MATHEMATICAL ANALYSIS OF RAXDOM NOISE 131 



The sinusoidal terms of 7 are obtained by squaring 



(3(1 + k cos pt) cos qt 



and multiplying by a. The remaining portion of / has a continuous power 

 spectrum given by 



fQ'[iv(f- 



Wcif) =cx'Q'\ wif - U) + w{f + /,) 





w{x)w(f — a:) dx 



00 



where /p denotes p/lir and/g denotes (7/27r. 



4.6 Two Correlation Function Methods 



As mentioned in Section 4.4 these methods for determining the output 

 power spectrum are based on finding the correlation function ^(r) for the 

 output current. From this the power spectrum, W(f), of the output cur- 

 rent may be obtained from (2.1-5), rewritten as 



Wif) = 4 [ ^(t) cos 27r/T dr (4.6-1) 



•'0 



It will be recalled that W(f)Af may be regarded as the average power which 

 w^ould be dissipated by those components of / in the band/,/ + A/if / were 

 to flow through a resistance of one ohm. 



The input of the non-linear device is taken to be a voltage V(t). It may, 

 for example, consist of a noise voltage F.v(/) plus sinusoidal components. 

 The output is taken to be a current I(t). The non-Unear device is specified 

 by a relation between V(t) and /(/). In this work /(/) at time / is assumed 

 to be completely determined by the value of V{t) at time /. 



Two methods of obtaining ^(r) will be described. 



(a) Integrating the two-dimensional probability density of V(t) and 

 V(t -f r) over the values allowed by the non-linear device. This 

 method, which is especially direct when applied to noise alone through 

 rectifiers, was discovered independently by Van Vleck and North. 



(b) Introducing and using the characteristic function, which for the sake 

 of brevity will be abbreviated to ch. f., of the two-dimensional prob- 

 ability distribution of V(t) and V{t -f r). 



