Kalman Filters in Time and e Frequency Analysis

Use of Kalman filters in time and frequency analysis John Davis1st May 2011

Overview of TalkSimple example of Kalman filter

Discuss components and operation of filter

Models of clock and time transfer noise and deterministic properties

Examples of use of Kalman filter in time and frequency

Predictors, Filters and Smoothing AlgorithmsPredictor: predicts parameter values ahead of current measurements

Filter: estimates parameter values using current and previous measurements

Smoothing Algorithm: estimates parameter values using future, current and previous measurements

What is a Kalman Filter ?A Kalman filter is an optimal recursive data processing algorithm

The Kalman filter incorporates all information that can be provided to it. It processes all available measurements, regardless of their precision, to estimate the current value of the variables of interest

Computationally efficient due to its recursive structure

Assumes that variables being estimated are time dependent

Linear Algebra application

Simple Example

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Filter Errors

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

11110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate

State Vector Extrapolation

Date

Offset

Step 1: State Vector Extrapolation

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)

Uncertainty Extrapolation (lower)

Uncertainty Extrapolation (upper)

Date

Error / Uncertainty

Uncertainty Extrapolation

Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset

Step 4: State Vector Update

Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Uncertainty Update (lower)

Uncertainty update (upper)

Date

Error / Uncertainty

Step 5 Uncertainty update

Sheet1

DateTruthMeasurementFilter EstimateUncertaintyErrorUncertainty (lower)Uncertainty (upper)MeasurementFilter EstimateState Vector ExtrapolationState Vector UpdateErrorUncertainty (lower)Uncertainty (upper)Uncertainty Extrapolation (lower)Uncertainty Extrapolation (upper)Uncertainty Update (lower)Uncertainty update (upper)

15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-182.06E-096.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Simple Example (Including Estimates)

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Filter Errors

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

0.0000000021110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset


Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty


Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset


Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-182.06E-096.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Estimation Errors and Uncertainty

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

0.0000000021110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset


Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty


Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset


Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-182.06E-096.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Operation of Kalman Filter

Provides an estimate of the current parameters using current measurements and previous parameter estimates

Should provide a close to optimal estimate if the models used in the filter match the physical situation

World is full of badly designed Kalman filters

What are the applications in time and frequency?

Analysis of time transfer / clock measurements where measurement noise is significant

Clock and time scale predictors

Clock ensemble algorithms

Estimating noise processes

Clock and UTC steering algorithms

GNSS applications

Starting point for iteration 11 of filter

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

111111

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty


Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset


Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-185.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Five steps in the operation of a Kalman filter

(1) State Vector propagation

(2) Parameter Covariance Matrix propagation

(3) Compute Kalman Gain

(4) State Vector update

(5) Parameter Covariance Matrix update

Components of the Kalman Filter (1) State vector x Contains required parameters. These parameters usually contain deterministic and stochastic components.

Normally include: time offset, normalised frequency offset, and linear frequency drift between two clocks or timescales.

State vector components may also include: Markov processes where we have memory within noise processes and also components of periodic instabilities

Components of the Kalman Filter (2) State Propagation Matrix F(t) Matrix used to extrapolate the deterministic characteristics of the state vector from time t to t+t

Steps in the operation of the Kalman Filter Step 1: State vector propagation from iteration n-1 to n

Estimate of the state vector at iteration n not including measurement n.

State propagation matrix calculated for time spacing t

Estimate of the state vector at iteration n-1 including measurement n-1.


Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

11110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty


Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset


Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-186.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Components of the Kalman Filter (3) Parameter Covariance Matrix PContains estimates of the uncertainty and correlation between uncertainties of state vector components. Based on information supplied to the Kalman filter. Not obtained from measurements.Process Covariance Matrix Q(t)Matrix describing instabilities of components of the state vector, e.g. clock noise, time transfer noise moving from time t to t+tNoise parameters sParameters used to determine the elements of the process covariance matrix, describe individual noise processes.

Steps in the operation of the Kalman Filter Step 2: Parameter covariance matrix propagation from iteration n-1 to n

Parameter covariance matrix, at iteration n-1 including measurement n-1 State propagation matrix and transpose calculated for time spacing t Parameter covariance matrix, at iteration n not including measurement n Process covariance matrix


Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

11110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty

Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset


Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-186.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Components of the Kalman Filter (4) Design Matrix HMatrix that relates the measurement vector and the state vector using y =H x

Measurement Covariance Matrix RDescribes measurement noise, white but individual measurements may be correlated. May be removed.

