MATHEMATICAL ANALYSIS OF ILiNDOM NOISE 117 



The reader might be tempted to associate the coefficient of ^o in (4.2-19) 

 with the continuous portion of the output power spectrum. However, this 

 would not be correct. It appears that the principal contribution of the 

 continuous portion of the power spectrum to Iff is aVo/Tr , just as in (4,2-7) 

 when k is zero. The difference between this and the corresponding term 

 in (4.2-19) seems to arise from the fact tliat the amphtude of the recovered 

 signal is not exactly aQk/ir but is modified by the presence of the noise. 

 This general type of behaAdor might be expected on physical grounds since 

 changing P, say doubling it, in (4.2-7) does not appreciably affect the Iff 

 in (4.2-7) (which is due entirely to the continuous portion of the noise 

 spectrum). The modulating wave may be regarded as slowly making 

 changes of this sort in P. 



4.3 Some Statistical Properties of the Output of a General 

 Non-Linear Device 



Our general problem is this: Given a non-linear device whose output / is 

 related to its input T" by the relation 



I = — [ F{iu)e'''" du (4A-1) 



27r Jc 



which is discussed in Appendix 4A. Let the input V contain noise in addi- 

 tion to the signal. Choose some frequency band in the output for study. 

 \Miat are the statistical properties of the current flowing in this band? 



It seems to be difficult to handle this general problem. However, it 

 appears that the two following results are true. 



1. As the output band is chosen narrower and narrower the statistical 

 properties of the corresponding current approach those of the random noise 

 current discussed in Part III (provided no signal harmonic lies within the 

 band). In particular, the instantaneous current values are distributed 

 normally. 



2. When the input V is confined to a relatively narrow band the power 

 spectrum of the output I is clustered around the (d.c), 1st, 2nd, etc. 

 harmonics of the midband frequency of T'. The low frequency output in- 

 cluding the d.c. is 



Id = AoiR) = ^ [ F{iu)MuR) du (4.3-11) 



2x Jc 



where R is the envelope of T'. 



The envelope of the nth harmonic of the output, when w > 0, is 



A^{R) =- [ F{iu)Jn{uR) du (4.3-1) 



