632 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



makes use of the contour in the complex plane for the polynomial , corre- 

 sponding to the 2;-plane circle \ z \ = 1. (This is like a Nyquist diagram 

 except that the contour for the variable, z, is different.) There will be 

 \ Za\ < 1 if and only if the contour for the polynomial encloses the 

 origin. 



Now the polynomial in (34), and (35), is merely a special case of the 

 function on the left in (28), and (30). For this special case (30) becomes 



E K,z"' = 1 + Hiz) (38) 



The polynomial cannot enclose the origin without passing thrcjugh some 

 negative real value. But this requires an ] H{z) \ > 1, at some point on 

 the contour in question, | ^ | = 1, which happens to be also our useful 

 interval. On the other hand, a — a = when ^ Kkz'''' = 1, and H{z) 

 is in the nature of a correction term, which is small in the useful interval 

 when a — a is small. 



The conclusion is: There will be no | 2, | < 1 unless the approxima- 

 tion, a to a, is so poor that a — a exceeds several db in the useful inter- 

 val. 



Besides the requirement \ z„ \ > 1, the z„ must meet physical restric- 

 tions, which we found to be the same as those limiting the natural 

 modes p„ . The Za may be calculated as follows: The z„ are roots of the 

 polynomial in (35), in terms of z~. All the roots in terms of z' are z„ , 

 except the two required roots at zl , which correspond to assigned gain a. 

 Each Za is a square root of a 2^ . There are two possible square roots, 

 however, differing only as to sign. As far as gain a is concerned, either 

 choice of sign is permissible; for a depends only on z^ . For a physical 

 network, however, the choice must be such that Re z^ < 0. This choice 

 is possible if, and only if -y/zi has a finite real part. A pure imaginary Za 

 corresponds to a negative real z„ , and thus negative real roots in terms 

 of z"^ are excluded by physical considerations. 



Table I lists both zl and z„ for a number of values of n. When n is 

 even, all roots are physical. On the other hand, when n is odd, one root 

 is always non-physical. In a sense, an odd n is not really appropriate 

 for the present illustrative problem, with any physical design. An odd n 

 must necessarily bring in a real natural mode, which merely increases 

 the sort of distortion we are trying to equalize — that is the distortion 

 due to unwanted real modes. 



The following argument substantiates the suggestion, and also illus- 

 trates manipulations of a sort which are frequently useful in more gen- 

 eral applications: The highest order coefficient in (34), Kn+2 , may be 

 set aside for adjustments to satisfy physical requirements. The rest of 



