18 06 2025 E TS Mida I

Page 1

MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2024/2025 Prof. Luigi Piroddi, Prof. Simone Formentin – June 18th, 2025 1. Multiple choice questions Check with an × the correct answer (wrong answers are penalized with a negative score; missing answers yield 0 points) Consider the stochastic process y(t) defined by the following equations: S: x(t+1) = 0.8 x(t) + v1(t) y(t) = −0.3 x(t) + v2(t) where v1(·) ~ WN(0,4) and v2(·) ~ WN(0,1) are independent white noise processes, uncorrelated with the initial state. 1.1) Which of the following statements is true?  a) The process y(t) is stationary, since v1(·) and v2(·) are independent.  b) The process y(t) is stationary only if the initial value of x(·) is 0. × c) The process y(t) is stationary, since S is an asymptotically stable (linear) system and the inputs are stationary processes.  d) The process is not stationary, since S is an unstable (linear) system. The system is asymptotically stable (eigenvalue in 0.8, inside the unit circle) and fed by stationary processes. Determine the value of the power spectral density Γ(ω) for ω = 0, π/2, π. 1.2) Γ(0) = ?  a) 3.6944  b) 0.5625  c) 1.1111 × d) 10.000 1.3) Γ(π/2) = ? × a) 1.2195  b) 1.1111  c) 0.2195  d) 3.2155 1.4) Γ(π) = ?  a) 0.1111  b) 1.2195 × c) 1.1111  d) 3.1276 In operatorial notation: y(t) = − 0.3 z−0.8 v1(t) + v2(t) Φ(z) = 0.3 z−0.8 0.3 z-1−0.8 4 + 1 = 0.36 1.64−0.8(z+z-1) + 1 = 1.64−0.8(z+z-1) 2−0.8(z+z-1) Γ(ω) = 2−1.6cos(ω) 1.64−1.6cos(ω) → Γ(0) = 10, Γ(π/2) = 1.2195, Γ(π) = 1.1111 Determine the canonical representation of the process, in the form W(z) = 1 + cz-1 1 − az-1 , ξ(·) ~ WN(0,λ2). 1.5) a = ? × a) 0.8000  b) 0.3000  c) 0.4000  d) 0.5000 1.6) c = ?  a) -2.0000 × b) -0.5000  c) -0.3644  d) 0.3333 1.7) λ2 = ?  a) 3.2927 …