Kalman Gain KDetermines the weighting of current measurements and estimates from previous iteration. Computed by filter.

Steps in the operation of the Kalman Filter Step 3 :Kalman gain computation

Parameter covariance matrix, at iteration n not including measurement n Design Matrix and Transpose Kalman Gain Measurement covariance matrix

Components of the Kalman Filter (5) Measurement Vector yVector containing measurements input during a single iteration of the filter

Identity Matrix I

Steps in the operation of the Kalman FilterStep 4 :State vector update

Estimate of the state vector at iteration n including and not including measurement n respectively. Kalman Gain Measurement Vector Design Matrix

Step 4 :State Vector Update

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

0.0000000021110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty

Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset

Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

111111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty


Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-182.06E-096.66E-095.80E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Steps in the operation of the Kalman Filter Step 5 :Parameter Covariance Matrix update

Kalman Gain Design Matrix Parameter Covariance Matrix, at iteration n, including and not including measurement n respectively Identity Matrix

Steps in the operation of the Kalman FilterIf Kalman gain is deliberately set sub-optimal, then use

Kalman Gain Design Matrix Parameter Covariance Matrix, at iteration n, including and not including measurement n respectively Identity Matrix

Step 5: Uncertainty Update

Chart1

0.0000000050.00000001010.0000000101

Truth

Measurement

Filter Estimate

Chart2

0.0000000050.00000001010.0000000101

0.00000000560.00000000460.0000000073

0.0000000048-0.00000000050.0000000045

0.00000000450.00000000230.0000000039

0.00000000410.0000000090.0000000051

0.00000000380.00000000950.0000000061

0.00000000330.00000000810.0000000065

0.00000000270.00000000360.0000000059

0.00000000370.00000000160.0000000051

0.00000000330.00000001330.0000000067

0.00000000420.00000000210.0000000058

0.00000000490.00000000310.0000000053

0.00000000560.00000001270.0000000066

0.0000000040.00000000910.0000000071

0.00000000540.00000000570.0000000068

0.00000000720.00000000790.000000007

0.0000000059-0.0000000050.0000000048

0.00000000560.00000000920.0000000056

0.00000000590.0000000080.0000000061

0.00000000530.00000000610.0000000061

Truth

Measurement

Filter Estimate

Chart3

-0.0000000051-0.0000000050.000000005

-0.0000000016-0.00000000360.0000000036

0.0000000002-0.0000000030.000000003

0.0000000006-0.00000000270.0000000027

-0.0000000011-0.00000000250.0000000025

-0.0000000023-0.00000000240.0000000024

-0.0000000033-0.00000000230.0000000023

-0.0000000032-0.00000000220.0000000022

-0.0000000014-0.00000000220.0000000022

-0.0000000034-0.00000000220.0000000022

-0.0000000016-0.00000000220.0000000022

-0.0000000004-0.00000000210.0000000021

-0.0000000011-0.00000000210.0000000021

-0.0000000031-0.00000000210.0000000021

-0.0000000014-0.00000000210.0000000021

0.0000000001-0.00000000210.0000000021

0.0000000011-0.00000000210.0000000021

-0-0.00000000210.0000000021

-0.0000000001-0.00000000210.0000000021

-0.0000000007-0.00000000210.0000000021

Error

Uncertainty (lower)

Uncertainty (upper)

