118 BELL SYSTEM TECHNICAL JOURNAL 



The mathematical statement is 



00 



/ = Z) MR) cos {noirnt + nd) (4.3-9) 



where fm = com/(2Tr) is the representative mid-band frequency of V and 6 

 is a relatively slowly varying phase angle. The results of Sections 4.1 

 and 4.2 are special cases of this. 



Middleton's result that the noise power in each of the output bands (in 

 the entire band corresponding to a given harmonic) depends only on \\- = 

 ^0 and not on the spectrum of Vu , where V^ is the noise voltage component 

 of V, may also be obtained from (4.3-9). We note that the total power 

 in the n^^ band depends only on the mean square value of its envelope 

 An{R), and that the probabiUty density of the envelope R of the input in- 

 volves Vn only through xpo . 



The argument we shall use in discussing the first result is not very satis- 

 factory. It runs as follows. The output current / may be divided into two 

 parts. One consists of sinusoidal terms due to the signal. The other con- 

 sists of noise. We shall be concerned only with the latter which we shall 

 call In . The correlation between two values of In separated by an interval 

 of time approaches zero as the interval becomes large. Let t be an interval 

 long enough to ensure that the two values of In are substantially 

 independent. Choose an interval of time T long enough to contain many 

 intervals of length r. Expand In as a Fourier series over this inten-al. 

 We have 



Lv = 2 + 2^ p« co^ ^^ + *« ^^^ ^Y~ 



71=1 L — 



(4.3-2) 

 1 Jo 



dt 



Let the band chosen for study be/o — - to/o + - and let 



T (fo -f) = 'h, T (fo + = ^2 (4.3-3) 



where Wi and «2 are integers. The number of components in the band is 

 (w2 — wi). We suppose /3 is such that this is small in comparison with T/t. 

 The output of the band is 



Jn = Z! \anCos^t + b„ sin -^ (4.3-4) 



