106 BELL SYSTEM TECHNICAL JOURNAL 



provided /b < f2— fi ■ The spectrum is confined to the region < f < fi — 

 /i . If /ft is equal to/2 — /i so that the output filter passes all the noise of 

 this type, we have 



WUh - /i) == Wnn = "^" .-"'-'"'" /^(T.TV2Pr„) (45) 



This result seems to hold approximately for a considerable range of input 

 spectra. For example, if we assume that the original noise is shaped like 

 an error function about /o , i.e., 



Wnif) = WnV^ g-(f-fo)' (46) 



with / taken from — sc to + =0 with small error for large /« , 



f 



00 



w 



(X)w(X +/) d\ = II'; Va/2:r e-"-'" (47) 



[ df [ wi\)ui\ ^ f)J\ = ir';/2 (48) 



which is in agreement with (45). 



The output of a half-wave linear rectifier contains fundamental compon- 

 ents and all even order modulation products. In general, the amplitudes 

 of the higher order products are small compared with the lower order. In 

 a particular problem some consideration of where the principal products 

 fall in the frequency band is required. The products just considered give 

 a fair approximation for the problem of detection of a radio frequency band 

 of signal and noise followed by audio amplification. Contain other products 

 should also be added to obtain higher accuracy. We have calculated the 

 products of order zero and two; the next ones of importance are the fourth 

 order, since the third order products vanish in a perfectly linear rectifier. 

 The fourth order products in this case which fall in the audio band are of 

 frequency 2po — pr — ps .po + pq — p,- — ps , and p„ + p,j — p,- — ps , where 

 the subscripts n, q, r, s refer to the original noise component frequencies. 

 The latter is, however, less important than the sixth order product 3po — 

 Pq — pr — ps , which involves only three noise components. Expressions 

 for the contributions from these products are given in Appendix III. 



Figure 4 shows computed curves for the noise produced in an audio band 

 by the various components. Curve A is Wsn + Tr„„ and includes what are 

 usually regarded as the principal contributors, the difference frequencies 

 between signal and noise, and between individual noise components. Curve 

 B is obtained by adding to Curve A, the contribution from the fourth order 

 products 2po — pr — ps and po + pq — pr — ps and the sixth order products 

 ipo — pq — pq ~ ps ■ Thus all products which include three or less noise 

 fundamental components are included. The curves are plotted in terms of 



