466 BELL SYSTEM TECHNICAL JOURNAL 



where i?i = 2 tanh {A,I2)R, 



Z2 = 4 sinh2 {A,l2)B}lzp, 

 and Rz = 2 sinh^ {A,I2)R{2R - coth {AJ2)R,)IR,. 



An equivalent form is 



su' = ^ ^ ^ + R/, (45) 



where R,' = 2 sech'^ (AJ2)RRJ{2R - tanh (^,/2)i?,,), 



22' = 4 sech2 iAc/2)R'R,y{2R - tanh {A,/2)R,yz„ 



, 2R{2R - coth (AJ2)R^) 

 and i<3 - ^2 coth iAj2)R - R,) 



There will be a physical network provided 



1 < coth {A, 12) ^ 2RlRp. (46) 



At the minimum Ac, Ri = R\ , Z2 = S2', and R3 = Rs = 0. 



It may be added that if (38) and (43) represent inverse networks of 

 impedance product 4R^, then another such pair is given by (40) and 

 (44), and still another by (41) and (45). 



An extension of these results may now readily be made to give 

 two-section composite networks whose attenuation constants are uniform 

 hut whose phase constants are not zero. It has been stated that to every 

 lattice type network having finite attenuation there usually corresponds 

 another one of the same structural form having the same attenuation 

 but a different phase characteristic. Hence, in either case above 

 where the two complementary sections giving a total uniform attenua- 

 tion are known, we may derive by reguiar methods the alternative 

 lattice sections, having, respectively, the same attenuation constants. 

 Since we would then have two sections to give the one attenuation 

 characteristic and two sections for the complementary characteristic, 

 it would be possible to obtain four composite networks of similar struc- 

 ture, all of which give the same uniform attenuation but four different 

 phase characteristics. One of these combinations would be the case in 

 which the phase constant is zero. Four more phase characteristics, 

 differing from the others by an amount tt, can obviously be obtained 

 by reversing the terminals of either section. 



2.9. Procedure for the Design of Distortion Correcting Networks 



It would be most gratifying to be able to obtain directly from a 

 desired propagation characteristic the corresponding form of network. 



