Full exam for Model Identification and Data Analysis in the Computer Engineering degree programme at Politecnico di Milano. The document covers: 1 MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Simone Formentin – July 15th, 2023 Surname Name University ID Number Signature ................................ ............................... .....................……… ………… .. ....................………
Full exam for Model Identification and Data Analysis in the Computer Engineering degree programme at Politecnico di Milano. The document covers: 1 MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2023/2024 Prof. Simone Formentin – July 15th, 2023 Surname Name University ID Number Signature ................................ ............................... .....................……… ………… .. ....................………
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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) +…
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