230 BELL SYSTEM TECHNICAL JOURNAL 



1 r r 



A±mn —~Y~2 I I /(^' y) •^os ("^•^' =•= ny)dydx, 

 B±mn =~YT I I /(^> 3') si" ('"^ =*= ny)dydx. 



(6) 



We now return to our origintil problem of representing the positive 

 lobes of a two-frequency wave as a sum of sinusoidal components. We 

 may apply the double Fourier series expansion oi f{x, y), which must 

 hold for all values of .v and v, to the special case in which x and y are 

 linear functions of the time. If we let 



X = pt -\r dp, \ 



y 



= ^qt + ej (^) 



the function /(.v, y) represents the rectified two-frequency wave as a 

 function of time. The values of x and y which are used lie on the 

 straight line, 



which is obtained by eliminating / from (7). A representation of 

 f{x, y) valid for the entire x^'-plane must of course hold for values of 

 x and y on this straight line. Hence we may substitute the values of 

 x and y given by (7) directly into the double Fourier series (5), and 

 the result will evidently be an expression for the rectifier output in 

 terms of discrete frequencies of the type {mp ± nq) Jlr. The phase 

 angle of the typical component is mQj, ± nQq and the amplitude is 

 expressed by (6). 



The solution is thereby reduced to the evaluation of the definite 

 double integrals of (6). Three different methods of reducing these 

 integrals have been investigated, and it appears that each has certain 

 peculiar advantages and points of interest. We shall consider them 

 separately. 



I. Straightforward Geometric Method 



In this method, which yields remarkably simple results in a direct 

 manner, we determine the boundaries of the region throughout which 

 /(x, y) vanishes and substitute appropriate limits in the integrals to 

 exclude this region from the area of integration. When this exclusion 

 has been accomplished, /(x, y) may be replaced in the integral by 

 cos X -\- k cos y. The boundary between zero and non-zero values of 

 /(x, 3') is the curve (4), which has two branches crossing the rectangle 



