MULTICHANNEL AMPLIFIERS BELOW OVERLOAD 609 



process when the number of channels is large. This is an especially 

 valuable simplification for products of high order for which the counting 

 becomes very tedious. Suppose there are D components in unit band 

 width in the range a to b. Let Xi, X2, • • • Xn be n frequencies such 

 that a — Xi < ^2 < • • • < Xn — b. The number of products of form 

 m\X\ + miXi + • • • + ninXn which can be formed by selecting funda- 

 mentals from the n bands of width dx\ at Xi, dx^ at x^, • • • dx„ at Xn 

 is D"dxidx2 • • • dxn- To count the total number of such products 

 which can be formed in the band a to be such that the resultant fre- 

 quency lies in the band xo to x, we form the integral 



/•6 r'Xn r'xi 



g{x, Xq) = D'^ I dXn I dXn-1 * * ' I <p{x, Xo, Xi, Xi ■ • • Xn)dXi, (6) 



where 



<p{x, Xo, Xi, Xi ' • • Xn) 



(1, Xo — WiXi + W2X2 + • • • + W„X„ ^ X \ 

 0, otherwise / 



A suitable representation of (p is furnished by 



n 



(fix, Xo, xi, X2 • • • x„) = -r— . I g-'^o^ — e*" e '■=1 — 

 27r? L V / z 



(8) 



where C is a contour going from z= — oot0 2=-f-oo and coinciding 

 with the real axis except for a downward indentation at the origin. 

 To obtain the number of products v{x)dx falling in band of width dx 

 at X, we write 



, . , . g{x, X — Ax) 

 Kx) = Lim ^ — -^ 



Ax = ^•'^ 



n 



= 2^ I e-'"dz I dxn 1 (/x„_i • • • / e '=^ dxi. (9) 



If the spacing between carrier frequencies is used as the unit band 

 width, we may set £> = 1, a = Wi, 6 = W2, x = ^, in (9) and obtain the 

 limiting forms of the results given in Table II as the number of channels 

 is made large. Evaluation of (9) is easily carried out by means of the 

 relation : 



r 27r4'"x'"~^ 



X 



e^"dz 



(m - IJ ! • ^~^ 

 10, X < OJ 



(10) 



