MATHEMATICAL ANALYSIS OF RANDOM NOISE 329 



If W is the average power dissipated in one ohm by /(/), 



W = Limit I f l\t) dt = I w{f) df 



= c [ A\f) df 

 Jo 



(2.8-5) 



which is an equation to determine c when W and A(f) are known. 



In using the representation (2.8-1) to investigate the statistical properties 

 of 7(0 we first find the corresponding statistical properties of the summation 

 on the right when the a's and b's are regarded as random variables distrib- 

 uted as mentioned above and / is regarded as fixed. In general, the time 

 / disappears in this procedure just as it did in (2.8-3). We then let iV ^ oo 

 and Af—>Oso that the summations may be replaced by integrations. Fi- 

 nally, the frequency range is extended to cover all frequencies from to oo . 



The usual way of looking at the representation (2.8-1) is to suppose that 

 we have an oscillogram of /(/) extending from / = to / = co . This oscil- 

 logram may be cut up into strips of length T. A Fourier analysis of I(t) 

 for each strip will give a set of coefficients. These coefficients will vary 

 from strip to strip. Our representation (TAf = 1) assumes that this varia- 

 tion is governed by a normal distribution. Our process for finding sta- 

 tistical properties by regarding the a's and i's as random variables while t 

 is kept fixed corresponds to examining the noise current at a great many 

 instants. Corresponding to each strip there is an instant, and this instant 

 occurs at t (this is the / in (2.8-1)) seconds from the beginning of the strip. 

 This is somewhat hke examining the noise current at a great number of 

 instants selected at random. 



Although (2.8-1) is the representation which is suggested by the shot 

 effect and similar phenomena, it is not the only representation, nor is it 

 always the most convenient. Another representation which leads to the 

 same results when the limits are taken is 



rf 



I{.i) = ^ Cn COS {0)nt - ^n) (2.8-6) 



n=l 



where (pi ,(p2 , • • • <p\ are angles distributed at random over the range (0, 2ir) 

 and 



Cn = [2w(Jn)Aff'\ O^n = 2irfn , /„ = wA/ (2.8-7) 



^^ This representation has often been used by W. R. Bennett in unpublished memoranda 

 written in the 1930's. 



