DESIGN OF REACTIVE EQUA/JZKRS 



729 



characteristic into two separate regions. The specification over the useful 

 bantl, < CO < coo , may be called the in-hand region while the specification 

 outside the useful band, coo < co < co, may be called the out-band region. 

 Thus, it is seen that the transfer characteristic over the in-band region 

 depends e.xclusively on the a = kf(oj) which is required to equalize a stated 

 fraction of the power loss between repeaters while the transfer characteristic 

 in the out-band region depends only on the specified resistance efliciency. 

 The first step in the derivation of the special transfer function for equali- 

 zation purposes is a normalization of the transfer characteristic of Fig. 13 

 in terms of eq. (7). As indicated in Fig. 14, a constant. A', is chosen so that 



Ke {K < 1) is equal to unity at — = x = 1. This choice of the transfer 



characteristic is convenient since the transfer characteristic is now e.xpressed 

 in a form similar to the familiar form of the transfer characteristic of a low- 



Fig. 14 — Normalized transfer characteristic of Fig. 13. 



pass filter and, hence, suitable for the addition of a Tchebycheff polynomial. ^^ 

 With the transfer characteristic appropriately specified, the next step is 

 to show the manner in which the denominator ^(.v-) of eq. (7), where this 

 equation is multiplied by the factor K, can be broken up into two functions 

 of x~ so that one of these functions approximates the reciprocal of the in- 

 band region of the transfer characteristic while the other produces the de- 

 sired cut-off characteristic. 



The derivation of the desired denominator, B{x-), begins by writing the 

 transfer characteristic of Fig. 14 for the in-band region as 



1 



B{x' 



Ke 



2k/M 15 



(10) 



" In order to make the following derivation clear, it is suggested that the discussion 

 of Tchebycheff polynomials, pp. 733-734, be examined at this time. 



'*The transmission a = ao + kf(x) will be written as kf(x) for the remainder of this 

 analysis. The general transmission level ao may be found in the final stages of a particular 

 design when the impedance level is adjusted to conform with the prescribed C„. 



