 477E+00
2. 865E+00
3, 333E+00

8

1, 000E+00
1, 000E+00
1,000E+00
9. 999-01
1, 000E+00
1, 001E+00
1, 004E+00
1,016E+00
1. 050E+00
1, 130E+00
1, 2B0E+00
1,.520E+00
1, 8696100
2. 350E+00
3,.003E+00
3, 886E+00
5.084E+00
6, 715E+00
8. 943E+00
1, 200E+01

BILAN, r, mu)

&

1.000E+00

16

1.000€+00
1,000E+00
1,.0006+00
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9. 999E-O1
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1.479640
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2. 7096+00
3. 881E+00
5.6736+00
8.418400
1.264601
1, 9196+01
2. 938E+01
4, 5336401

for p= SIZ

iC)

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6, 003E+00
1,127E+01
2. 1698401
4, 2446401
8, 410E+01
1, 684E+02
3. 40402
6.933602

1,.000E+00
1,001E+00
1,008E+00
1, 039E+00
1, 150E+00
1, 482E+00
2. 34GEH00
4. 394E+00
9, 042E+00
1, 948E+01
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9. 67S5E+01
2.202602
5.066E+02
1,.176E+03
2. 7526103

128

1,.000€+00
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1,043E+00
1. 170€+00
1, 577E+00
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1, 530E+03
4, 080E+03
1,.097E+04

1,674E+00
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7,062E+00
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5.077E+#02
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‘S12

3, 132E+01
1,0226+02
3. 421E+02
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4,.020E+03
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4. F32E04
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1024

1.001E+00
1,010€+00
1, 055E+00
1, 311€+00

INF

o

1, 000€+00
1,.000E+00
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1,000E+00
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1.020E+00
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1, 116E+00
1. 209E+00
1,331E+00
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1,667E+00
1. 889E+00
2, 156E#00
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2.865E+00
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1, 000E+00
1, 000E+00
1. 000E+00
1,.000E+00
1, 000E+00,
1.000E+00
1, 000E+00
1,003E+00
1,.0126+00
1,042E+00

1,118E+00.

1, 268E+00
1,512E+00
1, 864E+00
2,.347E+00
B.001E+00
3, BESEH00
5.084E+00
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8. 9436400
1, 200E+01

1, 000E+00
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5.670E+00
8, 416E+00
1, 2646401
1, 9196401
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4,SREH

1, 000E+00
1, 000E+00
1, 000E+00
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1,001E+00
1,004E+00
1,022E+00
1, 086E+00
1, 2708+00
1, 692E+00
2. 516E+00
3, 992E+00
6, SSBE+00
1, 101E+01
1,B76E+01
3, 236E+01
5. 042E+01
9. 926E+01
1, 760E+02

1,000E+00
1, 000E+00
1, 000E+00
1, 000E+00
1,.000E+00
1, 000E+00
1, 001E+00
1,005E+00
1,026E+00
1,107E+00
1, 353E+00
1,.967E#00
3.296600
5.973E+00
1,1258+01
2, 168E+01
4, 2436401
8, 410E+01
1, 684E+02
3, 40ME+02
6. 933E+02

1, 000E+00
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4, R301
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5.066E+02
1, 176E+03
2. TSEH3

26

1,.000E+00
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1,.000E+00
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1,.000E+00
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1. 000€+00
1, 001E+00
1, 007E+00
1, 0418400
1,261E+00


By definition, white noise has a power spectral density (PSD) that is constant
with Fourier frequency. Since random walk noise is the integral of white
noise, the power spectral density of a random walk varies as 1/f* (where f is
the Fourier frequency) [2]. We encounter noise models whose power spectral
densities are various power laws of their Fourier frequencies. Flicker noise
is very common and is defined as a noise whose power spectral density varies
as 1/£ over a relevant spectral range. If an oscillator’s instantaneous
frequency is well modeled by flicker noise, then its phase would be the
integral of the flicker noise. It would have a PSD which varied as 1/f*.

Noise models whose PSD’s are power laws of the Fourier frequency but not
integer exponents are possible as well but not as common. This paper
considers power-law PSD's of a quantity y(t); y(t) is a continuous sample
function which can be measured at regular intervals. For noises whose PSD's
vary as f* with a < -1 at low frequencies, the conventional sample mean and
variance given in eq (1) do not converge as N gets large [2, 3]. This lack of
convergence renders the sample mean and variance ineffective and often

pisleating in some situations.

Although the sample mean and variance have limitations, other time-domain
statistics can be convergent and quite useful. The quantities that we
consider in this paper depend significantly on the details of the sampling
procedures. Indeed, each sampling scheme has its own bias, and this is the

motivation for the bias functions discussed in this paper.
2. The Allan Variance

Recognizing that for particular types of noise, the conventional sample
variance fails to converge as the number of samples, N, grows, Allan suggested
that we set N = 2 and average many of these two-sample variances to get a
convergent and stable measure of the spread of the quantity in question [3].

This is what has come to be called the Allan variance.



1. Introduction

The sample mean and variance indicate respectively the approximate magnitude
of a quantity and its uncertainty. For many situations a continuous function
of time is sampled, or measured, at fairly regular intervals. Sampling is not
always instantaneous. It takes a finite time and provides an “average
reading." If the underlying process (or noise) is random and uncorrelated in
time, then the fluctuations are said to be "white" noise. In this situation,

the sample mean and variance calculated by the conventional formulas,

QQ)

= X (_-m)?,
1

provide the needed information. The "bar" over the y ‘in eq (1) above denotes
the average over a finite time interval. In time and frequency work, y is
defined as the average fractional (or normalized) frequency deviation from
nominal over an interval r and at some specified measurement time, As in
science generally, the physical model determines the appropriate mathematical
model, For the white noise model, the sample mean and variance are the

mainstays of most analyses.

Although white noise is a common model for many physical processes, more
general noise models are being identified and used. In precise time and
frequency measurement, for example, there are two quantities of great
interest: instantaneous frequency and phase. These two quantities by
definition are exactly related by a differential. (We are NOT considering
Fourier frequencies at this point.) That is, the instantaneous frequency is
the time rate of change of phase. Thus, if we were employing a model of white
frequency-modulation (white FM) noise, then the phase noise is the integral of
the white FM noise, commonly called a Brownian motion or random walk.
Therefore, depending on whether we are currently interested in phase or
frequency, the sample mean and variance may or may not be appropriate.

2



THE BIAS FUNCTION, Bz (r,.)

Figure 3. A three dimensional plot of the bias function B,(r,y), where
x = T/r, and the dead time is T- 7. The "fin" at r = 1 and p» = -2 approaches
zero width as the measurement bandwidth approaches infinity (see appendix ref.

(3]).

15


where o7(N,T,r) is the expected sample variance given in eq (1) and based on N
measurements at intervals T and averaged over a time r and r = T/r. In words,
B,(N,r,u) is the ratio of the expected variance for N measurements to the
expected variance for two samples (everything else held constant). The
variances on the right in eq (3) depend implicitly on the noise type even
though » or @ are not shown as independent variables. The noise-type
parameter, », is shown as an independent variable for all of the bias
functions in this paper, because the values of the ratio of these variances
explicitly depend on yp as will be derived later in the paper. Allan showed
that if N and r are held constant, then the a, » relationship shown in figure
2 is the same; that is, we can still infer the spectral type from the r

dependence using the equation a = -y-1, -2 < yw < 2 [3].
4. The Bias Function B,(r,p)

The bias function B,(r,y) is defined in [1] by the relation,

_ o(2,T,r) _ _o7(2,T,r
Bz (r,n) = TERS a “ie 9 : (4)

In words, B,(r,#) is the ratio of the expected two-sample variance with dead
time to that without dead time (with N = 2 and r the same for both variances).
A plot of the B,(r,u#) function is shown in figure 3. The bias functions B,
and B, represent biases relative to N = 2 rather than infinity; that is, the
ratio of the N sample variance (with or without dead time) to the Allan
variance and the ratio of the two-sample dead-time variance to the Allan

variance respectively.
5. The Bias Function B,(N,M,r,y)

Consider the case where a great many measurements are available with dead time
between each pair of measurements (T, > 1,). The measurements are averaged
over the time interval r,, the spacing between the beginning of one
measurement to the next is T,, and it may not be convenient to retake the
data. We might want to estimate the Allan variance at, say, multiples M of

5


BA(N,7,mu)

8, S33€-01
8, 581-01
8, B66E-O1
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9, 5696-01
1,.000E+00
1,050E+00
1, 107E#00
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1, 247E+00
1, 3336+00
1, 432E+00
1, 54E+00
1.677E+00
1, 827E+00
2. 000E+00
2, 198E+00
2, 426E+00
2, 6B8E+00
2. 9B8E+00
3, 333E+00

7.500E-01
7,827E-01
8, 221E-01
8. 700E-01
9, 2BAE-O1
1, 000E+00
1,.088E+00
1,197E+00.
1.3336+00
1.502600
1,714€+00
1. 982E+00
2,320E+00
2. TSOE+00
3, 2996+00
4, 000E+00
4, 900E+00
6,.059E#00
7, S55E+00
9. AIDE+00
1, 200E+01

BLN, mu)

4

1, 000E+00
DASSE-O1
9,098E-O1
9, 286E~01
9, 599E-01
1,000E+00
1, 048E+00
1. 104E+00
1, 168E+00
1.2436+00
1, 3296400
1, 428E+00
1, 541E+00
1,672E+00
1, 822E+00
1.995E+00
2, 194E+00
2.422600
2. 685E+00
2, 986EH00
3, 533E+00

