Exams Simulation

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Versione 3.3 Measurements Q: Is it possible to calculate the stationarity and the ergodicity from a single random signal? If yes, what is the mathematical formulation to do this and what is the physical meaning of stationarity and ergodicity? If not, explain why. Justify all the answers. A: A random signal can be seen as result of a stochastic process. The process could be stationary and even ergodic. Stationarity means that the statistic properties of the process do not vary with time (but the single signal does), ergodicity means that the acquisition position of the k-th signal is irrelevant (independence from the space domain). Ergodicity is a stricter property than stationarity; this means that only stationary processes can be ergodic. When only one signal is available, it is not possible to evaluate ergodicity. At least two records from different locations are required in order to assess the space invariance. To evaluate stationarity it is possible to split the record into sub records and perform for all of them the following calculations. The signal can be treated as stationary if results are not function of t1. 𝜇 = 1 𝑇 ∫ 𝑥𝑘(𝑡)∙ 𝑑𝑡 𝑡1+𝑇 𝑡1 𝑅𝑥𝑥 = 1 𝑇 ∫ 𝑥𝑘(𝑡)∙ 𝑥𝑘(𝑡 + 𝜏)∙ 𝑑𝑡 𝑡1+𝑇 𝑡1 T is the length of the k-th sub record. Q: Define the kurtosis coefficient and explain its use. A: The kurtosis is defined as follows. 𝛼4 = 1 𝑁 ∙ ∑(𝑥𝑖 − 𝜇)4 𝜎4 − 3 A normal distribution has a null kurtosis (due to the presence of -3). Positive (negative) values mean the function is sharper (flatter) than the Gaussian. In our field of interests, it is used to spot the presence of spikes and anomalous variation of signal shape (this coefficient tracks the time history of the record). Q: Calculate the cross correlation function at τ = 0.35s and τ = 2.20s between the following signals: 𝑥 =…