08 01 2024 E TS Mida I

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1 MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2022/2023 Prof. Simone Formentin – January 08th, 2024 Surname Name University ID Number Signature ................................ ............................... .....................……… …………. . ....................……… …………….. =========================================================================================================== - Write the solutions (including procedures and intermediate steps) in the blank areas (use the back of the page, if needed) - The number of pages is 4. Additional papers will not be considered. - Clarity, order and precision will be strongly considered for the final evaluation. 1. [Stochastic processes and prediction] Consider the stochastic process 𝑦𝑦(𝑡𝑡) generated according to the following scheme: where 𝑒𝑒(𝑡𝑡) ~ 𝑊𝑊𝑊𝑊(0,1), 𝑤𝑤(𝑡𝑡) ~ 𝑊𝑊𝑊𝑊(0,1) and 𝑒𝑒(𝑡𝑡) ⊥ 𝑤𝑤(𝑡𝑡). a. Is 𝑦𝑦(𝑡𝑡) a stationary process? Why? b. Write the canonical form of the process 𝑦𝑦(𝑡𝑡). c. Compute the optimal 2-step ahead predictor 𝑦𝑦 �(𝑡𝑡+ 2|𝑡𝑡) for the process 𝑦𝑦(𝑡𝑡). d. Compute the variance of the 2-step ahead prediction error. Solutions. a. Yes, the process 𝑦𝑦(𝑡𝑡) is stationary. In fact, it is the linear combination (sum) of two stationary processes: 1. The stationary white noise 𝑤𝑤(𝑡𝑡) 2. The process 𝜂𝜂(𝑡𝑡) generated by 𝑒𝑒(𝑡𝑡), filtered by an asymptotically stable system. b. Inspecting the transfer function that generates the process 𝜂𝜂(𝑡𝑡) it can be immediately seen that it is an all-pass filter, with a non-unitary gain. Thus, the process 𝜂𝜂(𝑡𝑡) is still a white noise with the following properties: 𝜂𝜂(𝑡𝑡)~ 𝑊𝑊𝑊𝑊(0,52) This means that the process 𝛿𝛿(𝑡𝑡) is also a white noise: 𝛿𝛿(𝑡𝑡)~ 𝑊𝑊𝑊𝑊(0,26) Eventually, the process 𝑦𝑦(𝑡𝑡) is 𝑦𝑦(𝑡𝑡) = 𝐺𝐺(𝑧𝑧)𝛿𝛿(𝑡𝑡), where: 𝐺𝐺(𝑧𝑧) = 1 1 + 1 2 𝑧𝑧−1 which is already written in…