NETWORK SYNTHESIS 657 



every Ck in two. All zeros of N = E -\- must be construed as natural 

 modes z, . Finally, the network must have as many infinite loss j)()ints 

 as natural modes, such that z, = —z„ . (Integer n is now the number 

 of" natural modes, rather than the total numl)er of singularities.) 



For physical networks, all the first /(. terms of the continued fraction 

 (108) must now be positi\'c. To meet this condition it may be necessary 

 to add linear phase to the assigned phase (by adding a negative cor- 

 rection to Ci). It appears that sufficient linear phase will always lead 

 to a physical design, provided the numlx'r of modes n is increased to 

 retain a reasonable accuracy. 



26. LINEAR PHASE 



When the assigned phase ^ is lineai', the calculations are relatively 

 simple. 



If a delay I) is to be appi'oximated ov(n- a fre(iuency inter\-al extending 



to CO = Wc , 



i~^ = -DcocTi (110) 



If delay D is to be realized without regard to gain variations, the ap- 

 propriate 0/E is 



g = tanh^ (111) 



A known continued fraction expansion of tanh A' may be applied to 

 (111), to obtain the coefficients of (108) without bothering with (105).t 

 The result may be arranged as follows: 



(112) 



Do3cZ 



Truncation of the continvied fraction gives 0/E, and then -{- E. The 



zeros Za- turn out to be proportional to - , and therefore root extraction 



techniciues are required only for one D, for each n. The zeros are tabu- 

 lated for sample n's, in Table II. 



t For tho expansion of tanh A', reference may t)e made to a text on continued 

 fractions by Wall", page 349, equation 91.6. 



