MATHEMATICAL ANALYSIS OF RANDOM NOISE 319 



In order to obtain the interpretation of w(f)df as the average power dis- 

 sipated in one ohm by those components of /(/) which he in the band /, 

 / 4- df, we set T = in (2.4-7) : 



Imit i [ fit) dl = f wU) dj (2.4-8J 



The expression on the left is certainly the total average power which would 

 be dissipated in one ohm and the right hand side represents a summation 

 over all frequencies extending from to <» . It is natural therefore to in- 

 terpret w(/)(//as the power due to the components in/,/ + df. 



The preceding sections have dealt with the power spectrum w{f) and corre- 

 lation function t^(r) of a very general type of function. It will be noted 

 that a knowledge of ivij) does not enable us to determine the original func- 

 tion. In obtaining iv{J), as may be seen from the definition (2.1-3) or from 

 (2.3-6), the information carried by the phase angles of the various compo- 

 nents of /(/) has been dropped out. In fact, as we may see from the Fourier 

 series representation (2.3-1) of /(/) and from (2.3-6), it is possible to obtain 

 an infinite number of different functions all of which have the same w(/), 

 and hence the same ^{t). All we have to do is to assign different sets of 

 values to the phase angles of the various components, thereby keeping 

 a„ + hn constant. 



2.5 Harmonic Analysis for Random Functions 



In many applications of the theory discussed in the foregoing sections 

 /(/) is a function of t which has a certain amount of randomness associated 

 with it. For example I{t) may be a curve representing the price of wheat 

 over a long period of years, a component of air velocity behind a grid placed 

 in a wind tunnel^ or, of primary interest here, a noise current. 



In some mathematical work this randomness is introduced by considering 

 /(/) to involve a number of parameters, and then taking the parameters to 

 be random variables. Thus, in the shot effect the arrival times /i , /2 , • - • Ik 

 of the electrons were taken to be the parameters and each was assumed to be 

 uniformly distributed over an interval (0, T). 



For any particular set of values of the parameters, /(/) has a definite power 

 sj)ectrum w{f) and correlation function i/'(r). However, now the principal 

 interest is not in these particular functions, but in functions which give the 

 average values of w(/) and ;/'(r) for fixed /and r. These functions are ob- 

 tained by averaging w(/) and i/'(r) over the ranges of the parameters, using, 

 of course, the distribution functions of the parameters. 



By averaging both sides of the appropriate equations in sections 2.1 and 



