Appendix C

WEIGHTING FUNCTION FOR DRIFT AND JITTER

Let the error in the Imaging instrument's Line of sight (LOS) due to drift and jitter at any time during the picture taking interval, T, be e(t). Then, assuming a zero-mean Gaussian distribution, the mean value of the LOS pointing error of a particular scene is given by

figure

The mean value of LOS error in the scene represents the actual drift of the LOS. As the drift occurs due to low frequency disturbances, the drift component may considered constant during the picture taking interval, T.

The error component, d(t), of the particular picture due to jitter effect is then given by

figure

The total jitter, D, is the RMS value of d(t) over the time, T, and its mean square value is given by

figure

figure

The above expressions for drift and jitter could be used directly, only if e(t) were a known function, which is usually not the case. In general, e(t) is a stationary random process, and the quantities of interest are the expected values of mean square drift and of mean square jitter. In order to compute these quantities, we proceed as follows:

We compute the autocorrelation function of e(t):

figure

The Fourier Transform of figure and its inverse function are given by

figure

figure

where figure is the spectral density, also called the power spectral density, of the process e(t), and is a function of the angular frequency figure. Setting r = 0 in the above relation yields

figure

which, from the fundamental definition of figure, gives

figure

figure

figure

figure

Simplifying the expression , we can write mean square drift as

figure

where,

figure

Similarly, the mean square value of the jitter can be expressed as:

figure


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