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1 MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Simone Formentin – July 15th, 2023 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. [True or false] Say whether the following statements are true (T) or false (F), by writing T or F in the small box preceding each statement. Every correct answer is worth 1 point, while 0.5 points are deducted for every wrong answer. No point is credited or subtracted for any unanswered item. 1. Consider a stationary process 𝑣𝑣(𝑡𝑡) with covariance function 𝛾𝛾𝑣𝑣(𝜏𝜏) and spectrum Γ𝑣𝑣(𝜔𝜔) = 9, ∀𝜔𝜔. Then: 𝛾𝛾𝑣𝑣(0) = 3 𝛾𝛾𝑣𝑣(1) = 0 𝛾𝛾𝑣𝑣(𝜏𝜏) is constant ∀𝜏𝜏> 0. 𝑣𝑣(𝑡𝑡) = 3, ∀𝑡𝑡. 2. Consider two uncorrelated stationary processes 𝑣𝑣1(𝑡𝑡) and 𝑣𝑣2(𝑡𝑡) with variances 𝜆𝜆2 and 𝜇𝜇2, respectively, and spectra Γ1(𝜔𝜔) and Γ2(𝜔𝜔). The process 𝑣𝑣(𝑡𝑡) = 2𝑣𝑣1(𝑡𝑡) − 2𝑣𝑣2(𝑡𝑡) has a variance of 4𝜆𝜆2 − 4𝜇𝜇2. The process 𝑣𝑣(𝑡𝑡) = 2𝑣𝑣1(𝑡𝑡) − 2𝑣𝑣2(𝑡𝑡) has a spectrum given by 4Γ1(𝜔𝜔) − 4 Γ2(𝜔𝜔). If both 𝑣𝑣1(𝑡𝑡) and 𝑣𝑣2(𝑡𝑡) are white noises, also 𝑣𝑣(𝑡𝑡) = 2𝑣𝑣1(𝑡𝑡) − 2𝑣𝑣2(𝑡𝑡) is a white noise. If 𝑣𝑣2(𝑡𝑡) = 1, ∀𝑡𝑡, then the variance of 𝑣𝑣(𝑡𝑡) = 2𝑣𝑣1(𝑡𝑡) − 2𝑣𝑣2(𝑡𝑡) is 4𝜆𝜆2. 3. C onsider an autoregressive process 𝑦𝑦(𝑡𝑡) = 𝑎𝑎 𝑦𝑦(𝑡𝑡 −1) +…