884 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



the quantity within the square brackets in (3.2): 



2Rv cos px cos py + R.^ sin px sin pij. (4.1) 



The integrations with respect to r may be performed with the help of 



f xPr+b cos 2irfT dr = ^^ Re e-'"^^*, (4.2) 



xl/r+b^l/r+c COS 2x/r dr 



-00 



(4.3) 



1 r 



= Re- / du w^{u)w^{f — u) exp { —i2v[hu + c(/ — u)]\ 



which follow from (2.2) and the fact that we have defined w{—f) to be 

 equal to w{f). In our notation the total power in a random noise function 

 is the integral of w{f) from / = to / = oo . 



The first order modulation term is obtained from (3.2) by replacing 

 the term within the square bracket by 2i?„ cos px cos py. When the ex- 

 pression (2.6) for /?„ is used, the integration with respect to r may be 

 performed with the help of (4.2) : 



f R, cos 27r/T dr = '-^ Re [(e"''*'^^ - l)(e'''^^ - 1)]. (4.4) 



^ ; J-co 2 



This leads to the following expression for the first order modulation 

 term in (3.2) 



w^U) I f dxgix)e-'^'>+'^' cos px(e""^'"'^ - 1) . (4.5) 



This is the quantity which is to be subtracted from We-e{f) to obtain 

 the interchannel interference spectrum wdf). 



The second order modulation term is handled in much the same 

 manner. With the help of (4.3) it may be shown that 



/OO -| /«00 



/?/ COS 27r/T rfr = Re - / du w^{u)w^{f — w) 

 -00 t: ^ — 00 



/ — 27r»xu 1 ^ / ~2irtx(/— u) -i \ \'i."/ 



From this it follows that the second order modulation term in (3.2) is 



1 /»00 /*00 



2T2 j_ duw^{u)w^if - u) j dxg{x)e'"''''^'''' sinpx 



(4.7) 



/ —ivixu , \ / — 2jrix(/— u) -i\ 



