Kalman Filter to Estimate a Scalar Random Constant

To estimate a scalar random constant denoted by a and we have the ability to take measurements at time k 0,1, 2, with observation noise v_k

Suppose we want to estimate a scalar random constant denoted by a and we have the ability to take measurements at time k  0,1, 2, observation noise process vk . corrupted with the additive noise

⦁ Write the time-varying Kalman filter formulation and the evolution of the Kalman

gain K at each instant k assuming Gaussian noise process v with R (k)  R|k| .

k k v

⦁ Fix a .134, R  1 and the initial estimate to be 0. Run the discrete time time-varying Kalman filter from k  0 to k 1000 for three values of the process noise parameter Q  102 ,104 ,106 , for three values of P  10,1,101 and for three values of the colored noise parameter   0.99, 0.5, 0.1and plot the estimate aˆk as a function of the index k . Comment on the results. What roles do the three parameters Q, P , play a role in the online estimation process?