246 TECHNIQUES FOR SIGNAL AND NOISE ANALYSIS 



The integral of the probability density function between any two values 

 will give the fraction of the elements of the process which lie between those 

 values. As an example, we consider the most common type of noise process, 

 a Gaussian process, so called because the probability density is Gaussian 

 or normal in form : 



Probability density function of a Gaussian noise process 



V2 



— exp;r-^ 



(5-22) 



This process has an average value of zero and a variance or mean square 

 value of 0-^. Most important, the probability density is independent of time. 

 For most of the noise processes which are of importance in engineering 

 applications the statistical parameters are independent of time, and such 

 processes are therefore called stationary processes. 



Fig. S-S shows the probability density function for a Gaussian process. 

 Most of the elements of the process have values in the neighborhood of the 



exp (x2/2(t2) 



Fig. 5-5 Gaussian Probability Density Function. 



origin. Only a very few of the noise functions will be very large or very 

 small at any particular time. If the process is stationary, the values of the 

 component functions will have the same distribution at any time. 



As previously noted, Gaussian noise processes are very common in 

 physical applications. They can be generated by the superposition of a 

 large number of time functions with random time origins. An example is 

 the shot noise generated in an electron tube. The random times of arrival 

 of electrons at the plate produce the shot noise fluctuations in the plate 

 current, which has the properties of Gaussian noise. A mathematical 

 example of Gaussian noise is produced by the superposition of a large 

 number of sinusoids of different frequencies and random phases. 



Also very useful and important is the joint probability density function 

 of values of the process at two different times. For a stationary Gaussian 

 process with zero mean, this joint probability density will have the following 

 form. 



Second-order probability density function of a Gaussian noise process 



27ra-Vl 



exp 



■vi- + 2p.Vi.V2 - 

 2<tH1 - p') 



(5-23) 



