1) The document describes projects to obtain timing figures for orbit tracking filters using different satellite data window lengths. 2) Timing runs show the Gauss-Newton filter takes 0.484 seconds while the Kalman and Swerling filters take around 0.5 seconds for a 500,000 data point run. 3) Results are shown for the Molniya and GPS-10 satellites using window lengths of 350, 650, and 1250 seconds, demonstrating that doubling the window length improves accuracy by a factor of around 2.8.

1. Norman Morrison Projects 1.1
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Chapter 1: Projects
Project 1.1
Obtaining a speed figure for your computer
Read the down-loadable document ProgramsSpeedDocumentsSpeed.pdf. Then run the
executable ProgramsSpeedProgramsSpeed.exe to obtain a speed figure for you
computer. Use the average of five runs.
Project 1.2
Timing figures
Open the down-loadable program 14_Orbit_Track. When run on my laptop1 it takes
about 0.48 sec to cycle the Gauss-Newton filter that extracts orbital parameters from a set
of observations on a near-earth satellite.2
Based on clock-speed differences, when Sputnik-1 was put into orbit in 1957, it would
have taken about an hour.
◄►
To obtain the Gauss-Newton, Kalman and Swerling execution times on your own
machine, start the program 14_Orbit_Track.exe
Then make the following selections:
Run Satellite Tracking : Run
Take the option | Use Q in Neither filter and run .. N |
When N = 500000 the run will halt. On my machine the total Kalman execution time
showed Kalman Tcume = 0.488 sec and the total Swerling execution time showed Swerling
Tcume = 0.492 sec.
1
A Dell® Studio XPS 1640 with dual Intel® processors. Rating Core™ 2 CP P8700 @ 2.53 GHz, Base
Score 5.9.
2
C runs about 10 times faster than compiled True Basic. It would therefore take even less time – about
56 milliseconds – if 14_Orbit_Track had been written in C, and even less if use was made of both
processors on my 2-core mother-board.

2. Norman Morrison Projects 1.2
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Press G for Gauss-Newton and then select Run Gauss-Newton.
Gauss-Newton now executes. On my laptop the display showed
Gauss-Newton Elapsed Time = 0.484 sec.
The Swerling and Kalman results are also displayed on the same screen.
Keep in mind that the timing figures are not comparable.
 The Kalman/Swerling filters provide Keplerian estimates almost as soon as they start
running, initially with poor accuracy but which improves steadily as their runs
progress.
 The Gauss-Newton filter on the other hand does not provide Keplerian estimates until
after the Kalman/Swerling filters have stopped running, at which point there is a
further delay of 0.484 sec before Gauss-Newton provides the results that have the
same final Kalman/Swerling accuracy. (See Project 1.3 below.)
Thus the Gauss-Newton execution time represents an additional latency or delay.
In 1957 when Sputnik-1 was put into orbit, computer clock rates were of the order of 1
megahertz, and that latency was as large as one hour.
Gauss-Newton therefore had to be ruled out and could not be used in satellite tracking.
Press M
You are now on the main menu. Select Exit to exit from the program.

3. Norman Morrison Projects 1.3
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Project 1.3
Timing and accuracy figures for two satellites
Obtain the times required to estimate Keplerians from a sequence of radar observations
on two satellites.
1. Molniya
1. Start the program 14_Orbit_Track.
Select Tracking window : Length : Specify and enter 350.
The first 50 sec of operation is taken up by the EMP filters which are initialising.
At the end of 50 sec the EMP output values are used to initialise the FMP filters as
well as the Kalman and Swerling filters.
The length of the Gauss-Newton/Kalman/Swerling data window is therefore 300 sec
and this is displayed on the screen.
Make the timing runs for Molniya and Kalman by following the instructions given in
Project 1.2 above.
Make and save a screen capture.
Then return to the main menu page.
2. Select Tracking window : Length : Specify and then enter 650.
Observe that the Gauss-Newton/Kalman/Swerling data window now has a length
of 600 sec.
Repeat the timing run for Molniya.
Make and save a screen capture.
Then return to the main menu page.
3. Select Tracking window : Length : Specify and then enter 1250.
Observe that the Gauss-Newton/Kalman/Swerling data window now has a length of
1200 sec.

4. Norman Morrison Projects 1.4
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Repeat the timing run for Molniya.
Make and save a screen capture of the results.
Then return to the main menu page.
2. GPS-10
Satellites : GPS-10
Repeat the same three runs – 350 sec, 650 sec, 1250 sec – for the satellite GPS-10. In
each case make and save the same screen captures as with Molniya.
Discussion of results
Molniya
We ran this project and obtained the three displays (extracted from the Molniya screen
captures) shown on the following pages as Figures A, B and C.

5. Norman Morrison Projects 1.5
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Figure A

6. Norman Morrison Projects 1.6
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Figure B

7. Norman Morrison Projects 1.7
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
Figure C

8. Norman Morrison Projects 1.8
Tracking Filter Engineering
The Gauss-Newton and Polynomial Filters
The figures show the following:
1. The Gauss-Newton, Kalman and Swerling sigma values in each case are all very
close. Thus the final accuracies of the three filters are the same.
2. The sigma values in Figures A, B and C are in the following ratios:
OMEGA: A:B ≈ 2.8:1 B:C ≈ 2.8:1
Inclin: A:B ≈ 2.9:1 B:C ≈ 2.9:1
omega: A:B ≈ 2.8:1 B:C ≈ 2.8:1
Eccent: A:B ≈ 2.8:1 B:C ≈ 2.8:1
A: A:B ≈ 2.8:1 B:C ≈ 2.8:1
tau: A:B ≈ 2.7:1 B:C ≈ 2.7:1
Thus doubling the length of the smoothing interval improves the accuracies of the
Keplerian estimates by a factor of about 2.8.
GPS-10
Compare your GPS-10 results to those of Molniya.