5-4] 



RANDOM NOISE PROCESSES 



245 



approximations to the constant function will give different impulse function 

 representations. A third representation of an impulse function can be 

 obtained in this way by using the triangular approximation shown in Fig. 

 5-3. As yf — > 00, this triangular function obviously approaches a constant. 



-2A 2A " 



Fig. 5-3 Triangular Approximation to a Constant Spectrum. 



The limit of the Fourier transform of this function will give the desired 

 representation: 



It is apparent that there are a variety of specific representations of 

 impulse functions. A familiarity with the forms of the representations, 

 so that they may be recognized when they arise during the course of an 

 analysis, is useful. A case of this kind occurs in Paragraph 5-5, where in an 

 example of a noise process the expression in Equations 5-21 turns up as part 

 of the power density spectrum (Equation 5-40). 



5-4 RANDOM NOISE PROCESSES 



In describing noise mathematically, it is useful to visualize a very large 

 group or ensemble of noise generators with outputs x{t), x'{t), x"(t), .... 

 The output of a specific noise 

 generator may be any one of the en- 

 semble functions with equal proba- 

 bility. The totality of all possible 

 noise functions is referred to as a 

 random process. Such processes are 

 described in terms of their statistical 

 characteristics over the ensemble. 

 Fig. 5-4 shows a few of the elements 

 of a noise process. At any time / the 

 mean value, the variance, or other 



statistical parameters can be determined. These parameters can all be 

 derived from the probability density Junction of the process at that time 

 which describes the distribution of values of the elements of the process. 



Fig. 5-4 Elements of a Noise Process 