Date

Error

Chart4

0.0000000050.0000000101

0.00000000560.0000000046

0.0000000048-0.0000000005

0.00000000450.0000000023

0.00000000410.000000009

0.00000000380.0000000095

0.00000000330.0000000081

0.00000000270.0000000036

0.00000000370.0000000016

0.00000000330.0000000133

0.00000000420.0000000021

0.00000000490.0000000031

0.00000000560.0000000127

0.0000000040.0000000091

0.00000000540.0000000057

0.00000000720.0000000079

0.0000000059-0.000000005

0.00000000560.0000000092

0.00000000590.000000008

0.00000000530.0000000061

Truth

Measurement

Date

Offset

Chart6

0.00000001010.00000001011

0.00000000460.00000000732

-0.00000000050.00000000453

0.00000000230.00000000394

0.0000000090.00000000515

0.00000000950.00000000616

0.00000000810.00000000657

0.00000000360.00000000598

0.00000000160.00000000519

0.00000001330.000000006710

0.0000000021110.0000000067

121212

131313

141414

151515

161616

171717

181818

191919

202020

Measurement

Filter Estimate


Date

Offset

Chart7

-0.0000000051-0.0000000050.00000000511

-0.0000000016-0.00000000360.000000003622

0.0000000002-0.0000000030.00000000333

0.0000000006-0.00000000270.000000002744

-0.0000000011-0.00000000250.000000002555

-0.0000000023-0.00000000240.000000002466

-0.0000000033-0.00000000230.000000002377

-0.0000000032-0.00000000220.000000002288

-0.0000000014-0.00000000220.000000002299

-0.0000000034-0.00000000220.00000000221010

111111-0.00000000260.0000000026

1212121212

1313131313

1414141414

1515151515

1616161616

1717171717

1818181818

1919191919

2020202020

Error

Uncertainty (lower)

Uncertainty (upper)



Date

Error / Uncertainty

Chart5

0.00000001010.000000010111

0.00000000460.000000007322

-0.00000000050.000000004533

0.00000000230.000000003944

0.0000000090.000000005155

0.00000000950.000000006166

0.00000000810.000000006577

0.00000000360.000000005988

0.00000000160.000000005199

0.00000001330.00000000671010

0.0000000021110.00000000670.0000000058

12121212

13131313

14141414

15151515

16161616

17171717

18181818

19191919

20202020

Measurement

Filter Estimate


State Vector Update

Date

Offset

Chart8

-0.0000000051-0.0000000050.0000000051111

-0.0000000016-0.00000000360.00000000362222

0.0000000002-0.0000000030.0000000033333

0.0000000006-0.00000000270.00000000274444

-0.0000000011-0.00000000250.00000000255555

-0.0000000023-0.00000000240.00000000246666

-0.0000000033-0.00000000230.00000000237777

-0.0000000032-0.00000000220.00000000228888

-0.0000000014-0.00000000220.00000000229999

-0.0000000034-0.00000000220.000000002210101010

-0.00000000161111-0.00000000260.0000000026-0.00000000220.0000000022

12121212121212

13131313131313

14141414141414

15151515151515

16161616161616

17171717171717

18181818181818

19191919191919

20202020202020

Error

Uncertainty (lower)

Uncertainty (upper)





Date

Error / Uncertainty

Sheet1


15.00E-091.01E-081.01E-08-5.06E-09-5.00E-095.00E-092.50E-171.01E-081.01E-08-5.06E-09-5.00E-095.00E-09

25.63E-094.56E-097.26E-09-1.63E-09-3.57E-093.57E-091.27E-174.56E-097.26E-09-1.63E-09-3.57E-093.57E-09

34.76E-09-4.52E-104.52E-092.39E-10-2.98E-092.98E-098.87E-18-4.52E-104.52E-092.39E-10-2.98E-092.98E-09

44.49E-092.30E-093.90E-095.98E-10-2.66E-092.66E-097.08E-182.30E-093.90E-095.98E-10-2.66E-092.66E-09

54.08E-099.00E-095.14E-09-1.06E-09-2.47E-092.47E-096.10E-189.00E-095.14E-09-1.06E-09-2.47E-092.47E-09

63.79E-099.51E-096.11E-09-2.32E-09-2.35E-092.35E-095.53E-189.51E-096.11E-09-2.32E-09-2.35E-092.35E-09

73.26E-098.12E-096.52E-09-3.27E-09-2.28E-092.28E-095.18E-188.12E-096.52E-09-3.27E-09-2.28E-092.28E-09

82.73E-093.62E-095.95E-09-3.21E-09-2.23E-092.23E-094.95E-183.62E-095.95E-09-3.21E-09-2.23E-092.23E-09

93.70E-091.63E-095.12E-09-1.41E-09-2.19E-092.19E-094.81E-181.63E-095.12E-09-1.41E-09-2.19E-092.19E-09

103.27E-091.33E-086.66E-09-3.39E-09-2.17E-092.17E-094.71E-181.33E-086.66E-09-3.39E-09-2.17E-092.17E-09

114.22E-092.06E-095.80E-09-1.58E-09-2.16E-092.16E-094.65E-182.06E-096.66E-095.80E-09-1.58E-09-2.60E-092.60E-09-2.16E-092.16E-09

