648 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Accordingly, the rule for Coa = C2k , k ^ m, now becomes, in place of 

 (56), 



AT *"* 



'J^ = 1:K,/' (84) 



This condition may be used to determine the coefficients K^ , A'^. of 

 A'' and I) (in combination with conditions of other sorts, when 

 m < )i" + n'). When the coefficients hav^e been calculated, the (z-plane) 

 natural modes z„ may be determined from the roots of N, exactly as 

 the Za of previous sections. The infinite loss points z„ may be calculated 

 from the roots of D, in exactly the same way except that Re 2„ need 

 not be negative. Signs of the z„ must be such that complex and imaginary 

 z„ are in conjugate pairs. Xote that there can be conjugate imaginarj^ z,, 

 only if D has a corresponding double negative real zero. 



When m = ti" + n', the following method may be used to calcvilate 

 the Kk and Kk determined by (84). The relation is first multiplied by 

 D, to get: 



Then algebraic manipulation is used to evaluate the power series equiv- 

 alent of the right hand side, through terms of order n" + n', using the 

 known values of the Kk , but general values of the Kk of D. Each coef- 

 ficient is a linear function of the unknowns, Kk . 



Now the polynomial N, in (85), has no terms of order k > n" . There- 

 fore (85) requires zero coefficients in the expansion of the right hand 

 side, from order n" + 1 to order n" + n'. Equating these coefficients 

 to zero gives 7i' linear equations in the n' unknown Kk . Solving for the 

 Kk determines polynomial D. The values calculated for the Kk may 

 then be used in lower order coefficients of the expansion of the right 

 hand side of (85), which are exactly the coefficients Kk of N. 



When n" — n' = or 1, a continued fraction method is likely to be 

 preferable. Various established techniques f may be used to convert the 

 series ^ Kkz' into a continued fraction of the form: 



22 KkZ^'' = ao + 



a 1 _. 1 



z'^ . 1 (86) 



a2 + 



z- ai 



t See, for example, Fry's applications of continued fractions to network de- 

 sign.' 



