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
 xu   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
2T
1
u  2 u

    x1 
 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.