124.87E-093.07E-095.30E-09-4.32E-10-2.15E-092.15E-094.61E-18

135.57E-091.27E-086.64E-09-1.07E-09-2.14E-092.14E-094.58E-18

143.97E-099.11E-097.09E-09-3.13E-09-2.14E-092.14E-094.56E-18

155.43E-095.66E-096.83E-09-1.41E-09-2.13E-092.13E-094.55E-18

167.17E-097.95E-097.04E-091.35E-10-2.13E-092.13E-094.54E-18

175.93E-09-5.03E-094.85E-091.09E-09-2.13E-092.13E-094.54E-18

185.60E-099.17E-095.63E-09-2.93E-11-2.13E-092.13E-094.53E-18

195.92E-097.99E-096.06E-09-1.39E-10-2.13E-092.13E-094.53E-18

205.34E-096.06E-096.06E-09-7.17E-10-2.13E-092.13E-094.53E-18

Sheet2

Sheet3

Constructing a Kalman Filter: Lots to think about! Choice of State Vector componentsChoice of Noise ParametersChoice of measurementsDescription of measurement noiseChoice of Design MatrixChoice of State Propagation Matrix Initial set up conditionsDealing with missing dataTesting out the filter Computational methods

Initialising the Kalman FilterSet the initial parameters to physically realistic values

If you have no information set diagonal values of Parameter Covariance Matrix high, filter will then give a high weight to the first measurement.

Set Parameters with known variances e.g. Markov components of Parameter Covariance Matrices to their steady state values.

Set Markov State Vector components and periodic elements to zero.

Elements of H, F(t), Q(t), R are pre-determined

Simulating Noise and Deterministic ProcessesUse the same construction of state vector components x, state propagation matrix F(t), parameter covariance matrix P, process covariance matrix Q(t), design matrix H and measurement covariance matrix R.

Simulate all required data sets

Simulate true values of state vector for error analysis

May determine ADEV, HDEV, MDEV and TDEV statistics from parameter covariance matrix

Simulating Noise and Deterministic Processes

Where

= Process Covariance Matrix

= State Vector at iteration n and n-1 respectively = State Propagation Matrix calculated for time spacing t = Vector of normal distribution noise of unity magnitude

Simulating Noise and Deterministic Processes Process Covariance Matrix at iterations n and n-1 respectively State Propagation Matrix and transpose computed at time spacing t. Process Covariance Matrix calculated at time spacing t

Simulating Noise and Deterministic Processes

where

Measurement Vector Design Matrix Measurement Covariance Matrix Vector of normal distribution noise of unity magnitude

ADEV, HDEV, and MDEV determined from Parameter Covariance Matrix and simulationz

Simulating Noise and Deterministic ProcessesTest Kalman filter with a data set with same stochastic and deterministic properties as is expected by the Kalman filter.

Very near optimal performance of Kalman filter under these conditions.

Deterministic PropertiesLinear frequency drift

Use three state vector components, time offset, normalised frequency offset and linear frequency drift

Markov Noise ProcessWhite NoiseRandom Walk NoiseMarkov Noise Flicker noise, combination of Markov processes with continuous range of relaxation parameters k Markov noise processes and flicker noise have memoryKalman filters tend to work very well with Markov noise processesMay only construct an approximation to flicker noise in a Kalman filter

Markov Noise Process

Describing Measurement Noise Kalman filter assumes measurement noise introduced via matrix R is white

Time transfer noise e.g TWSTFT and GNSS noise is often far from white

Add extra components to state vector to describe non white noise, particularly useful if noise has memory e.g. flicker phase noise

NPL has not had any problems in not using a measurement noise matrix R

White Phase Modulation

Flicker Phase Modulation 05101520-2.5-2-1.5-1-0.500.511.52x 10-10Simulated FPM NoiseMJDTime Offset

Flicker Phase Modulation

Measurement noise used in Kalman filter

Measurement noise ADEV, MDEV and HDEV

Clock NoiseWhite Frequency Modulation and Random Walk Frequency Modulation easy to include as single noise parameters.