9, 31E-01
1, 000E+00
1, 085E+00
1, 193E+00
1, 327E+00
1, 495E+00
1, 706E+00
1,972E+00
2, 3096100
2, 738E+00
3,.285E+00
3, 98SE#00
4, 885E#00
6, 045E+00
7. AE
9, 482E+00
1, 200€+01

for r= t

16

7, 083-01
7.A31E-01
7. B64E-01
8. 410E-01
9, 105E-O1
1, 000E+00
1,.117E+00
1,271E+00
1.4766+00
1, 7S4E+00
2. 133E+00
2, 658E+00
3, 391E+00
4, 428E+00
5. 894E+00
8, 000E+00
1, 1046401
1, 5466901
2. 191E+01
3, 1386401
4.533601

for r= 1.01

8.573€-01
9, 160-01
1, 000E+00
1,.1136+00
1,.265E+00
1, 468E400
1.743E+00
2, 1208400
2, 6426400
3.371400
4. 400E+00
5. BOAEHO0
7. 966E#00
1, 1006401
1.542601
2. 187EHOL
3, 15E+01
4.533901

2

6.875E-01
7.226E-01
7.672-01
8.245E-01
8,.997E-01
1, 000E+00
1,137E+00
1,327E#00
1,599E+00
1. 994E+00
2, 5B1E+00
3. 471E+00
4 BAGEH00
7,006E+00,
1, 045E+01
1, 600E+01
2, 506E+01
3, 999E#01
(6.4796+01
1,063E+02
1, 700E+02

cl)

6,771E-01
7122-01
7, S70E-01
8, 1S4E-01
8, 9336-01
1,000E+00
1. 150E#00
1,370E+00
1,700E+00
2..216E+00
3,.048€+00
44328100,
6, 801E+00
1, 096E+01
1.O41E+O1
3..200E+0!
5. 717E+01
1, 0ME+02
1. 938E+02
3, 646E+02
6. 33E+02

1, 000E+00
8.217E-01
8.009E-01
8. 3336-01
8. 99SE-01
1, 000E+00
1, 147E+00
1,362E+00
1, 689E+00
2. 2006100
3,025E+00
4, 400E+00
6, TSAE+00
1,069E+01
1, 80EH01
3, 185E+01
5. 695E+01
1. 041E+02
1, 934E102
3, 649E102
6, 9336102

21

128

6.7196-01
7.068E-01
7,517E-O1
8. 105E-01
8. 897-01
1, 000E+00.
4, 160E+00
1, 401E+00
1,7826+00
2.418E+00
3.528E+00
5.555E+00
9, 407E+00
1, 698E+01
3. 2306101
6, A00E+O1
1.308E+02
2.736402
5. 833E+02
1.260+03
2. TS2EH3

128

1, 000E+00
8, 181-01
7. 9636-01
8, 287E-01
8, 960E-01
1, 000E+00
1, 155E+00
1, 393E+00
1, 770E+00
2, 400E+00
3, 500E+00,
5.512610
9. 3K0EH00
1, 687E+01
32126401
6,.3696+01
1.9036+02
2, 730E+02
5.822602
1, 2596403
2.752E#03

6. 6996-01
7.0426-01
7.4906-01
8,0796-01
8,877E-01
1,.000€+00
1,1656#00
1.422690
1.847E#00
2,5996+00
4,.0166+00
6, 8586400
1, 267E+01
2, 614E+01
5, 6526401
1, 2806+02
2, 999102
7.212
1, 7626403
4.372803
1. 097E#04

26

1.000E+00
8, 1626-01
7, 9RE-OL
8, 262E-01
8, 940E-O1
1, 000E+00
1. 161E+00
1,414E+00
1, 835E+00
2.579E+00
3, 984E+00
6.804E+00
1.277E+01
2.597E+01
5.619601
1.2748+02
2, 987E+02
7. 180E+02
1, 738E+03
4.367E+03
1.097E104

312

6, 680E-01
7,028E-01
7.ATGE-01
8.065E-01
8, 866E-01
1,000E+00
1, 1696100
1, 438E+00
1,.898E+00
2, 7S9E+00
4, 509E+00
8.3469E+00
1, 7448401
4. 005E+01
9. B72E+01
2.580E+02
6,881E+02
1.897E+03
5. 331E+03,
1, S19E+04
4, S78E+04

8.249E-01
8.929E-01
1. 000E+00
1,165E+00
1.429E+00
1, 885E+00
2, 737E#00
4 ATZEHOO
8. 296E+00
1, 731E01
3. 9786401
9.814601
2, SA7E02
6..854E+02
1,892E+03
S.3216+03
1,518€+04
4. STRE+04

1024

6.673E-01
7.021E-01
7, 4686-01
8,058E-01
8, 850E-01
1.000€+00
1, 171E+00
1, M48E+00
1, 938E+00
2. 900E+00
5.005E+00
1, 010E+01
2, SS0E+01
. LI4E+01
1. 722EH02
5.120802
1,580E+03
5,.003E+03.
1,615E+04
5. 2B6E+04
1, 749E+05

8. 2426-01
8. 9246-01
1,000E+00
1, 167E+00
1, 40E+00
1, 924E+00
2.877E+00
4 6ZEH00
1,002E+01
2.326401
6.073E+)1
1, 7128402
5.095E102
15736403
4, 9BBE+03
1. 611E+04
5. 2B0E+04
1, 7A9EHOS:

INF

6. 667E-01
7.014E-01
7.AG1E-01
B.051E-01
8, 8546-01
1.000E+00
1, 175E+00
14706400
2, 065E+00
3.843100

INF
1..000E+00
8, 143E-01
7. 91%E-O1
8. 235E-01
8, 917E-01
1, 000E+00
1,1708+00
1. 461E+00
2,.050E+00
3.829800



UNITED STATES

DEPARTMENT OF COMMERCE
NATIONAL INSTITUTE OF STANDARDS
AND TECHNOLOGY

(nN

REFERENCE

100
-U5753

#1318

1990

| ALLIO3 396078

xa NIST Technical Note 1318

PUBLICATIONS

Variances Based on Data With
Dead Time Between the
Measurements

James A. Barnes
David W. Allan



B3(r,0) = Ste, (12)

as given in [1], and

M-1
2M + MeF(Mr) - Y (M-n)(2F(nr)-F((M+n)r) -F((M-n)r)]
n=1

By (2,M,r,p) = 2) (Fiz) + 2] ,

(13)

as indicated in the appendix.

For » = 0, eqs (11), (12), and (13) are the indeterminate form 0/0 and must be
evaluated by l’Hépital’s rule. Special attention must also be given when
expressions of the form 0° arise. We verified a random sampling of the table
entries using noise simulation and Monte Carlo techniques. No errors were
detected. The results in this paper differ some from those in [8], which
suggests that there may be some mistakes. Tables for the three bias functions
are listed at the end of the paper (note that the computer print-out did not
have a symbol for Greek mu = p).

7. Examples of the Use of the Bias Functions

The spectral type, that is, the value of uw, may be inferred by varying r, the
sample time. However, another useful way of determining the value of y is by
using B,(N,r,p) as follows: calculate an estimate of o% (N,T,r) and o} (2,27)

and hence B,(N,r,u); then use the tables to infer the value of p.

Suppose one has an experimental value for oe (Ny, Ty, 71) and its spectral
type is known, that is, » is known. Suppose also that one wishes to know the
variance at some other set of measurement parameters, N,, Tz, r2. An unbiased

estimate of o (N,,Tz,72) may be calculated by the equation:

— (t2)" (Bi (No. Tm) Bo (ro w)
03 (Na oTa ta) = (=) (Bs Giese Ba (eat } oF Ryn Tysta) (14)

where r, = T,/7, and r, = T,/r2.



Some useful asymptotic forms of B, can be found. In the case of large M and
M> xr, we may write that

B= +2, 1<ns2,
(8)
4 gn(2 _
By ~ aac) +3 we 0

One simple and important conclusion from these two equations is that for the
cases of flicker FM noise and random-walk FM noise, the r* dependence for
large r is the same whether or not there is periodically distributed dead
time, The values of the variances differ only by a constant, and in the
latter case the constant is 1. This conclusion is also true for white FM

noise, and in this case the constant is also 1.

In the cases r » 1 and -2 < » < -1, we may write for the asymptotic behavior
of B,

B, ~ M*, a= -p-1, (9)

as was determined empirically. In this region of power-law spectrum the B,

function has an M* dependence for an f* spectrum.

