New Results in the Calculation of Modulation Products 



By W. R. BENNETT 



A new method of computing modulation products by means of multiple 

 Fourier series is described. The method is used to obtain for the problem of 

 modulation of a two-freciuency wave by a rectifier a solution which is con- 

 siderably simpler than any hitherto known. 



THE problem of computing modulation products has long been 

 recognized as being of fundamental importance in communication 

 engineering. Heretofore certain quite fundamental modulation prob- 

 lems have been attacked by methods which are difficult to justify from 

 the standpoint of mathematical rigor and some of the solutions ob- 

 tained have been in the form of complicated infinite series that are not 

 easy to use in practical computations. In this paper these problems 

 are solved by means of a new method which is mathematically sound 

 and which yields results in a form well suited for purposes of com- 

 putation. 



The analysis here given applies specifically to the case of two fre- 

 quencies applied to a modulator of the "cut ofT " type; i.e., a modulator 

 which operates by virtue of its being insensitive to input changes 

 throughout a particular range of values. A simple rectifying charac- 

 teristic forms a convenient basis of approximation for study of such 

 modulators, and hence we consider in detail methods of calculating 

 modulation in rectifiers when two frequencies are applied. Applica- 

 ticvns to certain other types of modulation problems and to the case of 

 more than two applied frequencies are discussed briefly at the close. 



Half Wave Linear Rectifier — Two Applied Frequencies 

 We shall define a half wave linear rectifier as a device which delivers 

 no output when the applied voltage is negative and delivers an output 

 wave proportional to the applied voltage when the applied voltage is 

 positive. We may take the constant of proportionality as unity since 

 its only effect is to multiply the entire solution by a constant. Assume 

 the input voltage e{t) to be specified by 



e(/) = P cos ipt -\- dp) + Q cos (qt + Bg). (1) 



The output wave will then consist of the positive lobes of the above 

 function with the negative lobes replaced by zero intervals. It is 



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