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Extra info for Stationary Stochastic Processes for Scientists and Engineers
Example text
13 shows four realizations of the same process. 3 Power and average power In many applications of stationary processes, the variance V[X (t)] = r(0) or sometimes E[X (t)2 ] are quantities of main interest. For example, if X (t) represents the potential difference (“voltage”) over a resistance R, the electric power in the circuit is P = X (t)2 /R, and on the average it is proportional to E[X (t)2 ]. As another example, we can take X (t) as the varying height of the sea surface around the mean sea level.
The two-dimensional distributions can say something about the smoothness of the output. 17. If, on the other hand, there is a strong positive correlation between neighboring X (t1 ) and X (t2 ), the variations will be slower, but of the same magnitude, as in the lower plot. 13. One also needs the higher dimensional distributions to link several X (t1 ), X (t2 ), . . , X (tn ) to each other. 17 Noise with small (top) and large (bottom) positive temporal dependence. 18 Histogram with normal density (left) for wave measurements (right).
This is a “moving average” process, which is a topic in Chapter 7. 1, we can calculate its mean value and covariance function. 5 ·Ut−1 ] = 0. For the covariance function, we have to work harder, and to keep computations under control, we do separate calculations according to the size of t − s. 25σ 2 , where we used the fact that V[Ut ] = V[Ut−1 ] = σ 2 , and that Ut and Ut−1 are independent, so C[Ut ,Ut−1 ] = C[Ut−1 ,Ut ] = 0. 5σ 2 . The case s = t − 1 gives the same result, and for s ≥ t + 2 or s ≤ t − 2, one easily finds that r(s,t) = 0.