NETWORK SYNTHESIS 



025 



The luuctions of z luive the foUowitijj; properties: Because of tlie mild 

 restrictions, wiiich we have imposed on the singularities of a + /jS, the 

 series zL ^^~' defines a function which is analytic within, and on the 

 circle \z\ = 1. Then R{z), also, is analytic within, and on the circle. 

 Further, R{z) has no zeros anywhere in the same I'cgion. (Kiz), however, 

 may be more general than the rational fraction in (20).) Finally, because 

 of the even and odd symmetries, required of a and i^, (22) may be 

 broken into the following parallels of the equations (21): 



a = 2_/ CkTk 



T.Ckz' = \og[R{z)R{-z)] 

 i$ = Z C,T, 



Z C,z' = 1 



R{-z, 



k, even 



/,-, odd 



(23) 



In some applications, it is possible to express R(z) in closed form. In 

 all applications, it is possible to expand R{z) as a power series, convergent 

 in the region 1 ^ | ^ 1. The same is true of 1/R(z), since there are no 

 zeros in the region. Coefficients of either series (R{z) or 1/R(z)) may 

 readily be calculated by means which we shall examine a little later. For 

 the present we shall say merely that R{z) is a known function, corre- 

 sponding to an assigned a + i^. 



\). A DESIGN CRITERION 



When the gain and phase function, a + i,5, is to approximate a + ?'^, 

 the error in the approximation is (a — a) + i(l3 — j8). The error ma}' 

 be expressed in terms of * by taking the difference of corresponding 

 equations in (20), (22). The difference of the logarithms may be ex- 

 pressed as a single logarithm of a ratio. Alternatively, and also more 

 conveniently for our later piu'poses, it may be expressed as the negative 

 of the logarithm of the reciprocal ratio. Ayhen this is done, 



{a- a) + i((3 - ^) = E (C, - On 



z 



E Kc-.- - c<)z' 



= -log { ^ 



n 1 



n 1 



KW 



(24) 



Consider the following arbitrary requirement, as a design criterion: 

 The series 2_/ ^kTk is to match exactly the series iL CkTk , through 



