As this is concerning signal processing i guess this is the right place to post the question. Im Trying to learn how to use kalman filters. Ive reached some form of verry basic understanding of the state-space model but im still kindof confused. What im trying to do now is to understand an example that is using kalman recursion to find the steady state kalman gain.(adsbygoogle = window.adsbygoogle || []).push({});

We have an AR(1) process described by: x(n) = 0.5x(n-1) + w(n), where w(n) is white-noise with variance 0.64. we are observing a process: y(n) = x(n)+ v(n), where v(n) is white-noise with variance 1.

The state-space model becomes:

x(n) = 0.5x(n-1) + w(n)

y(n) = x(n) + v(n)

and we see that A(n-1) = 0.5, B(n) = 1 and C(n) = 1. From the variances we have Qw=0.64 and Qv=1.

We have the initial conditions: x'(0|0) = 0 and E{e^2(0|0)} = 1, where e(0|0) = x(0) - x'(0|0). (' = estimate) and im trying to use these formulas to perform the recursion: (im skipping some matrix-related stuff since the matrices in this example is just single numbers so transposing isnt doing anything)

x'(n|n-1) = Ax'(n-1|n-1)

P(n|n-1) = AP(n-1|n-1)A + Qw

K(n) = P(n|n-1)C[CP(n|n-1)C+Qv]^-1

x'(n|n) = x'(n|n-1) + K(n)[y(n) - Cx'(n|n-1)]

P(n|n) = [I-K(n)C]P(n|n-1)

We start with P(0|0) = E{e^2(0|0)} = 1.

Ok. Now to my problem. How do i get y(n) ? I get stuck on the first iteration when I want to calculate x'(1|1) and i need y(1), how to i get it?

what is I ? on the last formula-row "P(n|n) = [I-K(n)C]P(n|n-1)"? In the example it is equal to 1, but

where do the 1 come from?

Also if anyone have any good (simple!) tutorial suggestion on the net about kalman-filtering that would be appreciated.

Thanks alot!

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# Kalman recursion

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