644 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



If a sharp cut-off were not required, this approximate H{z) could be 

 made exactly zero, by using Kk — Kk for all coefficients. Then the actual 

 design error would be determined by the approximation inherent in the 

 use of (74) in place of (73). For high selectivity, however, Kn should be 

 much larger than Kn , as large as possible within assigned limits on 

 a — a in the useful range. (It is readily established that K„2" will deter- 

 mine a at asymptotically high frequencies.) The other Kk are then to 

 be adjusted so that a — a exhibits the desired "equal ripples." 



The following H{z) has the functional form (74), and also meets con- 

 ditions (45): 



K\ Kn 



H{z) = Gz 



(75) 



Ko + Krz' • • • KnZ 



Multiplying G/" into the numerator gives a rational fraction which is 

 obviously consistent with (74). The coefficients Kk of (74) which corre- 

 spond to (75) are: 



Kk = Kk + GKn-k (76) 



In the useful interval [X) Kk/z^^\ is [^ KkZ^''\*. Hence the polynomials 

 in z and \/z have identical magnitudes, in the useful interval; and, 

 since also \z\ = 1, | H{z) \ = ] (7 | in (75). The phase variation, over 

 the useful interval, is the same for H{z) as for z ", which yields the same 

 number of ripples in a — a as an ordinary Tchebycheff filter of like 

 degree, t 



The constant G is arbitrary, except that its sign must be properly 

 chosen to avoid non-physical natural modes. Increasing G increases the 

 filter selectivity, but also increases a — a in the useful interval. G and n 

 are to be chosen together, to realize an assigned selectivity within an 

 assigned limit on distortion. 



The above analysis may be related to the following filter problem: 

 Required to design a filter which has flat gain, in the useful interval, 

 but which has m assigned frequencies of infinite loss, in addition to n 

 arbitrary natural modes (m ^ n). The n arbitrary natural modes may 

 be regarded as compensating for gain variations due to the assigned 

 frequencies of infinite loss, in the useful interval, while reinforcing their 

 effects at other frequencies. Compensation of effects of the infinite loss 

 points is the same as simulation of the effects of natural modes at the 

 same (assigned) frequencies. The approximation in the useful interval 



t This assumes an a with the general characteristics described in Section 8, 

 which are such that the numerator and denominator of the fraction in (75) will 

 each have a net phase shift of zero, across the useful interval. 