Flicker Frequency Modulation may be modelled as linear combination of Integrated Markov Noise Parameters

White Frequency Modulation

Random Walk Frequency Modulation

FFM Noise

Flicker Frequency Modulation

Periodic InstabilitiesOften occur in GNSS ApplicationsOrbit mis-modellingDiurnal effects on Time Transfer hardwarePeriodicity decays over timeModel as combination of Markov noise process and periodic parameterTwo extra state vector component to represent periodic variations

GPS Time Individual Clock (Real Data)

Input data space clock prn 9

Input Data

Time Offset

DayD

-5

x 10

1.622

1.6215

1.621

1.6205

1.62

1.6195

1.619

7

6

5

4

3

2

1

0

GPS Time Individual Clock (Simulated Data)

ADEV, HDEV, MDEV Real Data

ADEV, HDEV, MDEV Simulated Data

MDEV Theory

HDEV Theory

ADEV Theory

MDEV Measurements

10

2

10

3

10

4

10

5

10

6

10

-14

10

-13

10

-12

10

-11

ADEV, HDEV MDEV Theory and Measurement

tau

ADEV

ADEV Measurements

HDEV Measurements

Kalman Filter ResidualsResiduals Residual Variance (Theory)

In an optimally constructed Kalman Filter the Residuals should be white!

Residual Variance obtained from Parameter Covariance matrix should match that obtained from statistics applied to residuals

Kalman Filter Residuals (Simulated Data)

Kalman Filter Residuals (Real Data)

Computational Issues Too many state vector components result in a significant computational overhead

Observability of state vector components

Direct computation vs factorisation

Direct use of Kalman filter equations usually numerically stable and easy to follow.

Issue of RRFM scaling if a linear frequency drift state vector component is used

Clock Predictor ApplicationMay provide a close to optimal prediction in an environment that contains complex noise and deterministic characteristics

If noise processes are simple e.g. WFM and no linear frequency drift then very simple predictors will work very well.

Clock Predictor Equations (Prediction) Repeated application of above equation.

Apply above equations with calculated at the required prediction length t

Clock Predictor Equations (PED)

Repeated application of above equation.

Apply above equations with calculated at the required prediction length t

Example: Predicting clock offset in presence of periodic instabilities ADEV = PED/Tau

Above equation exact in case of WFM and RWFM noise processes, and a reasonable approximation in other situations

ADEV, HDEV, MDEV Simulated Data

MDEV Theory

HDEV Theory

ADEV Theory

MDEV Measurements

10

2

10

3

10

4

10

5

10

6

10

-14

10

-13

10

-12

10

-11

ADEV, HDEV MDEV Theory and Measurement

tau

ADEV

ADEV Measurements

HDEV Measurements

(PED /Tau) Prediction of Space Clocks (simulated data)

ADEV, HDEV, MDEV Real Data

(PED /Tau) Prediction of Space Clocks (real data) Add plot

Kalman Filter Ensemble Algorithm Most famous time and frequency application of the Kalman filterInitially assume measurements are noiseless. Measurements (Cj C1), difference between pairs of individual clocksEstimates (T Ci), where T is a perfect clock.Parameters being estimated cannot be constructed from the measurements: Badly designed Kalman filter. Problem solved using covariance x reduction

Stability of Composite Timescales

2

2.5

3

3.5

4

4.5

5

-15.8

-15.6

-15.4

-15.2

-15

-14.8

-14.6

-14.4

-14.2

Log10()

Log10(y)

Clock 1 (FFM)

Clock 2 (WFM)

Clock 3 (WFM)

Composite

Simple Composite

Optimal Composite

Advantage of Kalman Filter based Ensemble AlgorithmPotential of close to optimal performance over a range of averaging times

Clasical Ensemble Algorithms provide a very close to optimal performance at a single averaging time

Combining TWSTFT and GPS Common-View measurementsTime transfer noise modelled using Markov Noise processes

Calibration biases modelled using long relaxation time Markov

Real Time Transfer Data Example

Kalman Filter Estimates

Kalman Filter Residuals

SummaryOperation of the Kalman filter

Stochastic and deterministic models for time and frequency applications

Real examples of use Kalman Filter

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Kalman Filters in Time and e Frequency Analysis