NETWORK SYNTHESIS 627 



polynomials contribute to gain. The following condition turns out to be 

 the eciuivalent of C-2k = C^k , k ^ m: 



Kl 



n('-a 



R{z)R{-z) = 1 (28) 



where = means approximation in accordance with a power series of 

 e\'en ordered terms, through order 2m. Correspondingly, only odd 

 ordered Tchebycheff polynomials contribute to phase. The following 

 condition is equivalent to C2a_i = C->k-\ ,^^— 1 to ni'. 



1 - -77 1 + 



1m 



n — ^ n — ^-#^ = 1 (29) 



i -f- // 1 — / 



The remaining sections (except the last) develop in more detail the 

 application of s-plane techniques to more specific synthesis problems, of 

 various sorts. Most of these (but not quite all) are based directly on 

 (27), (28), or (29). The exceptions use a modification of (28), in which 

 the function of z on the left is retained, but with the zeros and poles 



me 



adjusted for a different kind of approximation to unity, = but not = . 

 In all cases, unity is approximated with one of the functions appear- 

 ing in (27), (28), (29). It will be convenient to use H{z) to represent the 

 error in the approximation, or departure from unity. When gain only is 

 of interest, the function in (28) is used, and H{z) is defined by: 



. n (i - f) 



R{z)R(-z) = 1 + H(z) (30) 



In developing the specific techniques, we shall start with a very 

 definite, rather special example, in order to illustrate the techniques 

 with specific operations. This will be discussed in considerable detail in 

 Sections 10 through 14. Thereafter we shall examine how these specific 

 operations may be generalized, in a number of different respects. 



10. AX INTRODUCTORY EXAMPLE 



The example which has been chosen for detailed discussion is the 

 equalization of the gain distortion produced by two resistance-capacity 



