r5_randomproc - 13.42 Design Principles for Ocean Vehicles...

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13.42 Design Principles for Ocean Vehicles Prof. A.H. Techet Spring 2005 1. Random Processes A random variable, x () ζ , can be defined from a Random event, , by assigning values x i to each possible outcome, A , of the event. Next define a Random Process, x ( , t ) , a i function of both the event and time, by assigning to each outcome of a random event, , a function in time, x 1 t , chosen from a set of functions, x i ( t ) . A 1 p x () t 1 1 A 2 p x 2 2 t (6) M M M A p x t n n n This “menu” of functions, x i ( t ) , is called the ensemble (set) of the random process and may contain infinitely many x i ( t ) , which can be functions of many independent variables. EXAMPLE : Roll the dice: Outcome is A , where i = 16 is the number on the face of the : i dice and choose some function t x i = t i (7) to be the random process.
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3.1. Averages of a Random Process Since a random process is a function of time we can find the averages over some period of time, T , or over a series of events. The calculation of the average and variance in time are different from the calculation of the statistics, or expectations, as discussed in the previously. TIME AVERAGE (Temporal Mean) lim Mx () } = T →∞ T 1 0 T x () d t = { i t i t x t (8) TIME VARIANCE (Temporal Variance) t Vx i l i m 1 T {( t )} = T →∞ T [ x ( t ) M { x ( t ) } ] 2 d (9) i i 0 TEMPORAL CROSS/AUTO CORRELATION This gives us the “correlation” or similarity in the signal and its time shifted version.
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.22 taught by Professor Alexandratechet during the Spring '05 term at MIT.

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r5_randomproc - 13.42 Design Principles for Ocean Vehicles...

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