NETWORK SYNTHESIS 663 



least squares analysis may be applied to determine these func- 

 tions, f 



In applying least squares analysis, it must he borne in mind that the 

 network gain a does nt)t approximate the assigned gain a in the simple 

 least squares sense. When ^2^ = C-ik for k ^ n, a — a depends upon 

 two least sijuarcs approximations. The first n terms of ^ 02*^2*; represent 

 a least squares approximation to a, and are made identi{;al with the 

 first n terms of ^ C'>kT'>k , which represent a least squares approximation 

 to a. 



When (117) relates p to z, 2;-plane singularities z„ may be defined by: 



a -\- h Iz^ — 

 vl = ■ )- ^2 (118) 



1 + c 



{■-3' 



I 2. 1 > 1, 



Re Za to have same sign as Re p^ 



An additional singularity, Zo , is also needed, corresponding to the finite 

 poles of (117). It may be defined as follows: 



1 + c L - -Y = 



\ Zo/ (119) 



Uo 1 > 1 



When Pa — p", in a of (2), is expressed in terms of z and z, , (117) 

 introduces denominator factors (1 — z /zo) and (1 — I/zqz). As a re- 

 sult, a of (21) must be changed to the following, for band-pass intervals: 



OC = z2 C2kT2k 



11 { 1 — 72 I / Jl\n"-nl 



(120) 



When definite values have been chosen for a, h, c of (117) (in order 

 that the Ck may be calculated), (1 — z'/zo) in (120) is not subject to 

 arbitrary adjustment. This situation can be handled by defining N/D 

 as the rational fraction in the a ofiuatioiis of (21), as before, but re- 



t For general discussions of orthogonal functions and least squares approxi- 

 mations, see Courant and Hilhert^, and also a short text by Jackson.'^ 