6. The Bias Functions

The bias functions can be written fairly simply by first defining the
function,

F(A) = 2a"*2 - (A + 1)#t2 - [(A - 1)]4t?, (10)

The bias functions become

N-1 N-n
1+) + F(nr)
Bea) = —S BG (ty



112,070) for r= 40%

% 2 “ ia rd 312 104

fe\ rt 2 4

2} 1.000800 2.000600 4, 000E+00 1.600401 3. 2006+01 4,400E+01 1. 200E+02 2.56002 5,120E+02 1.026409
“1,0 © 1.000800 1. 7816+00 3, 0318+00 9.190E+00 1.000801 2.700E901 4.050E+01 8.4501 1.470602 2.5806102
1d 1 10006900 1.516E+00 2.297800 ‘S.278E+00 8.000E+00 1,2198+01 1.058E+01 2. 7066101 4.2226+01 6, 400E01
1.4 4 1,0008#00 1, 3206900 1.7416+00 ‘S.0B1E+00 4.000800 5.Z7NE+00 6. 904E+00 9.190600 1.213601 1. 600E+01 ©
1.2 1 1,000EH00 1. 149E*00 1.919600 1.516600 1, 7416400 2.000E+00 2,297E+00 2.6996+00 3.031E+00 3.482E+00 4.000E+00
“11 1.000800 1,0008+00 1.000E+00 1.000E*00 1.0008+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1. 000E*00
eB 1 1,000E+00 6, 704E-O1 7. 5B1E-01 6,601E-01 S.788E-O1 S.005E-01 4,50E-01 3.7ME-O1 3.206E-01 2.879E-01 2, 5008-01
sb | 1.000900 7.507E-O1 5. 7608-01 4, 576E-01 3.327E-01 2.550E-01 1,991E-O1 1.47ME-O1 1.129E-01 8.663E-02 6,6798-02
“14 t 1.000800 6.644E-Ol 4. 4408-01 2, 907E-O1 2,000E-01 1.97E-O1 9.033802 7.087E-~12 5279-02 4.085E~-02 3, 297E-02
v2 10008400 S.950E~01 2.649E-01 2, 300E~O1 1.578E~01 1.100E-01 9, 0078-02 .7.9ME-02 6.777E-02 4.310E-02 6,0428-02
© —-t L000E+00 5, 65E-01 9.496E-01 2.443E-01 1.9Z0E-O1 1.G67E-01 1, 599E-O1- 1.476801 1.440801 1.4286-01 1.4206-01
«2 1 1,0008+00 S.733E-O1 3.943E-01 3.109E-O1 2.670E-01 2.734E-01 2,675E-01 2,650E-O1 2.497E-01 2.6346-01 2.6328-01
+4 10008900 6,075E-01 4.406E-01 4. 178E-O1 3.994E-01 3.92KE-O1 9.902E-01 3.092E-01 3.009E-01 3.087E-01 3.087E-01
26 1 1.000800 6. 5308-01 S.400E-01 S.190E-01 S.024E-01 4.991E-01 4.900E-O1 4.977E-~01 4. 976E-01 4.975E-01 4. 975E-01
+B + 1,0008+00 7.024E-01 4.199E-01 S.97IE-Ol $.907E-01 S.OPIE-O1 5.087E-01 S.Q85E-0l S.G83E-01 5.985E-01 5, O89E-01
1 t  1,000E+00 7.500801 6.876E-01 6, 719E-O1 6,480E-01 6.470E-01 6,666E-01 6.667E-01 6, 667-01 6.667E-01 6.667E-01

1.2.4  1,000E*00 7979-01 7.S14E-01 7.40SE-01 7,300E-Ol 7.374E-01 7,372-01 7.372-01 7.372E-01 7.378-01 7.372E-01
1.41 1.0008#00 8.443E-01 8. 131E-Ol 8,0S7E-O1 9,080E-01 9,034E-01 8.030E-01 8.094E-01 8,0G4E-01 8.034E-01 8.034E-01
1.61  1,000E#00 8.9S8E-01 8. 743E-O01 8.696E-01 9.689E-01 6,482E~01 8.682E-01 8.481E-01 S.4G1E-01 8,681E-01 8,681E-01
1,01 1.000600 9.469E-01 9.3628-01 9.539E-O1 9.339E-01 9.302E-01 9.532E-01 9.3328-01 9. 3028-01 9.332E-01 9.3326-01
2 § — 1,0008+00 1,000E+00 1.000E*00 1,000E+00 1. 000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

09(2,N,rjeu) For r= 9192

4 8

4.000E+00 8, 000E+00
1.000E+00 1. 741E+00 3.031E+00 5, 278E+00 9, 190E+00 1,600E+01 2, 7B6E+01 4.8S50E+01 8.445E+0] 1.470E+02 2, 560E+02
1.000E+00 1.516600 2. 297E+00 3.482E+00 5.2786+00 8.000E+00 1.213E+01 1.838E+01 2. 7B4E+01 4.222E+01 6. 400EH0I
1.000E+00 1,320E+00 1, 741E400 2.277E+00 3. 031E*00 4.000E+00 5.278E+00 6.964E+00 9.190E+00 1.213640! 1.L00E+01
1,000E+00 1.1496+00 1. 320E+00 1, 516E+00 1. 741E+00 2. 000E+00 2.297E+00 2.639600 3.031E+00 3.482E+00 4,000E+00
1,000E+00 1,.000E+00 1.000E+00 1.000E+00 1,000E#00 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00
1,000E+00 8, 7066-01 7.SQ0E~01 6.579E-01 5.744E~O1 5.003E-01 4, 356E-01 3,793E-01 9.9036-01 2.8766-01 2,505E-01
1,000E+00 7, 5048-01 5. 7S4E-01 4, 3686-01 9.317E-01 2,522E-O1 1,9186-01 1.41E-O1 1, 115E-01 8.5236-02 6,533E-02
1,000E+00 4.634E-01 4. 418E-01 2,959E-O1 1.997E-01 1.369E-O1 9.444E-02 6,685E-02 4.9646-02 3.6436-02 2,871E-02
1,000E+00 5.920E-01 3.594E-01 2.266E-01 1.S06E-O1 1.071E-O1 8.211E-02 6,778E-02 5.955E-02 5,483E-02 5,212E-02
1.000E+00 S.5P1E-O1 3,429E-Ol 2,365E-01 1,639E-01 1. 5786-01 1.448E-01 1.383E-O1 1,351E-01 1.535E-01 1.327E-01
1,000E+00 5. 7046-01 3.899E-01 3.139E-01 2.B17E-O1 2.679E-01 2.620E-01 2.594E-01 2.583E-01 2.578€-01 2.576E-01
1,000E+00 6, 063E-01 4.669E-01 4, 159E-O1 3.974E-01 3.906E-O1 3.081E-01 3.872E-01 3,868E-01 3.867E-01 3,867E-01
1.000E+00 6,S46E-O1 S.463E-01 S.124E-O1 5.018E-01 4.965E-O1 4.975E-O1 4.971E-O1 4.970E-01 4.970E-01 4.970E-01
1.000E+00 7.023E-01 6, 198E-O1 5.970E-01 5S. 907E-01 5.890E-01 5.885E-01 5. 9B4E-01 5.904E-01 5.884E-01 5.8B4E-01
1.000100 7,500E~01 4,875E~O1 6.719E-O1 6.680E-01 6.670E-01 6,668E-01 6,667E-01 b.647E-01 6.667E-01 6.667E~01
1,0006+00 7.979E-01 7.S14E-01 7.405E~O1 7.3Q0E-01 7.374E-01 7,372E-01 7.372E-01 7.372-01 7.372-01 7.372E-01
1,000E+00 8,469E-01 8. 131E-O1 8.0S7E-01 8.040E-01 8.036E-01 8.035E-01 8.034E-01 8.004E-01 8,034E-01 8,034E-01
1.000E+00 8, 95GE-01 8, 743-01 8.A96E-01 8.685E-01 8.682E-01 6.682E-01 8.481E-01 8.481E-01 8.681E-01 8.681E-01
1,000E+00 9,4469E-01 9.362E-O1 9.309E-01 9. 353E-01 9,332E-01 9,332E-01 9.332E-01 9.532E-01 9.332E-01 9.332E-01
1,000E+00 1.000E#00 1.000E*00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00

40

US GOVERNMENT PRINTING OFFICE 1990 731-011/09185 Rogtan &


t
:
1
t
:
'
'
:
!
4

BL(N,reu) for r= INFINITY

1, 000E+00
1,000€+00
1, 000E+00
1.000E+00
1,.000E+00
1.000E+00
1, 000E+00
1,.0006+00
1,.000E+00
1, 000E+00
1. 000E+00
1,091E+00
1, 1986400
1.327E#00
1, 4826+00
1. 667E+00
1.8896+00
2. 156E+00
2, 477E+00
2.865E+00
3. 333E+00

1,.000€+00
1, 0006+00
1. 000E#00
1,000E+00
1,000E+00
1.000€+00
1.0006+00
1,0006+00
1,000E+00
1.000E+00
1.000E+00
1, 210E+00
1,487E+00
1. B54E+00
2. ASE+00
3.000E+00
3, BBSEH00
5.0B4E+00
6. 714E+00
8. 9436100
1, 200E+01

1, 000E+00
1,000E+00
1,.000€+00
1, 000E+00
1.000E+00
1, 000E+00
1, 000E+00
1,000E+00
1, 000E+00
1,000E+00
1, 000E+00
1.360E+00
1, BROE+00
2. 680E+00
3,867E+00
5.657E+00
8, A15E+00
1. 2646401
1 919EHO1
2. 938E+01
4, S33E+01

32

1, 000E+00
1.000€+00
1.000E+00
1..000E+00
1.000E#00
1,000E+00.
1,.0008+00
1.000€+00
1, 000E+00
1,.000E+00
1, 000E+00
1, 541600
2.441E+00
3. 958E+00
6, 343E+00
1, 1006103
1,.875E+01
3. 236E+01
5. 6426401
9. 926E#01
1, 760E+02

S12

1,000E+00
1.000€+00
1,000E+00
1,0006+00
1.000E+00
1.000E+00
1,000E+00
1,000E+00
1,0006+00
1, 000E+00
1,000E+06
1, 757E+00
3. 184E#00
5. 916E+00
1.123601
2.167401
4, 2036401
8, 409E101
1, 6846402
3, 404402
6, 933402

