NETWORK SYNTHESIS G45 



is to be no better than necessary, so that there may be a maximum 

 reinforcing of kjsses at other frequencies. In these terms, (58), and 

 ^ Kkz' in (73), correspond to the assigned natural modes (at the 

 same locations as the assigned frecjuencies of infinite loss). Then the 

 ideal ^Z KkZ~'' is itself a polynomial, of degree yn ^ ?i, and (74) is e.xact, 

 rather than an approximation to (73). Then (70) determines the n 

 arbitrary modes in such a way that the net filter gain approximates 

 zero in the Tchebycheff sense, over the useful interval. 



A different procedure for obtaining the same result is described in the 

 author's paper "Synthesis of Reactance 4-Poles".'* The above analysis 

 of the filter problem is of interest in relating the more general synthesis 

 techniciues, in terms of Tchebycheff polynomial series, to previous filter 

 theory. 



Similar filters have also been obtained by Matthaei , on a potential 

 analogy l)asis. He includes, however, somewhat more general filter char- 

 acteristics, for which he obtains only approximately equal ripple errors. 

 Analysis of the sort described abo\-e may be used to clarify Matthaei's 

 analysis of the conditions under which he obtains exactly equal-ripples. 



Equation (75) may be related to work of Bashkow . The (arbitrary) 

 amplitude of the (equal) maxima of | a — « | , computed from H{z) of 

 (75), depends only on \ G \ . The frequencies at which the maxima occur 

 correspond to phase H{z) = st, which is independent of \ G \ . Thus, 

 the locations of the maxima are invariant to the arbitrary amplitude, 

 ivithin the range where (75) applies. (75) applies only when (74) may be 

 used in place of (73). Generally, (74) only approximates (73), and the 

 approximation introduces small variations in the maxima of | a — a | 

 (when a corresponds to (7G)). If the maxima themselves are sufficiently 

 small, the small variations will be large percentage variations; and the 

 adjustments to compensate for the variations will yield significant 

 shifts in the location of the maxima. In other words, the locations of 

 the maxima of \ a — a\ , ret^uired for equal amplitudes, are largely 

 invariant to the magnitude of the equal amplitudes, but only to an 

 approximation which becomes worse as the amplitudes are decreased. 



Bashkow states the invariance of the freciuencies of maximum 

 I a — a I , as a more or less empirical conclusion, based on a quite dif- 

 ferent approach to the same synthesis problem. 



Equation (75) may be related also to work of Kuh. The natural 

 modes z^ are zeros of 1 + H(z). In other words, H(za) = —1. Using the 

 H{z) of (75) gives the following: 



o 0,, '>n I - -^1 Kn\ 



Ao + Kiz; + • • • Knz; = -Gz; <Ko -{- -j + • ■ • -2^} (77) 



Za Za \ 



