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1 MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2022/2023 Prof. Simone Formentin – January 29th, 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. [Analysis of stochastic processes and prediction] Consider the stochastic process defined by the following model: y(t) = 2e(t − 2) − 1 2 e(t − 4) e(t)~WN(0,1) Answer the following questions, adequately justifying your answers. a. Is the process stationary? b. Write the recursive equation of the process, written in canonical form. c. Compute and draw the diagram of the process Power Spectral Density Γy(ω). d. Write the recursive equation of the 1-step ahead predictor y�(t + 1|t). e. Compute the Power Spectral Density of the 1-step ahead predictor Γy�(ω). 2 Solutions. a. The process can be written in the transfer function form: 𝑦𝑦(𝑡𝑡) = 2𝑧𝑧2 − 1 2 𝑧𝑧4 𝑒𝑒(𝑡𝑡) = 𝑊𝑊(𝑧𝑧)𝑒𝑒(𝑡𝑡) Considering that: • The process transfer function is asymptotically stable, as its poles are all located within the unitary circle; • e(t) is a stationary stochastic process; then the process y(t) is stationary. b. The process transfer function is not in canonical form, as: • the numerator is not monic; • the order of the numerator and denominator are not the same. The process equation can be manipulated to get the desired properties as…