Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 6.4 Correlation Function for an Impulse Train
In the example we will derive of the correlation function for shot noise. This procedure is a two step
process as shown in the diagram below. The first part of the process, the differentiation of the Poisson
process, will be covered in this example and the transformation by the linear process represented by h(t)
will be covered in Example 6.7.
Shot noise is model by a process that is expressed by the block diagram shown below. We first apply a
Poisson noise source to a differentiator and then input the results of this process to a block that represents a
transfer function that has an impulse response, h(t), or
For the differentiation process, we now notice that
z t     t  t i 
i
When the input process is a Poisson process of example 5.3 or
x t    ut  t i 
i
The model for computation of the correlation function is shown below and we need the non-stationary
representation of the differentiation because the Poisson process is non-stationary
It is known that the correlation function for a Poisson process contains a min function. Let up plot the min
function as a function of t2 with t1=1
t1=1;
t2=0:.05:4;
for i=1:length(t2)
ym(i)=min(t2(i),t1);
end
plot(t2,ym)
axis([0 4 0 1.5])
xlabel 't2 - sec'
ylabel 'min(t1,t2)'
1.5
min(t1,t2)
1
0.5
0
0
0.5
1
1.5
2
t2 - sec
2.5
3
3.5
4
From the plot the of min function we find its derivative equal to min(t1,t2)/  t2 = u(t2)-u(t2-t1) for t2 > 0.
This plot shows that the derivative of a min function with respect to t 2 is a step function, u(t1 - t2). The
correlation function for a Poisson process is
R XX t1 , t 2   2  t1  t 2    min t1 , t 2 
and the derivative with respect to t2 of this step function is an impulse function or completing these two
differentiation operations upon RXX we have
RZZ t1 , t 2   2     t1  t 2 
The correlation function of the output variable, Z, is stationary in the wide sense since R ZZ is only a
function of. The input process, x(t), is, however, not stationary as we have already noted.
