IDEAL FILTERS 



235 



A second set of characteristics was obtained by choosing a = 1/6 

 and adding various groups of transition factors in a similar fashion. 

 The results are shown by Figs. 7 and 8. The characteristics are drawn 

 only for w^'s between 1 and 3 in this case, since with larger w's the 



10 





0.0. 



. < 



01 

 -lO 

 uJcc 



> L.. 



2 



0.3 



04 0.5 06 



FREQUENCY 



0/ 



0.8 



10 



Fig. 9 — Low pass filter with a = 1/4. Envelope of phase departures. 



transition interval becomes disproportionately wide in comparison 

 with the practical transmission range. Still a third set, corresponding 

 to a = 1/4 and m = 1 is shown by Figs. 9 and 10. 



80 



uj 20 



0.75 1.00 1.25 150 1.75 2.00 2.25 2 60 2 75 3.00 3.25 



FREQUENCY 



Fig. 10 — Low pass filter with a = 1/4. Attenuation. 



As an illustration of the accuracy to be expected from the approxi- 

 mate method, a comparison between the results obtained by this 

 method and the exact characteristics is shown in Figs. 11 and 12 for 

 the cases m = 1 and m — 2 oi Figs. 5 and 6. On the logarithmic 

 scales used for the figures, the curves appear to be in good agreement 

 almost up to the transition interval. The actual numerical de- 

 partures in the vicinity of that interval, however, are quite large. 



