636 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



amplitude, across the useful interval, is: 



|/^(^)|n,ax. ^ (n + 2)zl + (n + 1) 

 \H{z)Lin. (n+ 2)zl - (n+ 1) 



14. APPROXIMATION IN THE TCHEBYCHEFF SENSE 



(44) 



The above analysis suggests a way of improving the design deter- 

 mined l)y (37) (or the equiv^alent, (36)). An optimum a — a is commonly 

 one which has the following properties, in the useful interval: 



A maximum number of ^' ripples/^ 

 all maxima of \ a — a \ equal. 



(This usually minimizes the largest departure in the useful interval, 

 thereby yielding an "approximation in the Tchebycheff Sense.") Since 

 the variation in phase H(z) determines the number of ripples, while 

 I H(z) 1 determines the amplitudes of the ripples, the abo\'e conditions 

 will be met if H{z) has the following properties, in the useful interval: 



Phase H(z) as variable as possible, , . 



\ H(z) I constant. 



These conditions may be regarded as alternative design criteria, re- 

 placing C-2k = Cofc . They can in fact be applied to our special example, 

 and also to certain other special problems which will be noted later. 

 For more general applications, a suitable H(z) can be defined, but no 

 reasonably simple procedure has yet been found for calculating the re- 

 quired constants. (The difficulties will be particularized in a later 

 section.) 



For the present example, (38) may be used to replace (34), and hence 

 also the second eciuation of (33), by: 



E (C2. - Cn)z' = - log [1 + H{z)] (46) 



(33) requires H{z) to be a polynomial in z', of degree n + 2, with two 

 zeros of [1 + H{z)\ at z' = H . The object is to find an H{z) of this 

 sort, which also satisfies (45), at least to a good approximation. 

 The following H{z) does in fact exhibit the required properties: 



H{z) = Gz [1 _ J/,.] W') 



The function is a polynomial because the factor [1 — /""^V^"""*"^] is 

 divisible by (1 — J/z^). The constants J and G are to be chosen to 



