Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 5.7 Correlation Functions for a Poisson Process
In this example we will compute the correlation function for a Poisson process and the telegrapher
waveform which is based upon a Poisson density distribution. We will also show that we may use the
independent non-overlapping interval property of a Poisson process to compute the two-dimensional
density distribution for a Poisson random process. The computation of the two-dimensional distribution
function is based upon the one-dimensional density functions. The density function for a Poisson random
variable is
Pk 
  t k e  t
k!
The number of points in the interval t3 - t1, t3 > t1 is
expression
Pxt 3   xt 3   k  
xt 3   xt 3   k and is computed by the following
  t 3  t1 k
k!
e  t3 t1 
The mean and variance for a Poisson process are computed based upon the results of example 5.2
Ext 3   xt 3     t 3  t1 
and


E xt3   xt 3   2  t3  t1     t3  t1 
2
2
The correlation function is computed by introducing two new times t4 and t2 where t4 > t3 > t2 >t1. The
two intervals t4 - t2 and t3 - t1 are overlapping in region t3 to t2. The correlation function or
is computed by noting that
E[ (x(t4) - x(t2)) (x(t3) - x(t1)) ]
x(t4) - x(t2) = (x(t4) - x(t3)) + (x(t3) - x(t2))
x(t3) - x(t1) = (x(t3) - x(t2)) + (x(t2) - x(t1))
Now the entire parenthesized intervals on the right-hand side of the equation shown above are nonoverlapping. When intervals overlap these overlapping intervals contain the same elementary exponential
processes. We have already shown that a Poisson process for any time interval is the sum of these
exponential processes. When the intervals do not overlap they do not share any exponential time random
processes and hence the numbers of events in each of these Poisson intervals are independent. The twodimensional probability density function is
P[ x(t4) - x(t3) = m, x(t3) - x(t2) = n ] = P[x(t4) - x(t3) = m] P[x(t3) - x(t2) = n]
The right hand side of the equation is a product of two distribution functions because the time intervals do
not overlap. The expected value or correlation of the two intervals is just the product of the expected
values of each of the intervals. The expected value for overlapping intervals is computed by dividing up
the intervals and forming the product as
E[ (x(t4) - x(t2)) (x(t3) - x(t1)) ] = E[ (x(t4) - x(t3)) + (x(t3) - x(t2)) (x(t3) - x(t2)) + (x(t2) - x(t1))]
Forming the product and substitution of the single interval values we have
E[ (x(t4) - x(t2)) (x(t3) - x(t1)) ] =
syms l t4 t3 t2 t1
f=l*(t4-t3)*l*(t3-t2)+l^2*(t3-t2)^2+l*(t3-t2)+l*(t4-t3)*l*(t2t1)+l^2*(t3-t2)*(t2-t1);
pretty(f)
2
2
2
2
l (t4 - t3) (t3 - t2) + l (t3 - t2) + l (t3 - t2) + l (t4 - t3) (t2
- t1)
2
+ l
(t3 - t2) (t2 - t1)
Collecting terms we have
E[ (x(t4) - x(t2)) (x(t3) - x(t1)) ] =
syms x
collect(subs(factor(f),l,x))
ans =
(t4*t3-t3*t2-t4*t1+t2*t1)*x^2+(t3-t2)*x
The first term can be recognized as (t4-t2)(t3-t1) by expanding
expand((t4-t2)*(t3-t1))
ans =
t4*t3-t3*t2-t4*t1+t2*t1
The latter expression can be expanded to verify the collection process. If we take the overlapping region t 4
> t3 and t2 = t1 = 0 we obtain, after renaming t4 -> t2 and t3 -> t1
Ext     t
  t 2  2  t1  t 2
Ext1 xt 3   Rt1 , t 3   
2
   t1    t1  t 2
t1  t 2
t 2  t1
Using the min function we can write
Rt1 , t 3     min t1 , t 2   2  t1  t 2
The correlation Function for Poisson Process can now be plotted in two-dimensional space and we have
replaced the min function by the Matlab.
function y=f5_7(t1,t2,lam)
% correlation function for a Poisson process
[sx,sy]=size(t1);
for i=1:sx
for j=1:sy
mn(i,j)=min([t1(i,j) t2(i,j)]);
end
end
y=lam*mn+lam^2*t1.*t2;
[x,y]=meshgrid(0:.1:2);
R=f5_7(x,y,2);
mesh(x,y,R)
A is the autocorrelation function
xp=0:.1:2;
A=2*xp+2^2*xp.*xp;
plot(xp,A)
20
18
16
14
12
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
The autocorrelation function of the telegrapher waveform is now calculated. This telegrapher process is
formed by creating the waveform with +1 at the Poisson event interval if the number of events from t = 0 to
tk is even and -1 if the number of events is odd. The number of events in interval t k is Poisson distributed
and therefore the probability that we have the same value at t1 to t2 is
Pxt1   1 xt 2   1  Pxt1   1 xt 2   1
p0   p2  p4     e
  t
   t 2   t 4