28

9, 6726401
2. 2026402
5. 066E+02
1, 176E#03
2.752603

HHH

1,000E+00
1,0006+00
1,000€+00
1. 000#00
1, 0006+00
1.0006+00
1,.0006+00
3.033600
9, SS3E+00
3.080E+01
1.017E#02
3, 417E+02
1. 165E+03
4.020903
1, 4026404
4. 9326404
1, 7A9E#05

1,000E+00
1,000E+00
1.000E+00
1,0006+00
1,.000€+00
1.0006+00
1.000+00
1,000€+00
1.000600
1,000E+00



Bi(Neymu) for r= 1028

4 ® 6 2 r) 128 TM% 12 10
000100 1.000600 1.000400 1.000640 1. 0006+00 1.000600 1.0006#00 1.000400 1,.000€+00
000E#00 1.000600 1,0006+00 1.000600 1.000400 1.000+00 1.000900 1.000E+00 1,.000€+00
0006400 1,0006+00  1,0006+00 1.000600 1.000640 1,0006+00 1,0006+00 1,0006+00  1.000€+00
0006400 1.000600 1,0006+00 1.000600 1. 0006+00 1.000+00 1.000640 1,0006#00 1,.0006+00
0006400 1.000600 1,0006+00 1.000600 1.000400 1,0006+00 1.0006+00 1.0006+00  1.000€+00
0006400 1.000600 1,0006+00 1.000800 1..000€+00 1.00000 1.000610 1.0006+00 1, 000€+00
000600 1,0006+00 1.000600 1,.000E+00 1. 0006+00 1,0006+00 1.000600 1.0006+00 1.006400
OO1E100  1,0028+00 1,0028+00 1.003640) 1. 0036+00 1,0046¥00 1,0086#00 1.0086#00 1, 0046400
004500 1.009600 1.019600 1.017E#00 1.020640 1.024600 1.026100 1.02700 1, 031€+00
0176400 1.089590 1.054600  1.0736+00 1.090E400 1120600 1,1325+00 1.144E#00 1.206400
+049E+00 1.108100 1.175600 1.248600 1.324640 1. 1.469630 1.584600 1.645E+00

1126400 1,2596400 1.446100 1.688400 1.946400 2.285EH00 2.607600 3.025500 3, S07E+00

+208E+00 1,505EH00 1, 9246100 24978400 3.268100 4.297600 5.649E100 7.472600 9. 8426100

306100 1,9606#00 2.6906+00 9,980E100 5, F53E100 8,943EH00 1. SSHENOL 2.049601 3, 1026+01

AB2E+00 2,94SE+00 3,672E100 6.552100 1, 128EH01 1.926101 TLMVENOL S.SSEHOL 1,0186+02

1.667800 3.001E+00 5.668E+00 1.100601 2.167E+01 4.20101 8.569601 1, 7116402 3,.4186+02
1,889E+00 3,885E+00 8.414100 1.876601 4.243901 9.672601 2.213602 S.O7SEH02 1, 1658403
2.158400 S.OBAE¥00 1.260641 3.208641 8.409601 2.202602 S.7BIEHO2 1.5Z5EH09 4.020609
2.ATTE*OO 6.714E+00 1.919101 5,UA2E901 1,48HEH02  S.06HEH02  1.SWE03 4,6286+03 1.402804
2,065E+00 9.943EHOO 2. 9986101 9.926501 3.4OHEHO2 1. 17EH03 4, 0806103 1.418E+08 4, 9328108
3.339E+00 1.200601 4.53201 1, 7806102 6. 950EH02 2, 7S2EH03 1.0P7EHOA 437REHO4 1, 7496005

' BI(N,rjau) for r= 2088
mite 4 a 16 2 “ 128 9 512 1028 INF

“2 1.000E+00 1.000100 1. 000E+00 1.000600 1,000E+00 1.000600 1, 000600 1,000E#00 1, 000E+00 1, 000E+00

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“1.6 1 1.000E#00  1,000E+00 1,000E+00 1.000600 1, 000E+00 1.000600 1.000600 1.000E+00 1, 000E+00 1. 000E+00

“1.4 + 1,000E#00 — 1.000E+00 1.000E+00 1.000EH00 1.000600 1.000EH00 1, 000E#00  1,000E+00 1.000E+00 1, 0O0E+00

“1,2 + 1,000E+00 —1,000E100 1. 000E+00  1.000E+00 1.000600 1.000E+00 1. 000E+00 1.000E#00 1.00000 1, 000€+00
:

1.000E+00 1.000600 1.000 +00 1.000E+00 1,000E+00 1.000600 1.000E+00 1,.000E#00 1.000600 1. 000E+00
~1B = 1,000E+00  1,000E+00  1.000E#00 1. 000E+00 1. 000E+00 1.000600 1,000E100 1.000E#00 1.000600 1. 000E+00
1,001E+00 1, 001E#00  1.002E400 1,002E#00 1.002600 1.026100 1,003E+00 1.003600 1.003E+00 1, 003E+00
1,003E+00 1.007E+00  1.0106+00 1.012E#00 1.015E#00 1.017E+00 1. 0186100 1.019E+00 1.020600 1. 029E+00
1,0146+00 1.030600 1, 046E+00  1,061E+00  1.076E+00 1.089900 1, 101E+00 1, 1126100 1.122600 1, 184EH00.
1,OASE+O0 1, 100E#00 1.162400 1.229E+00 1. 2996400 1.372600 1, 4466400 1,S21E#00 1, 596E100.
1,208E+00 1, Z51E+00 1. 491E#00 1.648E+00 1. 906E+00 92. 2096100 2. SK0EH00 2. 9606100 3, 434E100
1, 2046400 1,S01E+00 1. P16E+00  2,483E+00 3. 247E+00 4, 2666400 5.620600 7,.412E+00 9. 780E+00
+6 L.329E100 1, 858E100 -2,688E+00 3,972E+00 5. 941E+00 B.942E+00 1.351EH01 = 2,044E+01 3, 09SEHL
1.482600 2,345E+00 9, 870E#00 6, SHBEH0O 1.120601 1. 941E101 3. 367EH01  S,E51E+01 1, 0186402
1.667E+00 9, 000E+00 5.647E400 1, 100E+01 2. 167E+01 4. 301E+01 8,568E+01 1.710602 3. 417E#02
1.889E+00 3.885E+00 GB. AISEH00 «1. 875E+01 «4. 2496401 9.672601 «2.219802 S.075E+02 1. 165E+03
2.156E+00  S.0B4E+00 1, 2646401 3. 236EH1 «8.409401 2.202802 5. 7BVEH02 1, S25EH03 4,.0206#03
2,ATTE+OO 6. 714E#00 1. 919EH01 = S.O42EHOL 1. GBAEHO2 « S.06KE102 1, S30E03 4.628603 1, OZEH04
2,865E+00 8.943E+00 2. 93RE+01 9. 926EH01  3.404EH02 «1.176E103 4, 080E103 1. 418EH04 4, 932104
3.333400 1, 200E+01 4, S33EH01 1, 7804026. 933602 2. 7523 1.097EH04 4, 378EH04 1. 749EH05

e

27



More specifically let us consider a sample function of time as indicated in
figure 1. A measurement consists of averaging y(t) over the interval r. The
next measurement begins at a time T after the beginning of the previous
measurement interval. There is no logical reason why T must be as large as r
or larger--if T <1, then the second measurement begins before the first is
completed, which is unusual but possible. When T= 1, there is no dead time

between measurements.

The accepted definition of the Allan variance is the expected value of a two-
sample variance with no dead time between successive measurements, In

symbols, the Allan variance is given by
02 (r) = HEL (Yn+i1-Yn)? 15 (2)

where there is no dead time between the two sample averages for the Allan

variance and the E[-] denotes the expectation operator.
3. The Bias Function B, (N,r,p)

Define N to be the number of sample averages of y(t) used in eq (1) to
estimate a sample variance (N = 2 for ‘an Allan variance). Also define r to be
the ratio of T to r (r = 1 when there is no dead time between measurements).
The parameter » is related to the exponent of the power law of the PSD of the
process y(t), If a is the exponent in the power-law spectrum for y(t), then
the Allan variance varies as r raised to the 4 power, where a and p are
related as shown in figure 2 [2-4]. We can use estimates of yw to infer a, the
spectral type. The ambiguity in a for p = -2 has been resolved by using a
modified of(r) [5-7].

Often data cannot be taken without dead time between sample averages, and it
is useful to consider other than two-sample variances. We will define the

bias function B,(N,r,#) by the ratio,

2
Byrn) = GOume | @)

a



S(M) = 4-M-R(O) - 4-M-R(r) ~ 2-M-G(MT)

+2 JY (M-n)[2-G(nT) - G((M+n)T) - G((M-n)T)].

Following the work cf Barnes and Allan [2,3], we can define the function U(r)

by the relation,

U(r) = 2-R(O) - 2-R(r),
and also define

F(nr) = G(nT)/U(r),

where r = T/r. The function U(r) for power-law power spectral densities has

the form,

ret?

U(r) = ,
4-22

which yields
F(nr) = 2-(nr)#*2 - (nr+1)#*? -[nr-1]4t?

Finally, the working relation can be written as

M-1
2-M+M-F(Mr) -Y (M-n) [2-F(nr) - F((M#n)r) - F((M-n)r)]
n=1

By (2,M,r,u) =
(2 + F(r)] - met?

