DISTORTION CORRECTION 459 



equal in number to the phase coefficients gives us, if this number is 

 equal to the number of independent network parameters, the desired 

 set of linear equations to be solved by the usual methods. In a net- 

 work where the number of phase coefficients is one less than the 

 number of network parameters an additional relation will be needed 

 to determine the network elements and this can be supplied from the 

 attenuation characteristic. Here the attenuation characteristic can 

 probably be lowered uniformly without altering the phase character- 

 istic. (See Section 2.82.) 



Method 3. Solutions from the Propagation Constant 

 Since it has been shown in Section 2.5 that any network of the type 

 considered in this paper can always be represented physically by a 

 lattice type having an equivalent propagation constant, we can 

 simplify the discussion here by dealing entirely with the lattice net- 

 work. From (13) the impedance ratio ZnjlR for this type is derived 

 in terms of its propagation constant as 



If = J^ = tanh (r/2), (25) 



which holds at all frequencies. Thus, a determination of the recurrent 

 network from its propagation constant {attenuation and phase constants 

 together) reduces to the solution of a two-terminal impedance network from 

 its impedance characteristic. The impedance ratio components 5 and 

 y in (19) will become definite known functions of frequency deter- 

 mined through (25) by the propagation constant of the given lattice 

 network. 



A method of solving for the impedance coefficients a^, bo, etc., and 

 hence the network elements from the components 5 and y, follows. 

 Instead of attempting to separate the impedance ratio expression 

 into its real and imaginary parts which can then separately be equated 

 to 5 and y, which is the usual method, let us multiply (19) by the b- 

 polynomial. Now equating separately the real and imaginary parts 

 we obtain a pair of equations which are linear in the coefficients and 

 hold at all frequencies. This pair of impedance linear equations are 

 formally 



ao - f^ai -[-■••- sbo -\- fybi + f-sb^ + • • • =0, 

 and (26) 



fa, - f'a^ + ybo - fsb, + f'yb^ + • • • = 0. 



By this means the formulae are put in a form such as to require in all 

 cases the solution of a set of equations linear in the coefficients, obtained 

 from (26) at different frequencies. A procedure for their solution 



