328 BELL SYSTEM TECHNICAL JOURNAL 



2.8 Representation of Noise Cureent 



In section 1.8 the Fourier series representation of the shot effect current 

 was discussed. This suggests the representation* 



N 



lif) = 2 {O'n cos a)„/ + 5„ sin w„/) (2.8-1) 



n=l 



where 



co„ = 27r/„ , /„ = 7/A/ (2.8-2) 



On and 5„are taken to be independent random variables which are distributed 

 normally about zero with the standard deviation \/w(/„)A/. w(/) is the 

 power spectrum of the noise current, i.e., wif) df is the average power which 

 would be dissipated by those components of /(/) which He in the frequency 

 range/,/ + dfii they were to flow through a resistance of one ohm. 



The expression for the standard deviation of a„ and 6„ is obtained when 

 we notice that A/ is the width of the frequency band associated with the «th 

 component. Hence w(/„)A/ is the average energ>' which would be dissi- 

 pated if the current 



a„ cos ccj + f>n sin wj 



were to flow through a resistance of one ohm, this average being taken over 

 all possible values of a„ and i„ . Thus 



w{fn)^f = o.n cos uj + 2fl,.6„ COS uj siu ioj + 6„ sin uj = an = bn (2.8-3) 



The last two steps follow from the independence of a„ and b„ and the identity 

 of their distributions. It will be observed that w(f), as used with the repre- 

 sentation (2.8-1), is the same sort of average as was denoted in section 2.5 

 by w(/). However, w{f) is often given to us in order to specify the spectrum 

 of a given noise. 



For example, suppose we are interested in the output of a certain filter 

 when a source of thermal noise is applied to the input. Let A(f) be the 

 absolute value of the ratio of the output current to the input current when a 

 steady sinusoidal voltage of frequency / is applied to the input. Then 



uif) = cA\f) (2.8-4) 



* As mentioned in section 1.7 this sort of representation was used by Einstein and 

 Kopf for radiation. Shottky (1918j used (2.8-1), apparently without explicitly taking 

 the coefficients to be normally distributed. Nyquist (1932) derived the normal distribu- 

 tion from the shot effect. 