18


Lb ft
“Lat
120

83(2,,r,au) for rs 4

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35

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2

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83(2,M,r,au)

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36

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1,
Zz.

Appendix..

Tables of Bias Functions

The Bias: Functor! By (Wp F 5H). wee ews ea ee ce Suereree vieeie WEIS Alege HHT G on oe & 4
The: Bias! PUASEL ON, Boe Wie sa sce cia 0.4 ifn wld Serre ws SATA TT HE AM ee ese goers 5
The: Bias ‘Function By. (NM 2jH)\s 2. je me weno oie alee ing Be et ewe wn eves a we 5
The Bias Functions............... dca alee 070 B08 ON eae ee ween ne eee eee news 7
Examples

Come kaart oases 25 5% 20. oseceigse wo Wis BS WE OS FS Yevwry mye ons HIN HONS BETS Teaneve care 2 9
RRERLOBEEE oie oi eee ore ws ee aS 0g TR ne emis ew DWE OE LEER cere ss oN Ba 10

iii



NIST-1144 U.S. DEPARTMENT OF COMMERCE |* PUBLICATION OR REPORT NUMBER
(REV. 3-88) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY NIST/TN-1318
2 PERFORMING ORGANIZATION REPORT NUMBER
BIBLIOGRAPHIC DATA SHEET —
March 1990

a, TITLE AND SUBTITLE
Variances Based on Data With Dead Time Between the Measurements

AUTHOR(S)
James A. Barnes and David W. Allan

6. PERFORMING ORGANIZATION (|F JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS) 7, CONTRACT/GRANT NUMBER
U.S. DEPARTMENT OF COMMERCE
NATIONAL merrrure OF STANDARDS AND TECHNOLOGY aT EEE paaaGE COED

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

10. SUPPLEMENTARY NOTES:

]_DocuMENT DESCRIBES A COMPUTER PROGRAM; SF-185, FIPS SOFTWARE SUMMARY, IS ATTACHED.

7. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBLIOGRAPHY OR
UTERATURE SURVEY, MENTION IT HERE)

The accepted definition of frequency stability in the time domain is the two-sample variance (or Allan
variance. It is based on the measurement of average frequencies over adjacent time intervals, with
no "dead time" between the intervals. The primary advantages of the Allan variance are that (1) it is
convergent for many encountered noise models for which the conventional variance is divergent; (2) it
can distinguish between many important and different spectral noise types; (3) the two-sample approach
relates to many practical implementations, for example, the rms change of an oscillator's frequency
from one period to the next; and (4) Allan variances can be easily estimated at integer multiples of
the sample interval.

In 1974 a table of bias functions which related variance estimates with various confiqurations of
number of samples and dead time to the two-sample (or Allan) variance was published [1]. The tables
were based on noises with pure power-law spectral densities.

Often situations occur which unavoidably have distributed dead time between measurements, but still
the conventional variances are not convergent. Some of these applications are outside of the time and
frequency field. Also, the dead times are often distributed throughout a given average, and this
distributed dead time is not treated in the 1974 tables.

This paper reviews the bias functions 8,(N,r,u), and B;(r4u) and introduces a new bias function, B,
(2,r,u), to handle the commonly occurring cases of the effect of distributed dead time on the computed
variances. “Some convenient and easy to interpret asymptotic limits are reported. A set of tables for
the bias functions are included in the Appendix of this paper.

12. KEY WORDS (6 TO 12 ENTRIES: ALPHABETICAL ORDER: CAPITALIZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)
Allan variance; bias functions; data sampling and dead time; dead time between the
measurement; definition of frequency stability; distributed dead time; two-sample

variance
13. AVAILABILITY 14. NUMBER OF PRINTED PAGES
UNUMITED 48

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

15. PRICE
ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,
XK | WASHINGTON, DC 20402,

ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

ELECTRONIC FORM


(9]

Stability Using Non-zero Dead-Time Counter Techniques," IEEE Trans. on
Instrum. & Meas., IM-34, 144-151 (1985).

D.W. Allan, "Should the Classical Variance Be Used as a Basic Measure in
Standards Metrology?" IEEE Trans. on Instrumentation and Measurement,
IM-36, 646-654, 1987.

11


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13


National Institute of Standards and Technology Technical Note 1318
Natl. Inst. Stand. Technol., Tech. Note 1318, 48 pages (Mar. 1990)
CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE
WASHINGTON: 1990

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325



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8.000E-01
8, 3646-01
8. 744-01
9.142E-O1
9. SHOE-O1
1,.000€+00_

2.560E+02
4, 8086401
1, 034E+01
2. 887E+00
1, 336E+00
1. 000E+00
9. 251-01
9. 096E-O1
9.090E-O1
9. 1298-01
9, 187E-O1
9, 2336-01
9. 326E-O1
9, 4036-01
9, 483E-01
9. SOSE-O1
9, CAIE-01
9, 735E-01
9. 822E-01
9. 911E-O1
1. 000E+00

5. 922E-01
‘5. 917E-01
6, 1236-01
6, 3936-01
6,687E-O1
6. 996E-01
7,318€-01
7, 652-01
8, 000E-O1
8, 363E-01
8. 744E-01
9. 142-01
9, 560E-01
1,.000€+00

5.120E+02
8, 311E+OL
1, 5196901
3.512E+00
1, 398E+00
1,.000E+00
9. 228-01
9. OBSE-01
9.087E-O1
9, 128E-O1
9, 1B6E-O1
9. ZSHE-01
9. 326E-01
9. M03E-01
9. 483E-01
9, S65E-O1
9. GASE-C,

9, 5E-01
9822-01
9. P11E-O1
1.000E+00

3.3198+01
8. 6ME+O0
2. 507E+00
1.000E+00
6, 483E-01
5.828€-01
5.881E-01
6, 1106-01
6, 369E-01
6, 686E-O1
6.996E-O1
7. 318-01
7.652E-01
8. 000E-01
8. 363-01
8. 7ME-01
9142-01
9. 360-01
1.000E+00

1, 024€+03
1, MIEHO2
2. 2548401
4, 336E+00
1,470E+00
1. 000E+00
9. 2028-01
9.077E-01
9. 085E-O1
9. 128E-01
9, 1BGE-01
9, 2536-01
9. 326E-01
9.403E-01
9, ABSE-O1
9, 565E-01
©. GAIE-O1
9. 735E-O1
9, 822E-01
9. SLIE-O1
1, 000E +00

1.024603
2. 267EH02
5. 010E+01
1. 126E+01
2.810€+00
1, 000E+00
6. 3255-01
5.757E-01
5.857E-01
6, 1036-01
6, 387E-01
6, 686E-O1
6. 990E-01
7.318E-01
7.0S2E-01
8.000E~01
8.363E-01
8. 7ME-O1
9. 142E-01
9. S60E-01
1,000E+00


BLNr,au) for r= 1d

2 1,0006+00  1,0006+001,0006+001,0006+00 1.000E+001,000€+00 1, 0006+00 1. 0006+00 1, 0006+00 1. 0006+00
“1.8 1 9.S27E-O1 9.243E-01 9.08201 8.994E-01 8.948E-01 8,924E-O1 8.911E-01 8.905E-01  8,902E-01 8, 898E-01
“1,6 1 Q,AISE-OL 9,048E-01 8,831E-01 8,709E-O1 8.642E-01 8,407E-01 8, 588E-01 8.578E-01 8,574E-01 8. 549E-01
<1.4 9,A93E-01 9.1S7E-OL 8.948E-O1 8.825E-01  8,755E-O1 8.717E-O1 8.69601 8.485E-01 8.480E-01 8.474E-01
“1.2 1 9,696E-O1 9,482E~O1 9.941E-01 9.254E-01 9,201E-O1 9. 17IE-OL 9.1SHE-01 9.145E~O1 9. 1406-01 9. 1348-01
+11 1,0006+00 1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1, 000E+00
1B t 1.080E+00 1.072400 1.096E+00 1, 1146400 1.126E#00 1. 194E#00 1.139400 1.142400 11446400 1. 147E+00
-6 + 1,088E+00 1, 168E#00 1.230400 1, 2BIE#00 1.3196#00 1.348640 1.366400 1.379640 1.389E#00 1. 408E+00
4 tL IATE#OO 1, 288E+001.416E+00 1.528540 1. 617E+00 1.692400 1. 7SIE#00 1. 797E+00 1, 832E+00 1. 948€+00
2 1 1217EHOO 1.4448+001.671E+001.889E+00  2.071E+00 2.275400 2. 440E+00 2, 586E+00 2. 7148400 3. 5938+00
Ot 1,298E+00  1.643E+00  2,024E+00 24356400 2.8636+00  3.30HE#00 3, 752E+00 4.205E+00 4. 661EH00

