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Unformatted text preview: 4 RANDOM PROCESSES 23 4 RANDOM PROCESSES From the essential aspects of probability we now move into the time domain, considering random signals. For this, assign to each random event A i a complete signal, instead of a single scalar: A i −→ x i ( t ). The set of all the functions that are available (or the menu) is call the ensemble of the random process. An example case is to roll a die, generating i = [1 , 2 , 3 , 4 , 5 , 6] and suppose x i ( t ) = t i . In the general case, there could be infinitely many members in the ensemble, and of course these functions could involve some other variables, for example x i ( t, y, z ), where y and z are variables not related to the random event A i . Any particular x i ( t ) can be considered a regular, deterministic function, if the event is known. x ( t o ), taken at a specific time but without specification of which event has occurred, is a random variable. 4.1 Time Averages The theory of random processes is built on two kinds of probability calculations: those taken across time and those taken across the ensemble. For time averages to be taken, we have to consider a specific function, indexed by i : 1 T m ( x i ( t )) = lim x i ( t ) dt (mean) T →∞ T V t ( x i ( t )) = lim 1 T [ x i ( t ) − m ( x i ( t ))] 2 dt (variance on time) T →∞ T 1 T R t ( τ ) = lim [ x i ( t ) − m ( x i ( t ))][ x i ( t + τ ) − m ( x i ( t ))] dt (autocorrelation) . i T →∞ T The mean and variance have new symbols, but are calculated in a way that is consistent with our prior definitions. The autocorrelation is new and plays a central role in the definition of a spectrum. Notice that is an inner product of the function’s deviation from its mean, with a delayed version of the same, such that R (0) = V t . Consider the roll of a die, and the generation of functions x i ( t ) = a cos( iω o t ). We have T m ( x i ( t )) = lim a cos( iω o t ) dt = 0 T →∞ V t ( x i ( t )) = lim 1 T a 2 cos 2 ( iω o t ) dt = a 2 T →∞ T 2 1 T a 2 R i t ( τ ) = lim a 2 cos( iω o t ) cos( iω o ( t + τ )) dt = cos( iω o τ ) . T →∞ T 2 In this case, the autocorrelation depends explicitly on the event index i , and has a peak of a 2 / 2 at iω o τ = 2 πk , where k is an integer. These values for τ are precisely separated by the period of the i ’th harmonic in the ensemble. When the functions line up, we get a positive R t ; when they are out of phase, we get a negative R t . 4 RANDOM PROCESSES 24 4.2 Ensemble Averages The other set of statistics we can compute are across the ensemble, but at a particular time....
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.
- Spring '05