182 BELL SYSTEM TECHNICAL JOURNAL 



The network can be designed from equation (6) or by making use of 

 both (3) and (7). Both methods will be illustrated in two types of 

 networks giving filter characteristics. 



Filters with Characteristics Similar to the "Constant K" 



Type of Filter 



As the simplest form of F{\) take F{\) = [(X)/(i)]2". The poles of 

 (3) are then simply the 2n roots of (— 1)"~S which may be written 

 ± cos (wtt/w) rb i sin {rmrln) if n is odd, and ± cos \_{2m — l)7r/2w] 

 ± i sin \^{2ni — l)7r/2w] when n is even. In the first case m varies 

 between zero and {n — l)/2 and in the second case between unity 

 and njl. For the case of n being odd, equation (6) may then be 

 written 



^ m=(n— 1)/2 * 



g-(ai + »A) = -— L^ IT • (8) 



1 + A'' + 2 COS A 



n 



where the polynomial D is unity in this case. The last equation 

 expanded is in the form 



ga, + ,-3l ^ 1 + ^^X + ^2X2 + • • • + ^„X", (9) 



which is the form for the ratio EojEx for the network shown in Fig. 3. 

 By writing out the ratio EojEi for this network and comparing terms 

 with equation (8) expanded in the form of (9) the values of the a's 

 may be found to be * 



ai = sm ;r- , 



2n 



. dnr . IT 



Sm ;r- Sm ;r- 



2n 2n 

 02 = 



O1 COS'' TT- 



2n 



. 2m — 1 . 2m — 3 



sm — -^ TT sm — X 



2n 2n 



am = 



(10) 



,w — 1 

 2n 



TT 



2n 



* By the evaluation of the finite sums and products of the trigonometric terms. 

 No short method has been found for obtaining the results. 