-2 1 1.393640 1,898E+00  2,521E+00  3.272E+00 4, 160E#00 1996100 6, 404E#00 7. 797E+00 9. 4026400

<4 1.504600 2.223600 3.219600 4.575E+00 6.398400 8.828400 1. 2056401 1.6328401 2, 197EHOL

+6 + 1,6336#00  2,040E+00 4, 2195+00 6, 699E+00 1.035E+01 1. G07EH01  2.4496401 3. 771EH01 5. 7SHEHOL

+E 1,7836+00 3,177E+00 5.637EH00 9,95KEH00 1. 7SOEH01 3, 068101 S.S65EH01 9, 34KEH01 1. 6948402

1 1, 957E+00  3.8706+00 7.6966+00 1.S35E+01 3. 085E+01 6, 1288401 1.225602 2.449E02 4.898EH02

1.2 1 2,1586+00 4, 7686400 L.O49EH01 2,4226#01  5,S21EHO1 —1,2636402 2.894402 6, 640802 1,524E+03

1.4  2,391E+00 5.929400 1,508E+01 3.897EH01 1.016502 2,666E+02 7.011E#02 1.847E403 4,.8706+03

1.6 +  2,660E+00 7,443E+00 2, 1S4E401 6, 366EH01 1.904802 5, 79061021. 731E403 5.237603 1. SB6E+04

1.8 1 2,9726+00 9.418400 3,111EHO1 1.053602 3.414EH02 1.249603 4.330603 1, 50KE404 5. 2386404

2 + 3,33E400 1. 2006401 4,5336+01 1. 7606402 6.9396402 2.752403 1.097404 4, 378E+04 17496405

BLIN,r,mu) for r= 2
mi We 4 8 128 256 512 1024 INF

1,000E+00 1, 000E+00 1, 000E+00 1.000E+00 1. 000E+00 1.000600 1.000600 1.000E+00 1.000001. 000E+00

2
“1.8 9.901E-01 9, 836E~01 9, 796E-01 9. 774E-01 9. 761E-01 9. 7S5E-01 9. 752E-01 9. TS0E-01 9. 7A9E-01 «9, 748E-01
“hb 9,B45E-01 9. 737E-01 9.669E-01 9. 630-01 9. 607E-01 9.S9SE-01 9. S89E-01 9. 584E-01 9.SR4E-01 9. 582E-01

9.827E-01 9, 719E-01 9. 641E-01 F.S9BE-01 9, 565E-01 9. SS0E-01 9. SHIE-O1 9. SG7E-01 9. SHME-01 9, 532E-01
9,886E-01 9. 799E-01 9. 738E-01 9.699E-01 9.675E-01 oP. Gb1E-01 9. 659-01 9. O48E-01 9. G46E-01 9, ASE-01
1.000E+00 1. 000E+00 1. 000E+00 1. 000E+001.000E+00 1.000600 1.000E+00 1.000E+00  1,000E+00 1, 0O0E+00
1.019E+00 — 1.035E+00 1. 048E+00 1.058E+00 L.064E100 1.069600 1.072600 1.074810 1.075E+00—1.077E+00
1,086E+00  1,090E+00 1,126E100 1. ISSEHO0 1. 178EH00 1. 195E+00 1.207600 1.215E400 1, 221E+00 1. 2336100
1,084E+00 1, 168E+00 1. 246E+00 1, 315E+00 1.373600 1.420600 1. 457E+00 1. 487E400 1. S09EH00 1. S83E+00

=.2 2 1.133E+00 1.277600 1.425E+00 1. 568E+00 1. 702EH001.824E+00 1. 934E00 2.031E100 2, 117E400 2. 704E+00
0 —& — L,195E+00 14276400 1, 6886100 1,971E+00 2, 267E+00 = 2.573E400 2, BBHEHO0 3. 198E100 3. SISEHOO
+2 =  1,2736+00  1,629E+00 -2,075E+00  2,615E#00 3. 256EH00  4.005E00 4.876E+00 95. 882EH00 7. 042E 100
ht 1.369600 1, 90IEHOO-2,.644EH00 3. 659E+00  5.025EH00 6, BATEHOO «9.267600 1. 247E#01 1, 670E +01
+b 1 1,4B6E#00  2,2646+00 -3.485EH00 5.371E+00 §8.262E+001.267EH01 1. 937EH01 2, 9S5EH01 4, 498E+01
«8 t 1.628400 2, 750EH00 4, 753E+00 8. 215EH00L.4SIEHO1 «= 2,A9MEH01 «4. SATEHO1 «7, S76EH01 «1.320602
1 1,800E#00  3.400E+00 6, 600600 1,300E+01 2. SB0EH01 5. 140E#01 1.026602 2, 050E102 4.098102
1,2! 2.007600 4,271E#00 9.409600 2.114601 4, 800EH01 1. 096EH02 2, 510E102 5, 75KEH02 1, S21EH03
1.4 ¢  2,255E#00  5.439E+00 1. 366E+01 3, 512E+01 9, 142E401 2. 395EH02 6. 29984021. OS9EHO3 «4.3 74E#03
1.6 1 2,5526+00  7,012E+00  2,014E+01 5. 935E¥01 1.773602 5. R4EH02 1. O11EH03 «4, 87481031, 476EH04
1,8 + 2.908600 9, 132E+00  3,006E+01 1.017E*02 3, 487E402 1.205EH03 4. 179103 1. ASSEH04 5, 0535404
2 $9, 3B3E+00 1.200601 4.533801 1. 76081026. 933EH2 2. 7S2EH03 1. 097E+04 4, 37864041. 7A9EH05.

22


Variances Based on Data With
Dead Time Between the
Measurements

James A. Barnest
David W. Allan

Time and Frequency Division

Center for Basic Standards

National Measurement Laboratory

National Institute of Standards and Technology
, Boulder, Colorado 80303-3328
:

tAustron, Inc.
Boulder, Colorado 80301

yet oF Coy,
ows

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary
NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W Lyons, Director

Issued March 1990


:
'
t
'
oot
:
$
t

1, 000E+00

1,.000E+00

1,010E+00
1.011E*00
1.012600
1.013€+00
1.014E+00
1.015E+00
1.016E+00
1, 017E+00
1,.018€+00
1.019E+00
1.020E+00

8, 708E-01
9, 2B1E-01
9.69SE-O1
1.000€+00
1,024€+00
1,.043€+00
1..060E+00
1. 075E+00
1, 0B7E+00
1, 102E+00
11146400
1, 126E+00
1, 138E+00
1, 150E+00
1,162E+00
1.174800
1, 186E+00
1, 198E+00
1, 210E+00

6.667E-01
7. 196E-O1
7, 787-01

1, 091E#00
1. 192E+00
1, 304E+00
1, 4296100
1, S66E+00
1.7186+00
1,886E+00
2.,071EH00
2.2700
2. S00E+00
2. 747EH00
3. 018E+00
3, 310E+00
3, CAZEHO0
4,000E+00

6. 667E-01
7,062E-01
7, S61E-O1
8.192E-01
8, 990E-01
1, 000E+00
1, 128E+00
1, 290E+00
1, 494E+00
1, 752E+00
2.078E+00
2. 4B9E+00
3,.007E100
3,.658E+00
4, ATSE+00
5. S00E+00
6, 7BAEH00
8. 3B9E+00
1, 039E+01
1, 2896401
1, 600E+01

4.47900
6.051E+00
8. 297E+00
1. 1SOE+01
1,607E+01
2. 2596401
3, 188E+01
45136401
6. 400E+01

6. 667E-01
7,018€-01
7472-01
8. 071-01
8.879E-01
1,.000E+00
1, 159€+00
1,392E+00
1, 7386+00
2,.265E+00
3.082E+00
4, 36SE+00
6. 404E+00
9. 673E+00
1. 49SE*01
2. S50E+01
3, 741E*OL
6, 008E+01
9. 706E*01
1, 574E+02
2. SEH

4, 6676-01
7.0148-01
7.461€-01
8.051E-01
8, 855€-01
1,000E+00
1.173800
1, 4555400
1,957E+00
2, 945E+00
5, 082E+00
1,0096401
2,2596+01
5.S21E+01
14296702
3,835E+02
1,.052E+03
2. 9286403
8.215803
2,317E+O4
6, SSHE+04

6.667E-01
7,014€-01
7,4B1E-01
8.051£-01
8, 854€-01
1.000E+00
161748100
1,460E+00
1, 983E+00
3, 064E+00,
5, SB2E+00
1, 209E+01
3. 031E+01
8, 4186401
2, 493E+02
7. 67SE+02
2, 4196+03
7.727EH3,
2, A90E904
8. 067E+04
2,621E+05

6. 667E-01
7,014€-01
7, 461-01
8,051E-01
8, 8546-01
1,000E+00
1.174800
1, 463600
2.003E+00
3, 167E+00
6, 082E+00
1, 439E+01
4, 050E+01
1, 2B1E+02
4 NGEH02
1, 536403
5, SS6E+03
2,039E+04
7. SAIEHO4
2, BO9EHOS
1, 049E+08

6. 067E-01
7.014E-01
7, 401E-O1
8.051E-01
8.854E-01
1,.000E+00
1,174€+00
1, 465E+00
2.018E+00
3. 257E+00
6, S82E+00
1, 7036401
5.3946+01
1, 947E+02
7.S71E#02
3,072E+03
1,277E+04
5. 382E+04
2. 2886405
9, 7E2E+OS
4 194E+06

6. 067E-01
7. 014€-01
7, 461E-01
8. 051E-01
8. 854E-01
1, 000E+00
1, 174E+00
1, 467E+00
2.0296+00
3, 336E+00
7.062E+00
2, 006E+01
7. 167E+01
2, 95SE+02
1.3196+03
6, 1MEHO3
2. 9S3EH04
1, 4206905
6, 937EHOS,
9. 406E+06
1,678€+07

6, 667E-01
T,014E-O1
7, 461E-O1
8.051E-01
8.854E-O1
1, 000E+00
1, 175E+00
1.470€+00
2, 065E#00
3.863600

30

1, 418E+00
1,817E+00
2.4720
3.582E+00
‘S.S14E+00
8, 951E+00
1.S16E+O1
2. 653E+01
4. TSOE+01
B.OMEHOL
1, 590E+02
2, 947EH02
5, 485E+02
1.0246+03

