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ANALYSIS of PERIODIC ERROR
The analysis performed on this page is done with permission of John Welsh who has provided the data. He has an ALTER D6 mount fitted with AWR Technology Intelligent Drive System. The primary worm wheel reduction is 200:1 and so the period of rotation of the worm at sidereal rate is 7 minutes 10.9 seconds.There is an internal gearbox of 7:1 and 2:1 ratios so there are other periods of 61.5 and 30.8 seconds. There are further mechanical periods possible considering tooth to tooth times on the two sets of geared reductions. On top of this it is driven by a motor in Microstep mode with a step to step sequence of 64 microsteps or 0.154 seconds. Data gathering was performed by a Starlight Xpress SXV Camera and Autoguider. For this technique to work there must be several cycles of data present, so logging was performed for over 14 minutes being two periods of the longest time. Here is the data set used for recording the raw data traces of the RA and DEC.

The dataset details are:

Date 28/12/2005
UT Time 21:18
Star Used hip 18993
RA of Star 04:04:09
DEC of Star 02:49:36
Altitude of star (Degrees) 39.1
Azimuth of star (Degrees) 171.5
Camera pixel size (arc second)
(binning 1 x 1)		 3.88

If you are going to try this analysis please make sure the X axis of the camera is exactly in line with the RA direction, else there will be a resolved component of the RA periodic error in the Y data stream as well as the X data stream. Here are the RA and DEC traces.

Looking at just the RA component the data can be fitted with a best fit sine wave. A programme called DPLOT was used which has very powerful data analysis and curve fitting procedures. It requires a CSV file which is easily produced from the SXV error data stream.

The DEC component shows a steady drift which means that the polar alignment is out slightly. As the star was near the central meridian the drift (of 0.496 arc second per minute) indicates an error in azimuth of the polar axis of 1.9 arc minutes. It is not clear if the drift was north or south so the required azimuth rotation direction of the polar axis is not known.

This shows a periodic error sine wave component of 4.2 pixel peak to peak with a correlation coefficient of 0.99. Note the amplitude is an RMS value of 1.48 pixel or 5.7 arc seconds RMS error.

It is a more complex procedure to do a Fast Fourier Transform as the input data needs to be equally spaced. There is a DPLOT function to do this using linear interpolation of the available data. Greatest sensitivity is obtained when there are only complete periods present of the predominant sine wave.

The Fast Fourier Transform (FFT) is a powerful method of analysis which uses all the data to work out the RMS value (Root-Mean-Square or power!) for each frequency component present. There is a minimum and maximum frequency that can be plotted relating to the input data set. The analysis assumes the data set repeats ad-infinitum in both directions of time. All repetitive waveforms can be decomposed into an infinite series of sine waves of varying amplitude, analysed elegantly in the FFT.

Here is the result of the FFT analysis on this data. It is plotted on logarithmic scales so you can see the data.

The periods you can pick out are:

Frequency
(Hz)		 Period
(second)		 Value
(arc second RMS)
0.002268441.07.87
0.004535220.51.65
0.0158763.00.79
0.0328830.40.69
So we can even pick up the motor errors! The periods pircked out match very closely the mechanical periodicities picked out at the start of this article. The fundamental frequncy is dificult to determine accurately because there are not enough periods present. Note the noise floor is about 0.1 pixel which is about the accuracy of the camera in its positioning. You can get better results of this analysis by taking more periods or by taking the data faster to get more high frequency resolution.
© January 2006 AWR Technology HOME PAGE www.awrtech.co.uk