Assume that a random variable x E N(0,X.) is repeatedly measured. The measurement process involves noise effects. Assume that the system can be described by y(k) = x + v(k) where y(k) is the k-th measurement, and the v(k) is the measurement noise (zero mean) which a white Gaussian process independent from x, i.e. E[v(k)v(j)] = V8kj, E[x V(k)] = 0 a) Obtain the general form of the least square estimate (k) = E[x]y(0), y(1), ..., y(k)]. Also, find the general form of the estimation error covariance X(k) := X3(k)x(k). Hint: Using the recursive formula for the MMSE estimator, (k) can be written as (k) = f(k 1) + Et(k 1)]K|0,1...k1] Firstly, show that ijo = f(0) + v(1), 2|0,1 = (1) + v(2), ..., Secondly, show that Xx(o)ujo = Xx(0)x(0), and Xyzjoijo = Xx(o)&(0) + V = X(0) + V Xx(1)2|0,1 = Xx(1)x(1), and Xz|0,12|0,1 = Xx(1)8(1) + V = X(1) + V : Thirdly, from the above results, show that (k) = f(k 1) + F(k)(y(k) (k 1)) = (1 F(k))&(k 1) + F(k)y(k) where F(k) is some correction term that can be represented by X(k 1) and V (find it). From this dynamics equation, find the dynamics equation for t(k) and X(k). After that, try to deduce the general forms of (k) and X(k). b) Show that in the limit of Xo approaching to co, i.e. no prior information on x, f(k) and the estimation error covariance, respectively, are asymptotic to lim (k) = (y(0) + y(1) + ... + y(k)), Xo90 k +1 V lim E[||(k)||2] = Xo4 k+1 th Assume that a random variable x E N(0,X.) is repeatedly measured. The measurement process involves noise effects. Assume that the system can be described by y(k) = x + v(k) where y(k) is the k-th measurement, and the v(k) is the measurement noise (zero mean) which a white Gaussian process independent from x, i.e. E[v(k)v(j)] = V8kj, E[x V(k)] = 0 a) Obtain the general form of the least square estimate (k) = E[x]y(0), y(1), ..., y(k)]. Also, find the general form of the estimation error covariance X(k) := X3(k)x(k). Hint: Using the recursive formula for the MMSE estimator, (k) can be written as (k) = f(k 1) + Et(k 1)]K|0,1...k1] Firstly, show that ijo = f(0) + v(1), 2|0,1 = (1) + v(2), ..., Secondly, show that Xx(o)ujo = Xx(0)x(0), and Xyzjoijo = Xx(o)&(0) + V = X(0) + V Xx(1)2|0,1 = Xx(1)x(1), and Xz|0,12|0,1 = Xx(1)8(1) + V = X(1) + V : Thirdly, from the above results, show that (k) = f(k 1) + F(k)(y(k) (k 1)) = (1 F(k))&(k 1) + F(k)y(k) where F(k) is some correction term that can be represented by X(k 1) and V (find it). From this dynamics equation, find the dynamics equation for t(k) and X(k). After that, try to deduce the general forms of (k) and X(k). b) Show that in the limit of Xo approaching to co, i.e. no prior information on x, f(k) and the estimation error covariance, respectively, are asymptotic to lim (k) = (y(0) + y(1) + ... + y(k)), Xo90 k +1 V lim E[||(k)||2] = Xo4 k+1 th