6, 667E-O1
7.015E-O1
7.462E-01
8,083E-O1
8.859E-01
1.000E+00
1, 170€+00
1, 436E+00
1,877E#00
2. 652E+00
4.082E+00
6. 834E#00
1. 231E+01
2, 3486401
4. b69E+OL
9. SSOEH0!
1, 991E#02
4, 201E+02
8,.937E+02
1. 9106+03
4. 096EW03

6,667E-01
7014-01
7462-01
8,.052E-01
8.8S56E-01
1,.000E+00
1, 1726400
1, 447E+00
1, 9736400
2.809600
4. $82E+00
8, S1E+00
1.6746+01
3,.609E#01
8, 1796+01
1, 915E+02
4. S79EHO2
1, 1096+03
2.708403
6, 653€+03
1, 638E+04


$3(2,H,r,u) for r= 1024
fu\ Heo 2 4 8. 16 2 “ 128 b Si2 10246

000E+00 2.006100 4.000E+00 80006100 1.4005+01 3.200E+01 4.400EH0I 1.2B0E+02 2.560E+02 S.120E2 1.024619
000E+00 1. 741E+00 3.031E+00 5. 27BE+00 9. 190E100 1.G00E+01 2.7BKEH01 48506401 8. 4455401 1.470E+02 2.S60E+02
0O0E+00 1.S14E+00 2.297600 3.482E+00 5.278100 8.000E+00 1.21360] 1,090E+01 2. 7B4E+O1 4.228401 6. 400810!
000E+00 1.3196+00 1. 7H1E00 2.297600 S.CGIEHO0 4.000E+00 S.27BE+00 6.946100 9, 167E*00 1.213601 1.600E+01
000E+00
000E+00
000E+00

1
1
1
1
t 1,1496+00 1.319E+00 1,516E+00 1. 741E+00 2.000E+00 2. 297E+00 2.699E100 3.031E+00 3.482100 3. 9996100
1 1.000E+00 1.000E+00 1.000E100 1.000E100 1.000E+00 1.000E+00 1,000E#00 1.000E+00 1.000E*00 1.000E+00
1 8.708E-01 7. 569E-01 4. G07E-01 5. 7545-01 S.01SE-O1 4.370E~01 3.907E-O1 3.5206-01 2.094E-O1 2.524E-01
+ 1,000E+00 7.596E-01 S.7B1E-O1 4.406E~01 3.3546-01 2.575E-01 1.977E-01 1,5ME-O1 1.181E-01 9.2106-02 7.2386-02
1 1,000E+00 6.683E-01 4.S06E~01 S.O76E-O1 2.194E-01 1,SI4E-DL 1.105E-O1 O.988E-02 6.569E-02 S.9ME-~02 4. 6226-02
1 1,000E+00 6.031E-01 3.780E-01 2.497E-01 1.709E-O1 1.551E-OL 1.1128-01 9.7ME-02 6.95KE-02 8.904E-02 6, 2446-02
© —t — 1,0006+00 5, 737E-01 3.658E-01 2,639E-01 2.19801 1.090E-01 1,767E-01 1.700E-Ol 1.675E-01 1.659E-01 1. 6528-01
1 1,000E+00 S.811E-01 4.059E-O1 3.329E-01 3,0128-01 2,000E-01 2,0238-01 2.790E-01 2.708E-01 2.783E-01 2. 791E-01
1 1,000E+00 6,133E-01 4. 740E-O1 4, 2386-01 4.056E-01 3.990E-01 3. 965E~01 3.75HE-0l 3. 999E-01 3,951E-01 9. 951E-01
1 1,0008+00 6.S64E-01 5. 4866-01 5, 190E-01 5.O45E-01 5.012E-01 S,O01E-~O1 4.998E-01 4. 997E-01 4.997E-01 4. 796E-01
1 1,000E+00 7.028E-01 6. 2046-01 S.977E-01 5. 91SE-O1 S.097E-01 5.O7GE-01 S.OP1E-01 S.G71E-O1 5.891E-01 5. @71E-01
1 1,000E+00 7,502E-01 6.877E-01 6. 721E-01 6.662E-01 6.072E-01 6,670E-01 6.669E-01 6.669E-01 6.649E-01 6. 6698-01
+ 1,000E+00 7,979E-01 7.S14E-01 7,406E-01 7.3R0E-01 7.374E-O1 7.373E-O1 7.372-01 7.372-01 7.372-01 7.372-01
1 1,000E+00 8.463E-01 8, 131E-01 @,057E-01 8.040E-01 8.036E-01 8.0G5E-O1 8.005E-01 8.0G5E-O1 G.005E-01 8.005E-01
1 1,000E+00 6. 958E-01 8. 7436-01 8.496E-01 8.405E-01 6.682E-01 8. 6826-01 B,681E-01 8.681E-01 8681-01 8. 681E-01
+ 1,000E+00 9.469E-01 9.362E-01 9. 399E-O1 9.333E-01 9.302E-01 9.302E-01 9.532E-01 9.302E-01 9,552E-01 9.352E-01
+ 1,000E*00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.008E*00 1.000E*00 1.000E+00 1.000E+00

H 83(2,4,r,mu) for r= 2048

4

1. 4, 000E+00

1,000E+00 1. 741E+00. 3.031E+00 5.278E*00 9.190E+00 1.400E+01 2. 786E+01 4.850E+01 8.445E+01 1.470E+02 2, S00EH02
1,000E+00 1,516E+00 2,297E+00 3,482E+00 5. 278E+00 8.000600 1. 2136101 1.808E*01 2. 7BKE+0] 4.222E+01 6, 400EH)1
1,000E+00 1.320E+00 1.741E+00 2. 297E+00 3.031E#00 4.000E+00 5.278E+00 4.964600 9.189600 1.213E+01 1.600E+0i
1,000E+00 1. 1496+00 1.3196+00 1.516E+00 1. 741E+00 2,000E+00 2, 297EH00 2.4396100 3.031E+00 3.482E+00 4,.000E+00
1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00
1,000E+00 8.707E-O1 7. 5825-01 6.603E-01 S.751E-Ol 5.009E-01 4.363E-01 3.900E-01 3.311E-O1 2.885E-01 2.514€-01
1,000E+00 7. 5P1E-01 5, 768E-01 4.388E-01 3,342E-01 2,549E-01 1.949E-01 1.4ME-O1 1.149E-01 8.981E-02 6, 901E-02
1,000E+00 6.662E-01 4.468E-01 3.025E-01 2.075E-01 1.448E-01 1,0GSE~O1 7.6286-02 5.831E-02 4.646€-02 3,864€-02
1,000E+00 5. 987E-O1 3.706E-01 2.407E-01 1.664E-01 1.240E-O1 9.962E-02 8.565E-02 7. 7636-02 7.303E-02 7. 038-02
1,000E+00 S.681£-01 3,570E-01 2.S24E-OL 2,0236-01 1.770E-01 1.644E-O1 1.582E-01 1. 551-01 1,535E-01 1,527E-01
1,000E+00 5.768E-01 3, 995E-O1 3.Z50E-01 2.934E-01 2.900E-O1 2.742E-01 2.717E-01 2. 706-01 2.701E-01 2,699E-01
1,000E+00 6, 111E-01 4. 709E-01 4, 2046-01 4. 021-01 3, 959E-01 9.929E-01 3.920E-01 3.914E-01 3.915E-01 3. 9146-0!
1,000E+00 4,555E-01 5.A7S5E-01 5.198E-01 5.032E-01 4, 9996-01 4, 988E-O1 4.96SE-01 4.984E-01 4.984E-01 4, 9B4E-01
1,000E+00 7.026E-01 4. 201E-01 5. 973E-01 S.911E-Ol 5.694E-01 5,889E-01 5.G88E-01 5.887E-01 5.987E-01 5.887E-01
1,000E+00 7,S501E-01 6,876E-01 6.720E-01 6.681E-O1 6.671E-01 6.669E~01 6,668E-01 6.668E-01 6.648E-01 6. 666E-01
1,000E+00 7. 979-01 7.S14E-01 7.405E-Ol 7.3B0E-01 7.374E-01 7.372E-01 7.372%-01 7372-01 7,372%-01 7.372-01
1,000E+00 8.463E-01 8, 13IE-01 8.057E-O1 8,080E-01 8,.0G4E-01 8,035E~01 8,035E-01 8.034E-01 8.0304E-01 8. 034€-01
1,000E+00 8, 958E-01 8. 749E-01 8.696E-01 8.685E-O1 8.462E-01 9,682E-01 8.6B1E-01 8.681E-01 8.681E-01 8, 681E-01
1,000E+00 9.469E-01 9.362E-01 9.339E-O1 9.333E-Ol 9.332E-01 9.332E-01 9.3G2E-01 9.332E-01 9.332E-01 9,332E-01
1,000E+00 1,000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1.000E+00 1,000E+00

39


BI(Wrjau) for r= 64

MN we O4 8 16 R td 128 ca) 512 1024

“2 — 1,000E+00  1,000E+00 1. 000E+00 1, 000E+00  1.000E+00 1. 000E+00 1.000E100 1.000600 1, 000E#00.

