# About orthogonal projection theorem for Kalman filtering

In Kalman filtering, the minimum variance estimator can be found by orthogonal projection of X(k) on the space spanned by linear
combinations of observations Y(0), Y(1),...Y(k).
Is this estimator unbiased? How to show, if it is?
Because if I were to look at the best estimator from another way, e.g. a conditional expectation approach i.e. minimum variance estimator E{X(k)|Y(0), Y(1),...Y(k)}, then taking expectation on both sides, I can show that the estimator is unbiased. What about in the orthogonal projection case?
Regards
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Beihai wrote:

Kalman filters are usually only described in detail, in DSP algorithms, in my experience. If you find a good, detailed reference on Kalman filters, that is not deeply embedded in DSP mathematics and techniques, please let me know.
I prefer Kalman Filters (implemented with algebra) but, I'm still looking for that detailed, simplistic reference.
James
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James wrote:

in
and
What do you mean by DSP? (dynamic stochastic programming?).
An old "favourite" reference is the book:
Jazwinski AH (1970) Stochastic Processes and Filtering Theory. Academic Press
A more recent thing, more embedded in a statistical time-series background is :
Harvey AC (1989) Forecasting, structural time series models and the Kalman filter. Cambridge University Press
David Jones
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I think he means Digital Signal Processing (DSP)
beihai

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Beihai wrote:

For a linear system excited by white Gaussian noise, the optimal estimator is the linear system that gives the minimum variance. The solution is unique -- while the implementation may differ there is no other optimal solution that doesn't give exactly the same results.
The Kalman filter in this case is that linear system (it's time-varying, but it's linear).
So I think it's safe to say that the estimate is unbiased.
--

Tim Wescott
Wescott Design Services