142 BELL SYSTEM TECHNICAL JOURNAL 



Section 11 summarizes specific results on the single-frequency biased 

 rectitier case. The general expression for the amplitude of the -typical 

 harmonic is evaluated in terms of a hypergeometric function for the power 

 law case with arbitrary exponent. 



Section HI takes up the evaluation of the two-frequency modulation 

 products. It is found that the integer-power-law case Tan be expressed in 

 finite form in terms of complete elliptic integrals of the first, second, and 

 third kind for almost all products. Of these the first two are available in 

 tables, directly, and the third can be expressed in terms of incomplete 

 integrals of the first and second kinds, of which tables also exist. No direct 

 tabulation of the complete elliptic integrals of the third kind encountered 

 here is known to the author. They are of the hyperbolic type in contrast 

 to the circular ones more usual in dynamical problems. Imaginary values 

 of the angle /3 would be required in the recently published table by Heuman . 



A few of the product amplitudes depend on an integral which has not 

 been reduced to elliptic form, and which is a transcendental function of two 

 variables about which little is known. Graphs calculated by numerical 

 integration are included. 



The expressions in terms of elliptic integrals, while finite for any product, 

 show a rather disturbing complexity when compared with the original 

 integrals from which they are derived. It appears that elliptic functions 

 are not the most natural ones in which the solution to our problem can be 

 expressed. If we did not have the elliptic tables available, we would prefer 

 to define new functions from our integrals directly, and the study of such 

 functions might be an interesting' and fruitful mathematical exercise. 



Solutions for more than two frequencies are theoretically possible by the 

 same methods, although an increase of complexity occurs as the first few 

 components are added. When the number of components becomes very 

 large, however, limiting conditions may be evaluated which reduce the 

 problem to a manageable simplicity again. The case of an infinite number 

 of components uniformly spaced along an appropriate frequency range has 

 been used successfully as a representation of a noise wave, and the detected 

 output from signal and noise inputs thus evaluated . The noise problerri 

 will not be treated in the present paper. 



I. The General Problem 

 Let the biased rectifier characteristic, Fig. 1, be expressed by 



/ 0, E < b\ 



I = (1.1) 



\f{E -b), b < eJ 



1 Carl Heuman, Tables of Comi)letc Ellii)tic Integrals, Jour. Math, and Phvsics, Vol. 

 XX, No. 2, pp. 127-206, April, 1941. 



. ^ W. R. Hcnnctt, Response of a Linear Rectifier to Signal and Noise, Jour. Acous. Soc. 

 Amer., Vol. 15, pp. 164-172, Jan. 1944. 



