458 BELL SYSTEM TECHNICAL JOURNAL 



When, as sometimes occurs, the number of coefficients is one greater 

 than the number of independent network parameters, it means that 

 one relation exists between the coefficients and hence any one of the 

 latter may be assumed dependent. The dependent relation can be 

 found from the formulae for Po, <2o, etc., in terms of ao, ^o, etc. How- 

 ever, in some such networks it is possible to use the attenuation con- 

 stant at a particular frequency, say zero or infinite frequency, and 

 thereby reduce the number of remaining coefficients and independent 

 network parameters to equality, when the case is readily solvable. 

 If this does not produce the desired reduction, it is usually best to first 

 transfer the dependent coefficient to the right-hand member of (23) 

 and after forming the set of linear equations solve them for the inde- 

 pendent coefficients in terms of the dependent one. Substitution of 

 these values in the dependent relation gives a polynomial in the 

 dependent coefficient which can be solved by Horner's method. Its 

 solution then determines the independent coefficients. This procedure 

 might be extended similarly to cases where the number of coefficients 

 is two or more greater than that of the linear equations, but obviously 

 the process becomes quite involved. 



The values of the attenuation coefficients Po, <2o, etc., are unique 

 when determined from linear equations. The impedance coefficients 

 Oo, &o, etc., derived from them are also single-valued to give a physical 

 solution in most types of networks, meaning that only one such 

 physical network has the particular attenuation characteristic. How- 

 ever, in the lattice type, it has been found that there are usually possible 

 two or more physical solutions for the impedance coefficients from the 

 attenuation coefficients, which correspond to two or more similar 

 appearing physical structures having identically the same attenuation 

 characteristic but different phase constants. 



Method 2. Solutions from the Phase Constant 



This method is applicable particularly to phase networks which 

 ideally have no attenuation and to other networks where the number 

 of phase coefficients equals the number of independent network 

 parameters. The procedure is the same as in the previous method 

 where now we operate with the phase constant formula (22). Multi- 

 plying the latter by its iV-polynomial, we obtain formally the phase 

 linear equation, true at all frequencies, 



JMi -f pUi + . • • - //TVo - pIIN^ - • • • = 0. (24) 



Fixing the phase constant, and hence //, in this equation at frequencies 



