1416 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



1 °° 



coth 2 = - + ^ 



2z 



z 1 2^ + nV^ ' 

 we obtain 



1 +°° C"^ 1 



^•W4g 1 ^2^"°"' 



(p + jnojs) + 



" + m)1 



Hence 



i —00 



, = .= ^™*'>(2Sc) 



Thus finally 



-^ ^ '''^^ = £ ^ 1 (27) 



This last result obtained from the theory developed above is now going 

 to be checked directly. Referring to Fig. 6, where the notation is defined, 

 and noting the periodicity of the boundary conditions, we get 



{E - Vr)e-''"''' = E - {V, + A). 



Noting that ei{t) = (Vi + A)e''"''^, and solving for T'l and A we fin- 

 ally get 



— t/RC 



By definition 



64 





E RCl - r'"^'' 



T 1 + e-^/«^ ' 

 or 



RC 1 



e, = E 



coth I - 



\2RC/ 

 This last equation checks with (27). 



VII. NUMERICAL EXAMPLES 



Consider the network of Fig. 7. The cutoff of both Ni and A''2 occurs 

 at w = 1. In view of the sampling theorem good transmission recjuires 



