Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 7.11 Response to Noise Input
In the example we will determine the variance of the output of a system excited by white noise. We will
first input band limited white noise and then extend the results to the case where we apply white noise. We
will alternately evaluate the integrals involved directly and using the formulas in Table 7.6-1. Let us
assume that the input power density spectrum is a constant
 A if   0
Si    
0 otherwise
The output power density spectrum, So(), for a system transfer function G() using Equation 7.6-5 is
syms om T A
So=1/(1+i*om*T)/(1-i*om*T)*A;
pretty(So)
A
------------------------(1 + i om T) (1 - i om T)
So=A/expand((1+i*om*T)*(1-i*om*T));
pretty(So)
A
---------2 2
1 + om T
When the system transfer function is
Gom=1/ (1+i*om*T);
The variance of the output of the filter using Equation 7.6-6 and integrating using Matlab and using mpi to
prevent the numerical substitution for pi by maple
syms omc mpi
sig2=1/2/mpi*int(So,om,-omc,omc);
sig2s=subs(sig2,mpi,pi)
sig2s =
1/pi*atan(omc*T)/T*A
The output to white noise can be found by letting c -> 

maple('assume',sym('T>0'));
limit(sig2s,omc,inf,'left')
ans =
1/2/T*A
We observe that the variance of the output for white noise input becomes A / 2 T. the same result can be
obtain using Table 7.6-1 and Equation 7.6-7
For Example, when G ( s) 
A
the coefficients become
1 sT
c0  A d 0  1 d 1  T
and by substitution we obtain
 
2
A
c02
1 A
I1 

 
2  d 0  d1 2  1  T 2 T
We obtain the same result as the direct integral evaluation.
When the transfer function is more complex Table 7.6-1 should be used. For example when
G ( s) 
A
1  s  T 
2

A
1 2 sT  T2
We perform the substitution using Matlab
syms c0 c1 d0 d1 d2
I2=(c1^2*d0+c0^2*d2)/2/d0/d1/d2;
I2s=subs(I2,[c0 c1 d0 d1 d2],[A^(1/2) 0 1 2*T T^2])
I2s =
1/4/T*A