SOLVED CH5115: Parameter and State Estimation Assignment 2

  1. (a) Determine if the random process v[k] = A cos2 (2πf k + φ), where φ is a constant but A is a random variable with zero mean and unit variance, is covariance stationary. (b) The random walk process v[k] = v[k − 1] + e[k] is known to be variance non-stationary. Assuming v[0] = 0, prove this result. Verify your finding numerically using Matlab. 2. A process evolves as y[k] = y ? [k] + e[k], where y ? [k] = b 0 2 q −2 1 + f 0 1 q−1 u[k], u[k] is a known signal and y[k] is the measured version of y ? [k]. The measurement noise is e[k] ∼ WN(0, σ2 e ) and u[k] ∼ WN(0, σ2 u ). Assume σeu[l] = 0, ∀l. (a) Develop expressions for σ 2 y , σyy[1], σyu[1], and σyu[2] in terms of the variances of u[k] and the white-noise sequences, i.e., σ 2 u and σ 2 e respectively. (b) Generate N = 500 observations of y[k] with σ 2 u = 2. Adjust σ 2 e such that the SNR σ 2 y ? /σ2 e is set to 10. Estimate the quantities (variance, auto-covariance and cross-covariance) in (2a) and compare their closeness with the theoretical answers in (2a). 3. For the series given in a2_q3.mat, (a) Determine the presence of any integrating effects. (b) Fit a suitable ARIMA model. Report all the necessary steps and the final model. 4. (a) For a GWN process y[k] ∼ N (µ, σ2 ), where 0 ≤ µ < ∞, derive the ML estimate and Fisher information of µ given N observations and known σ 2 . (b) Consider the linear regression problem Y = aX +b+ε. Determine the Fisher information of parameters a and b contained in N observations {(y[k], x[k])} N k=1 assuming X is free of randomness and ε ∼ N (0, σ2 e ). Verify your analytical answer (for the ML estimate) with simulation in Matlabby plotting the likelihood functions and locating the maximum. Choose N = 100, σ 2 e = 1, a = 2, b = 3 and µ0 = 1 (true value).