Random Signals for Engineers using MATLAB and Mathcad Copyright 1999 Springer Verlag NY Example 6.3 Variance of an Averaging Process In the example we will compute the variance of the output process when an averaging transformation is applied to the input process. The input random process is assumed to be an exponentially correlated. An example of such a process would be the telegraph waveform. In example 5.7 we have shown the correlation function of such a process to be to be in the form RXX e 2 The transformation process that we assume is an integral transform. Since the averaging interval is fixed the output of the processing is a single random variable. The values calculated for this output random variable are the mean and variance. The output random variable is given by the following integral T 1 y xu du 2 T T The mean value of the output is zero since the input has a zero mean value. The output variance is computed using Equation 6.3-12 as 2T 1 u 2 u x1 du e T 0 2 T 2 Y Performing the integration using Matlab we obtain syms u lam T vary=1/T*int((1-u/2/T)*exp(-2*lam*u),u,0,2*T); 'siy2=' pretty(vary) ans = siy2= exp(-4 T lam) 4 T lam - 1 1/8 ------------- + 1/8 ----------2 2 T lam T lam ----------------------------------T Collecting on the subexpressions and simplifying we obtain pretty(factor(vary)) exp(-4 T lam) + 4 T lam - 1 1/8 --------------------------2 2 T lam This result can also be used to show that the telegraph process is ergodic in the mean as we will develop in Section 6.8. We will also show that the integration process can be made into an averaging process and the correlation function can be evaluated for the output of this running averaging process. This will also be done in Example 6.8.