“1.8 1,000E+00 1.000E+00 1.0006+00 1.0006+00 1.00000 1.000100 1.000100 1.000E+00 1, 000E#00 1. 000E+00
“1.6 ' 9,999E-O1 9.999E-01  9.999E-01 9..9996-01 9.999E-01 9,9996-01 9.998E-01 9. 998-01 9, 998-01 9. 998E-01
“1.4 1 9,999E-01 9.998E-01  9.997E-01 9.997E-01 9. 997E-01  9.997E-O1 9997-01 9.99KE-01 9, 996-01 9. 994E-01
“1.2 1 9.998E-O1 9,997E-O1  9.996E-01 9.995E-01 9.995E-01 9,.995E-01  9.995E-01  9,995E-01 9. 995E-01 9. 995E-01
-1$  1,000E+00 1.00000 1.0008+00 1,000E+00 1.000E+00  1.0006#00 1.000400 1.000400 1.000400 1. 0006+00
=.8 1 1,001E+00 1.00200 1.0036+00 1.00300 1,004E+00  1,004E+00 1,004E¥00 1.,.004E+00 1.00E#00 1, 004E+00
16 1,005E#00 1.009600 1, 0196+00 1.016600 1.018400 1.020400 1.021E+00 1.027400 1.022400 1.024E+00
4 © LO1MEHOO 1.028400 1,042E+00 1.054800 1.064+00 1.072400 1,078E+00 1.083400 1.087400 1, 100E+00
2 10356400 1.073400 1.128400 1, 1SIE¥00 1. 187E#00 1.220400 1.249400 1.275E+00 1.298640 1. 454E+00
0 1,0736#00 1.181600 1, 261E#00 1.369600 1.4826+00 1.600400 1,719E+00 1.840100 1. 9616400
+2 t  1.135E+00 1.313400 1,535E+00 1.808600 2.1296#00 2, 508E+00 2,944E+00  3,4506+00 4, 0336100
+41 1.22KE+00 1, 54BEH00 — 2,0036#00 2,624E#00 3, 461E+00 4,578E+00 6.080100 8.073400 1.062E+01
+6 + L,341E#00 1.889400 2.749E+00 4.080100 6.119400 9.227400 1.39SEHOL 2.114401 3.2036401
8 1 1.489640 2.3696+00  3,909E+00 6,6246400 L.IGTEHOL 1.966401 TALIEHOL 5.929E+01 1. 0316402
11 1.67000 3,0106+00 5.69100 1.105EH01 2. 177EHO1 4.322EH01 B.BIIEHOL 1. 7196402 3. 4955402
1.2 1 1.890400 3,891E¥00 8.429E+00 1.879401 4. 251E+01 9. 690E+01 2.218402 5.085E402 1. 167E+03
1.4 +  2.156E+00 5.087400 1.265401 3.238601 8. 415E401 2. 204E#02 5.793402 1,524E#03 4. 0226403
1.6 t 2.477EHOO 6.716400 L.DISEHOL 5.644E¥01 1, 6856402 S.047EH02 1.590403 4,6906403 1, 4026404
1.8 + 2.865E+00 8.949600 2.938401 9.927EH01 3.404E402 1.176603 4, 0806H03 1. 4186404 4, 9336404
2 3,333E+00 1, 20064014, 5336401 1.760602 6, 9336402 2.752403 1.097EH04 4,378E+04 1.749E+05
} BL(N,rjmu) for r= 128

4 8 16 R ca) 128 256 512 1024 INF

1,000E+00 1,000E+00 1, 000E+00 1, 000E#00 1. 000E+00 1, 000E+00 1.000600 1,000EH00 1. 000E+00 1. 0006 +00
1,000E+00  1,000E#00  1,.000E+00  1,000E+00 1, 000E+00 1, 000600 1.000E+00 1.000600 1, 000E#00 1. 000E+00
1,000E+00 1, 000E+00 1, 000E+00 1.000600 1.000E+00 1, 000EH00 9. 999E-01 9. 9F9E-01 9, 999-01 9. 9798-01
1,000E+00 9, 999E-01 9.999E-01 9.999E-01 9. 999E-01 9.999E-01 9.999E-01 9.999E-01 9. 999E-01 9. 999E-01
9.999E-01 9.999E-01 9, 998E-01 9. 998E-01 9, 998E-01 9. 998E-01 9, 99BE-01 9. 998E-01 9. 998E-01 9. 99BE-01
1,000E+00 1, 000600 1.000E+00 1,.000E+00 1. 000E+00 1. 000E+00 1. 000E+00 1,.000E+00 1.00000 1. 000E+00
1,001E+00 1, 001E+00 1. 002E+00 10026100 1,.002E+00 1.002600 1, 002E+001.002E+00 §1.002E+00 —1,.002E+00
1,003E+00 1, 006E#00  1.008E+00  1,010E+00 1.012E#00 1.013600 1,014E+00 1,014E+00 1, 015E#00 1, O1SEH00
1.010600 1, 0216400 1,031E+00  1.040E+00 1, 047E#00 1,053E+00  1.058E+00 1.062E+00 1. 065E+00 1, 074E+00
1,029E+00  1,060E+00 1,092E+00  1,124E+00 1, 153E+00 1, 181EH00 1. 2056100 1.226EH00 1. 245EH00 1, 375E+00
1,065E+00 1, 1446400 1,.232E+001.329E+00 1. 490EH001,SSHE+00 1. 640E+00 1. 748E+00 1. BSKEHO0
1,127E+00 1.294600 1, SO4E+00 1. 759E+00 2.062500 2.416600 2,.827E+00 3.303600 3. SSIEH00
1,217E+00 1, 532E#00 1, 973E+00 2.5 76E+00 3, 3B8E+00 4. 471E+00  5.909E+00 7.813400 1. 033E+01
1,336E+00 1,877E#00  2.725E+00 4,037E+00 6, 048EH00 9. 115EH00§1.378E+01 2.086601 3, 160E#01
1,486E+00 2, 354E+00  3.891E+00 4, 587EH00 L.ISIEHOL «1. SSSEHOL «3. SPIE] S.893EH01 1, 02502
1,668E+00  3,005E+00  5.679E+00  1,103E+01 2.172401 4/311EH01 «8, SOVEHO «1. 7148402 3. 426E102
1, 889E+00 3,8B7E400 8. 421E+00 1.877E101 4, 246EH01 9, 6B0E401 = 2, 215EH02 5.079802 1, 166EH03.
2,156E+00  5.085E+00 1. 265E+01 3, 237E+01 8, 411EHO1 2, 2036102 5. 790EH02 1.S25EH3 4, 021E03
2.477E+00 —b.715E#00 1. 9198401 5. b4SEHO1 «1. OB4EH02 «5.067E402 1, 5Q0EH034.6296403 «1, 402E 404
2,865E+00 8. 94GEH00 2. 738E+01 9. 926EH01 3, 404E+02 1, 176EH03 4, 0B0EH03 1. 4I8EH04 4.932104
3,333E+00 1, 2006101 4, SS3EH011.700EH02 6, 933602 2, 7S2EH03 1, 097EH4 4,378E+04 1, 7A9EH0S.

25


Since the time-domain definition for frequency stability is the Allan
variance, it behooves us, where possible, to relate other variances to the
Allan variance. If we have an N-sample variance on data with dead-time T-r

and we know the power-law spectral type (the value of y), then we may write

of (N,T,17)
By (N,r,#) Ba (r,p)

o2(r) = (15)

If we have an N-sample variance where each data entry is an average of M
samples with distributed dead time, then we may write

2
Bes. <2 o2 (N,M,T, 1)
29 (7) = Bryn) By (x,a) By OME n)” <18)

8. Conclusion

For some important power-law spectral density models often used in
characterizing precision oscillators (Sy (£) ~ £%, a = -2, -1, 0, +1, +2), we
havg studied the effects on variances when there is dead time between the
frequency samples, and the frequency samples are averaged to increase the
integration time. Since dead time between measurements is a common problem
throughout metrology, the analysis here has broader applicability than just to
time and frequency. Specifically, this kind of analysis has been used with
gage blocks and standard volt cells--showing that the classical variance may

be non-convergent in some cases [9].

Heretofore, the Allan variance has been shown to have some convenient
theoretical properties in relation to power-law spectra as the integration or
sample time is varied (if o,?(r) ~ r#, then a = -y -1, -2<y <2). Since
oy(r), by definition, is estimated from data with no dead time, the sample or
integration time can be unambiguously changed to investigate the r dependence.
From our analysis, we have concluded that for the asymptotic limit of several
samples being averaged with dead time present in the data, the r dependence of
the variances is the same. The a = -p -1 relationship still remains valid for
white FM noise (uz = -1, a = 0), flicker FM noise (u = 0, @ = -1), and for
9


B, (2,0,p)
B, (N,xr,2)
B, (N,1,1)
B, (N, 1,4)

B, (N,1,4)

B, (N,r,-1)
B, (N,r,-2)
B, (0,m)
B, (1,4)
By (r,2)

By (r,1)

By (r,-1)

B, (r,-2)

B, (2,M,1,4)
B,(2,M,r,-2)
B, (2,r,4)

By (2,M,r,2)
By (2,M,r,-1)

Table 1.

1
(N(N+1))/6
N/2

Table of some bias function identities

(N(L-N*))/[(2(N-1) (1-2")] for px0

N in(N)/[2(N-1) 1n(2)]
1 for p<0

N-1
(2/(N(N-1)] Y (W-n)

n=1
Lif net
lifrzior0oO
(0)
1
x
(3r - 1)/2 if r=1
rifoOsresi
lifr2tl
0 if r=0
1 if r=1
2/3 otherwise
1
M
1
1
1 for r2=1

12

for p=0

> m for p>O0d