NETWORK SYNTHESIS 0)29 



sentatioii of the network <;;uii (v. In (121 ), [\\c z„ ('orrespond to finite^ fie- 

 ciueiu"i(>s of infinite loss, and aic to We omitted when there are to l)e 

 natural modes only. What is left is the lo<;ai'ithm of the rcM-iprocal of a 

 polynomial, which is of course the u(»f>;ative of the lojj;ai'ithm of the 

 polynomial itself. Tims a may he described as follows, for this particular 

 application: 



Z Co,^' = -h)s K:U(i -K) <T = \,---, n (31) 



(For convenience, 2„ has been written for z^ , and K, has Ixhmi redefined 

 to avoid the 1/Kz required if it is defined as in (21).) 



The assigned gain a is even more special. In this particular problem, 

 a has most of the properties of a network gain a. Specifically, it is the 

 negative of the gain to be equalized, which in fact corresponds to a 

 finite network. As a result, R(z) of (22) may be expressed in closed 

 (rational) form. (Later on, we shall modify the methods appropriate for 

 this very special situation, so that R(z) need be representable only by 

 series.) 



The specific representation of our present a may be very similar to 

 the representation of a in (31), as follows: 



a = 2^ Cok-T'ik 



zcur' - +iog^;(^i - ly 



(32) 



(Both (31) and (32) apply only to the useful interval, \ z \ = 1.) 



The constant Zu is the 2:-plane mapping of the assigned unwanted 

 natiu'al modes at p = pn , and may be calculated therefrom by (8). In 

 (32), 2n determines the C-ik- , which in turn determine a. The constants 

 Za , in (31), are the ^-plane mappings of the arbitrary natural modes of 

 the eciualizer. They are to be adjusted to make a approximate a. Then 

 the network natural modes p„ mav be calculated from them, bv means 

 of (8). 



Taking the difference of corresponding eciuations in (31) and (32) 

 gives the following, analogous to (24): 



« — « = zJ (C'2/t — C-ii)T2k 



E(C^.-C.)/'=-,o,{^;«(l-|yn(i-|)} '''' 



This differs from (24) in two n^gards. It relates to the gain erroi', {a — a), 



