DETECTION OF TWO MODULATED ir.H7-:.S 5 



It is to be observed thai eacli of I lie lernis of this series, except the 

 first, contains time in the denominator and hence further expansions 

 are necessary. The denominators of the various terms can be ex- 

 panded by the binominal theorem in such a way as to put all the 

 expressions containing time in the numerators, the expansions being in 



powers of 



{ME cos pt + »J& cos qt)l{E + e). 



By the proper trigonometric transformations it is possible to reduce the 

 final expression for S to frequencies in p, q, u and the sums and differ- 

 ences of the various multiples of these quantities. An additional 

 discussion of this analysis is given in an appendix. In order that the 

 various series involved may converge with a manageable degree of 

 rapidity it is necessary to limit the relative amplitudes of the interfering 

 carriers and the degrees of modulation as well. Consecjuently the 

 solutions are restricted to intensities of the interfering carrier of 0.1, or 

 less, of the desired carrier and to degrees of modulation of either signal 

 ranging from 0.1 to .5. These limits are suitable also because we are 

 interested chiefly in interference by a relatively weak signal, the inter- 

 ference caused by a signal, the carrier amplitude of which is greater 

 than 0.1 of that of the desired carrier amplitude being near the tolerable 

 limit in the majority of cases. The upper value for the modulation of 

 0.5 is approximately equal to the average degree of modulation of a 

 station employing as deep modulation as is practical, only the peaks 

 running up to nearly unity. The value of 0.1 for the lower limit is of 

 course transgressed by soft passages in speech or music. However, the 

 range here specified is sufficiently large to give an excellent idea of 

 what may be expected from various degrees of modulation of desired 

 and interfering signals and the results of more extreme cases may be 

 inferred from the data here developed. Under these limits it is found 

 that the only audio frequencies of any importance which appear in 

 the output are: 



5 = ( ME - eg ia,M - ai + a,^\ + '"''"^.^^' ) cos pt 



, / / ciiMm meg\ .^e~fboni\ 

 + ( me - egi aom ^ ) jj^ — ) cos qi 



, / / auM m-eg\ h^e^g^ , i ox \ , 



+ [eg (go ^ Y^ ] + -j^ (2 + nr) j cos ut 



7^ cos 2ui 