 1 

 
2!
4!


Similarly for the opposite sign
Pxt1   1 xt 2   1  Pxt1   1 xt 2   1
   t   t 3   t 5

p1  p3  p5    e  t  


 
3!
5!
 1

The expected value of x(t) is the value x probability that it has this value or
    t 2

   t   t 3

Ext   e  t  1  1 
   1  

 
2!
3!
 

 1

syms lam k t
kfac=sym('k!');
tel=exp(-lam*t)*symsum((-lam*t)^k/subs(kfac,k,k),k,0,inf)
tel =
exp(-lam*t)^2
The correlation function can be computed by noting that P(x 1,x2) can be expanded using the conditional
expression
Px1 , x2   Px1 x2  Px2 
Since x1 and x2 can only take two values we have four cases
Ext1 xt 2   12  Px1  1 x2  1 Px2  1   1  Px1  1 x2  1 Px2  1
1   1  Px1  1 x2  1 Px2  1   1  1  Px1  1 x2  1 Px2  1 
2
e
  t 2
   t 2 2
 1 
2!

 2    t 2  t1 2


  
1  1 

2!
  t2 t1   


   e


3

 1  1     t 2  t1     t 2  t1   



1
3!



   t 2   t 2 3
 e  t2  

1
3!

 2    t 2  t1 2


  
1  1 

2!
  t2 t1   


   e


3

 1  1     t 2  t1     t 2  t1   



1
3!



From the above calculation we have computed P[x(t2)] and the P(x1| x2) is the even expression when x1 =
x2 = 1 or -1 and the odd expression when x1  x2 . Combining the results from the two terms in the above
expression and recognizing that the bracketed terms are just the series expansions of the exponential terms
we have
Ext1 xt 3   e
2  t1 t 2
The absolute value can be used because the results depend only on the positive interval between t 1 and t2
and therefore we may use the absolute value sign to keep the interval positive if t 2 > t1.
Strictly speaking this derivation assumes at t =0 the telegrapher waveform starts at +1or A = 1. When we
allow the starting value to be +1 or -1 with equal probability = 1/2 the average values of E[x(t)] and R()
must be averaged again to remove the positive value assumption. This can be accomplished by the results
of Section 5.5 or
Ext   EExt  x0  A  1  e 2t  1  e 2t  0
Similarly the correlation function was computed for x(t) but because of the initial value of x(t) we can
change variable to y(t) = A x(t) and let A be a random variable with +1 ,-1 with equal probability
E[ y(t1) y(t2) ]= E[ A2 x(t1) x(t2)] = E[A2] E[ x(t1) x(t2) ]
 1
Since E A
2
2
 1 2   1 1 2  1 we have
2
 1
2 1
 2  
 2  
R   12    1    e
e
2
 2
We see that the expectation expressions are constant and depend only upon = (t1 - t2) respectively, the
telegrapher process is wide sense stationary in the first and second order.
