k a Write down the state evolution equations That is write down the 4 4 matrix

K a write down the state evolution equations that is

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k ) . (a) Write down the “state evolution equations”. That is, write down the 4 × 4 matrix F k explicitly (it should depend on t k +1 - t k ). (b) The file kalman data.mat contains 499 measurements (in the vector y ) for different times (in the vector t ). Implement a Kalman filter to track the truck, and plot each of your 500 estimates of the position in the ( p x , p y ) plane from ˆ x k | k on a set of axes. To plot discrete points rather than a connected line, use something like plot(pxhat,pyhat,’o’) (c) Construct and solve the (large) system to estimate all 500 positions with knowledge of all of the measurements. That is, find ˆ x 0 | 499 , ˆ x 1 | 499 , . . . , ˆ x 499 | 499 . Plot the corresponding position estimates on another set of axes (maybe using ’x’ for the plot command), and overlay your answer from the previous part (maybe using ’o’ for the plot command again). Comment on what you see, in particular on how it related to your answer in the previous part. 7. (Optional) The file LMS data.mat contains the input (sequence u ) and output (sequence y ) of an unknown linear time-invariant system whose impulse response is supported on n = 0 , . . . , 18. (a) Implement the LMS algorithm and use it to estimate the impulse response. Turn in a plot of your estimated filter coefficients. (You can tell how well you did by seeing how well their convolution with u agrees with y . (b) Try different values of the stepsize μ and comment on the effect it has on the speed of convergence (and whether or not it converges). You might gauge this by how many iterations it takes to get a relative error k h n - h * k 2 / k h * k 2 0 . 01. (c) Now add a little bit of noise to the observations, using yn = y + sigma*randn(length(y),1)}. Comment on the convergence speed when sigma = 0.05 . You might have to adjust your step size to ensure convergence. 4 Last updated 11:14, November 26, 2019
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