Kalman filtering to suppress spurious signals in Adaptive Optics control Page: 6 of 54
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2. Incorporation of colored noise
In the standard formulation of the Kalman filter state space model, the measurement
noise is temporally white. This means that any specific instant of measurement noise
is temporally uncorrelated with all others. This is a reasonable assumption when
considering WFS noise due to standard CCD noise sources (i.e. photon noise and
read noise). However, noise due to another source may not be temporally white.
The AO state space model can be modified to deal with colored sources of noise. To
do so, we follow the procedure detailed in Candy,  (Section 5.8.2). We now present
a summary of these derivations.
The AO state space model is composed of two equations: one which describes the
temporal evolution of the state variables (AO phase to be corrected) and one which
describes the measurement process with white noise. In a very general form, the first
x[t + 1] = Ax[t] + Bw[t], (1)
where x is the state variables, A describes the temporal evolution, w[t] is temporally
white driving noise and B describes how that noise is used to generate the state. The
second equation is
y[t] = Cx[t] + Du[t] + v[t], (2)
where y[t] is the measurement (by the WFS) of the state, as described by C and v[t]
is temporally white measurement noise. The vector u[t] and matrix D are used to
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Poyneer, L & Veran, J P. Kalman filtering to suppress spurious signals in Adaptive Optics control, article, March 29, 2010; Livermore, California. (digital.library.unt.edu/ark:/67531/metadc835045/m1/6/: accessed July 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.