IDEAL FILTERS 



223 



The logarithm of Q is, of course, the sum of the logarithms of the 

 individual factors of this expression. Expanding these as series of 

 powers of 1/(1 — /) or 1/(1 +/), and collecting terms of like degree 

 in 1/(1 -/) and 1/(1 +/), we obtain 



+ { Ci - C2 + Cs - 



± 



2 """7 [ 1 - 



+ 



- - ( Ci^ - c^ + ci 



+ 3! cx^ - ci^ci 



f 1+/ 



=^2'"0 [(T^^^+cTTTPJ 



(20) 



where the sign of Cm is plus or minus according as m is odd or even." 

 As the terms of (20) are similar in form to those of (17),^^ we can make 

 the first m terms identical. This leads to the equations 



Ci 



C2 + C3 — 



3^ ry Cm 



Bia, 



± 2 Cm' = 0, 



(21) 



Ci^ — C<^ + Ts' — 

 Cx' - C2' + C3* - 

 Cl^ - C2^ + C3^ - 



whose simultaneous solution gives the desired transition factors. 



The number m of transition factors used will depend upon the desired 

 approximation to ideal characteristics In the practical transmitting and 



*' When the c's are evaluated it will appear that these series are all absolutely 

 convergent— so that their termwise sum correctly represents log Q — at all positive 

 frequencies outside the interval (1 — Cm, 1 + Cm). As the complexity of the network 

 is increased, in the approach toward ideal characteristics, the interval of non-con- 

 vergence closes on the reference frequency 1, and is contained by the given transition 

 interval. 



1* The first term of (17) does not contain/, and may therefore be neglected for the 

 same reason which led us to neglect K in (19). 



