20260212 MIDA1 solution

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MODEL IDENTIFICATION AND DATA ANALYSIS – Module 1, A.Y. 2024/2025 Prof. Luigi Piroddi, Prof. Simone Formentin – February 12th, 2026 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) represented by the following block diagram: z+2 z +(t) 4 + 1 z2 + y(t) where (ꞏ) ~ WN(2, 1). Compute the mean of the process. 1.1) E[ y(t)] = ?  a) 2 × b) 5  c) 1  d) 3 The system can be reformulated as an MA(1) with an additive constant process equal to 5. 1+2z1(t) 5 + + y(t) where (ꞏ) ~ WN(0,1). Since the MA(1) has zero mean, it follows that E[y(t)] = 5. Compute the variance of the process. 1.2) Var[ y(t)] = ?  a) 1  b) 1/2 × c) 5  d) 4 The variance of the process equals that of the MA(1), which amounts to Var[y(t)] = (1+22)ꞏ1 = 5. Compute the correlation function  ~ () for  = 0. 1.3)  ~ (0) = ?  a) 5 × b) 30  c) 10  d) 25 Since () = Var[y(t)] =  ~ () – E[y(t)]2, it follows that  ~ (0) = 5 + 52 = 30. The power spectral density has the form () = a + b cos() + c cos(2) + dꞏ(), where () is a Dirac function. Determine the values of parameters a, b, c, and d. 1.4) a = ?  a) 25  b) 0  c) 10 × d) 5 1.5) b = ?  a) 1  b) 2  c) 5 × d) 4 1.6) c = ? × a) 0  b) 1  c) 2  d) 1 1.7) d = ? × a) 25  b) 5  c) 4  d) 10 Consider first the MA(1) process. Its spectrum can be easily calculated as follows: (z) = (1+2z1)(1+2z)ꞏ1 = 5 + 2(z+z1)  () = 5 + 4 cos() An impulse at the origin (with amplitude equal to 25) must be added, due to the non-zero expected value, resulting in the following expression: () = 5 + 4 cos() + 25ꞏ() 2 Consider the stochastic process y(t) given by the following equation: y(t) = y(t1) –…