BAND WIDTH AND TRANSMISSION PERFORMANCE 585 



where: 



c = (/i -/2)//^6. X = Nfr/Ft>, y = ^/Ft, (29) 



An = 2\J,nx-y(y) " - [(-)" + sin (2;/X - >07r], (30) 



mr 



Ao = 2\y^yiy) + (1 - 2X) cos Try. (31) 



The function Jy (y) is Anger's function:''^ 



J^(y) = 1 f cos (vd - y sin 6) dd (32) 



T Jo 



It is equal to Jy(y), the more famiUar Bessel function of the first kind, only 

 when V is an integer. The values of v = 2n\ — y appearing in this solution 

 are in general not integers and hence the ordinary tables of Bessel functions 

 are inapplicable. 



The baseband filter accepts the components of the error which have fre- 

 quencies in the range : 



-Fb < Fb (c+ n\) < Fb (33) 



or 



-l±^<n<- -'. (34) 



X X 



The interfering wave in the baseband filter output is then 



8,{i) =^Fb iJ; (c + iiX)An cos [lirFbic + n\)l - 6] (35) 



where rh is the smallest integer not less than - (1 + c)/\ and no is the largest 

 integer not greater than (I — c)/A. It would be convenient at this point to 

 assume that M- is expressible directly in terms of 55(0- However, there is 

 reason to believe that such an assumption is pessimistic especially at the 

 higher bandwidths where the disturbance 5o(/) may never reach its maximum 

 values in the neighborhood of the actual slicing instant. A complete investi- 

 gation requires a study of the instantaneous wave form of 5o(/) in the neigh- 

 borhood of the slicing instant. We note that if the sheer operates at the 



traiUng edge, the unperturbed slicing instant is / = 2^ + ^^^'f' ' ^""^ ^^^^ 

 value of the disturbance at that instant is: 



80 (~ + ni/f) = ^^Fbi: {c-\- n\)An 



\Zrb / ^ n = ni 



" Watson, Theory of Bessel Functions, Chapter X. 



