220 BELL SYSTEM TECHNICAL JOURNAL 



and hence that ^^ 



tanh - = t \h^f Vae, (4) 



Zi = -^KxKy^ac-^, ^' (5) 



fli • • • flg 



where the a's are shorthand notation for 



^ n' -\ ^ 



J> Ji 



«;■= 1-772' «/ = 1-772- (6) 



We shall have frequent occasion to distinguish between those critical 

 frequencies of (4) which lie in the practical transmission band and 

 those which lie in the transition interval. To this end we shall denote 

 that interval by (/aj/b), where obviously /,i < fc < Jb, and shall write 

 P for the group of factors 



P-iJ¥f "-'"'"'''"' (7) 



\K,/ flifla • •• a A 



which lie in the wanted band, and Q for the remainder 



Q = Vac (8) 



(2.44-2 ' ' ' ttr 



which lie in the transition band. Then, obviously, (4) becomes 



tanh|=P(2. (9) 



Requirements on Transition Factors 

 With these formulae before us, we are now prepared to attack the 

 problem of meeting the double requirement of linear phase shift in 

 the practical transmission band and infinite loss in the practical 

 attenuation band. Expressed analytically, these requirements are 

 simply 



tanh ^ ^ tanh i ^ ^ i tan^ , f < J a, (10) 



2 la la 



tanh|= 1, />//., (11) 



" It should be noted that, except for the cut-off factor ac, (4) and (5) are entirely 

 independent. That is, the frequencies /i, . . . /r may be chosen as we desire, in 

 order to control the transfer constant, without in any way affecting the image im- 

 pedance; an d /i ', . . ., / ,' can be chosen at will without affecting e. Similarly the 

 constants ^KjjKy and '^K^Ky may be chosen at will. 



